Cyclic covers of an elliptic curve with two branch points

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson constructs the curves that carry the congruences of the course. Starting from an elliptic curve $E\colon y^2=h(x)$, a parameter $t$ gives a quadratic twist $B_t$ of $E$ with a rational point $R_t$. A rational function on $B_t$ with zeros and poles at $\pm R_t$ and $\pm[n]R_t$ defines cyclic covers $C_{m,t}\to B_t$ of every degree $m\mid n$. They have genus $m$, they are branched at exactly two points, and they carry an involution that inverts the deck transformations. As $t$ varies, the covers form a smooth proper family over an open subset of the affine line, and the same family can be written with the fixed curve $E$ and a moving point $R\in E$.

Prerequisites. Elliptic curves over imaginary quadratic fields and their automorphy. From Galois representations: The ℓ-adic Tate module of an elliptic curve, Section 1.1, in particular the criterion (G4) for principal divisors. From The étale fundamental group: The étale fundamental group, the Riemann–Hurwitz formula (5.5) in characteristic zero.

Conventions. Throughout, $k$ is a field of characteristic zero and $n\ge3$ is an odd integer. Curves are smooth, projective and geometrically integral unless stated otherwise. For a point $P$ of an elliptic curve, $(P)$ denotes the corresponding prime divisor, and sums of points are taken in the group law.

1. Twists and the parameter set

Let $h\in k[x]$ be a separable cubic polynomial and $E\colon y^2=h(x)$, with origin the point $O$ at infinity. For $t\in k$ with $h(t)\neq0$ put $\begin{gathered} B_t\colon\ h(t)\,y^2=h(x),\\ R_t=(t,1),\qquad S_t=[n]R_t .\quad(1.1) \end{gathered}$ The curve $B_t$ is an elliptic curve with origin $O$. Over $k\bigl(\sqrt{h(t)}\bigr)$ the map $(x,y)\mapsto\bigl(x,\sqrt{h(t)}\,y\bigr)$ is an isomorphism $B_t\to E$ of elliptic curves, so $B_t$ is the quadratic twist of $E$ by $h(t)$; it is isomorphic to $E$ over $k$ when $h(t)$ is a square.

Lemma 1.1. Let $U\subset\mathbb A^1_k$ be the set of $t$ with $h(t)\ne0$ for which the five points $O,R_t,-R_t,S_t,-S_t$ of $B_t$ are pairwise distinct. Then the complement of $U$ in $\mathbb A^1_{\bar k}$ is finite, and $U$ is a nonempty open subset of $\mathbb A^1_k$.

Proof. Write $R=R_t$. The point $R$ is affine, so $R\ne O$, and $-R=(t,-1)\ne R$. The remaining conditions say that $[m]R\ne O$ for $m=n$ (for $S_t\ne O$), $m=n-1$ and $m=n+1$ (for $S_t\ne\pm R$) and $m=2n$ (for $S_t\ne-S_t$). Under the isomorphism $B_t\to E$ over $\bar k$, the point $R$ goes to $\bigl(t,\sqrt{h(t)}\bigr)$, and the isomorphism respects multiplication by $m$. The torsion subgroup $E[m](\bar k)$ is finite, so the set of $x$-coordinates of its points is finite. Hence $t\notin U$ exactly when $h(t)=0$ or $t$ is the $x$-coordinate of a point of $E[m]$ for one of the four values of $m$. This finite set is stable under $\operatorname{Gal}(\bar k/k)$, so its complement is open and defined over $k$. $\square$

2. The Kummer function

Lemma 2.1. Let $t\in U(k)$. There is a unique $f_t\in k(B_t)^\times$ with $\begin{aligned} \operatorname{div}(f_t)&=(S_t)-(-S_t)\\ &\quad-n(R_t)+n(-R_t),\\ f_t(O)&=1,\quad(2.1) \end{aligned}$ and it satisfies $f_t\circ[-1]=f_t^{-1}. \tag{2.2}$

Proof. The divisor $D$ on the right of (2.1) has degree $1-1-n+n=0$. Its sum in the group law is $\begin{aligned} &S_t+S_t-[n]R_t-[n]R_t\\ &\quad=2[n]R_t-2[n]R_t=O . \end{aligned}$ By the criterion (G4) for principal divisors, $D=\operatorname{div}(g)$ for some $g\in\bar k(B_t)^\times$. The point $O$ is not in the support of $D$, so $g(O)\in\bar k^\times$, and $f=g/g(O)$ has divisor $D$ and value $1$ at $O$. Any two functions with the same divisor differ by a nonzero constant, so $f$ is unique among functions in $\bar k(B_t)$ with these properties. For $\sigma\in\operatorname{Gal}(\bar k/k)$, the function $\sigma(f)$ has divisor $\sigma(D)=D$, because the four points are $k$-rational, and value $\sigma(1)=1$ at $O$. By uniqueness $\sigma(f)=f$ for all $\sigma$, so $f\in k(B_t)$.

The function $f\circ[-1]$ has divisor $[-1]^*D$$=(-S_t)-(S_t)-n(-R_t)+n(R_t)$$=-D$, so $f\circ[-1]=c/f$ for a constant $c$. Evaluating at $O=-O$ gives $1=c$. $\square$

Remark 2.2. The normalization $f_t(O)=1$ matters. Any other choice $cf_t$ gives the cover $Y^m=cf_t$, which differs from $Y^m=f_t$ by the twist by the class of $c$ in $k^\times/k^{\times m}$. The normalized function is the one compatible with base change and with the identifications of Section 6.

3. The cyclic covers

Let $m>1$ be a divisor of $n$; it is odd. For $t\in U(k)$ let $C_{m,t}$ be the curve with function field $k(C_{m,t})=k(B_t)[Y]/(Y^m-f_t),\quad(3.1)$ that is, the normalization of $B_t$ in this field. We write $Z$ for a point of $B_t$, so that the cover is "$Y^m=f_t(Z)$".

Proposition 3.1. Let $t\in U(k)$.

The polynomial $Y^m-f_t$ is irreducible over $\bar k(B_t)$. Hence $C_{m,t}$ is geometrically integral and $C_{m,t}\to B_t$ has degree $m$.
The cover is totally ramified over $S_t$ and $-S_t$ and unramified over every other point of $B_t$.
$C_{m,t}$ has genus $m$.

Proof. (1) Let $K=\bar k(B_t)$ and $v$ the valuation of $K$ at $S_t$, so $v(f_t)=1$. Let $L=K(y)$ with $y^m=f_t$, and let $w$ be an extension of $v$ to $L$, with ramification index $e$, normalized by $w|_K=e\,v$. Then $m\,w(y)=w(f_t)=e$. So $m$ divides $e$, and $m\le e\le[L:K]\le m$. Hence $[L:K]=m$, the polynomial is irreducible, and $S_t$ is totally ramified.

(2) The same argument applies at $-S_t$, where $v(f_t)=-1$. Let $P\in B_t(\bar k)$ be any other point and $\pi$ a uniformizer at $P$. Then $f_t=\pi^{a}\varepsilon$ with $\varepsilon$ a unit at $P$ and $a\in\{0,n,-n\}$, so $m\mid a$. Put $y'=y\pi^{-a/m}$; then $y'^m=\varepsilon$. In the completed local ring $\bar k[[\pi]]$ the unit $\varepsilon$ has $m$ distinct $m$-th roots, by Hensel's lemma, because $\varepsilon(P)\ne0$ and $m$ is invertible in $\bar k$. So $Y^m-\varepsilon$ splits into $m$ distinct linear factors over the completion, and $P$ has $m$ distinct preimages, each unramified.

(3) By (2) the two branch points contribute ramification index $m$ each, and every other point is unramified. The Riemann–Hurwitz formula (5.5) of The étale fundamental group, applied over $\bar k$, with $g(B_t)=1$, gives $2g(C_{m,t})-2=m\cdot0+2(m-1),$ so $g(C_{m,t})=m$. The genus does not change under extension of the ground field, because the curve is geometrically integral. $\square$

4. The family over $U$

The covers vary in a smooth proper family. Let $\begin{gathered} \mathcal B\subset\mathbb P^2_U,\\ \mathcal B\colon\ h(t)Y_0^{\,2}Y_2=h^{\mathrm{hom}}(Y_1,Y_2) \end{gathered}$ be the projective closure of $h(t)y^2=h(x)$ over $U$, where $h^{\mathrm{hom}}$ is the homogenization of $h$ and $x=Y_1/Y_2$, $y=Y_0/Y_2$. Since $h(t)$ is invertible on $U$ and $h$ is separable, $\mathcal B\to U$ is an elliptic curve over $U$: smooth and proper with geometrically connected fibres of genus one and the zero section $O$ at infinity. The points $R=(t,1)$, $-R$, $S=[n]R$ and $-S$ are sections $U\to\mathcal B$, and by the definition of $U$ the five sections $O,\pm R,\pm S$ are pairwise disjoint.

Lemma 4.1. There is a rational function $f$ on $\mathcal B$ whose divisor is $\begin{aligned} \operatorname{div}(f)&=S(U)-(-S)(U)\\ &\quad-n\,R(U)+n\,(-R)(U),\quad(4.1) \end{aligned}$ which is regular and equal to $1$ along $O(U)$, and whose restriction to every fibre $\mathcal B_t$, $t\in U$, is the function $f_t$ of Lemma 2.1.

Proof. Apply Lemma 2.1 over the field $k(t)$ to the generic fibre of $\mathcal B$, with the point $R$. This gives $f\in k(\mathcal B)^\times$ whose divisor on the generic fibre is the generic fibre of (4.1), with $f=1$ at the generic point of $O(U)$. The surface $\mathcal B$ is regular, so $\operatorname{div}(f)$ is defined, and it equals the right side of (4.1) plus a vertical divisor $\sum_ua_u\mathcal B_u$ over finitely many closed points $u\in U$. The automorphism $[-1]$ of $\mathcal B$ over $U$ maps each fibre $\mathcal B_u$ to itself, and $f\circ[-1]=f^{-1}$ by (2.2) on the generic fibre. Comparing the multiplicity of $\mathcal B_u$ in the divisors of both sides gives $a_u=-a_u$, so $a_u=0$. Thus (4.1) holds. The section $O(U)$ is disjoint from the four sections in (4.1), so $f$ is regular and invertible near it, and its restriction to $O(U)\cong U$ is a function on $U$ that equals $1$ at the generic point, hence everywhere. Finally, the sections meet each fibre transversally in one point, so the restriction of $f$ to $\mathcal B_t$ has the divisor (2.1) and value $1$ at $O$; by uniqueness it is $f_t$. $\square$

Proposition 4.2. For each divisor $m>1$ of $n$, let $\mathcal C_m$ be the normalization of $\mathcal B$ in $k(\mathcal B)[Y]/(Y^m-f)$. Then $\mathcal C_m\to U$ is smooth and proper, and its fibre over $t\in U$ is the curve $C_{m,t}$ of Section 3.

Proof. Normalization of a scheme of finite type over a field in a finite extension of its function field is finite, so $\mathcal C_m\to\mathcal B$ is finite and $\mathcal C_m\to U$ is proper. Smoothness is local on $\mathcal B$. Let $P\in\mathcal B$ be a closed point over $u\in U$, and let $\tau$ be a local parameter of $U$ at $u$.

If $P$ lies on none of the four sections, $f$ is a unit at $P$. The algebra $\mathcal O_{\mathcal B,P}[Y]/(Y^m-f)$ is finite étale over $\mathcal O_{\mathcal B,P}$, since the derivative $mY^{m-1}$ is invertible ($Y^m=f$ is a unit and $m$ is invertible). An étale algebra over a regular local ring is regular, hence normal, so it is the local ring of the normalization, and it is smooth over $U$ because $\mathcal B$ is.

If $P$ lies on $\pm R(U)$, let $\pi$ be a local equation of that section near $P$; the section is a Cartier divisor on the regular surface. Then $f=\pi^{a}\varepsilon$ with $a=\mp n$ and $\varepsilon$ a unit, and $Y'=Y\pi^{-a/m}$ satisfies $Y'^m=\varepsilon$. The previous case applies to $Y'$, which generates the same field.

If $P$ lies on $\pm S(U)$, let $\pi$ be a local equation of the section. Because the section is a section of a smooth morphism of relative dimension one, $(\tau,\pi)$ is a regular system of parameters at $P$, and $\pi$ restricts to a uniformizer on the fibre $\mathcal B_u$ at $P$. Write $f^{\pm1}=\pi\varepsilon$ with $\varepsilon$ a unit. After the étale base change that adjoins $e=\varepsilon^{1/m}$, the element $Y''=Y^{\pm1}/e$ satisfies $Y''^m=\pi$. The algebra $\mathcal O[Y'']/(Y''^m-\pi)$ is regular with parameters $(\tau,Y'')$, and étale locally over $U$ it is the relative affine line with coordinate $Y''$, mapping to the relative affine line with coordinate $\pi$ by $Y''\mapsto Y''^m$. It is therefore normal and smooth over $U$. Smoothness and normality descend along the étale base change.

So $\mathcal C_m$ is smooth over $U$. In each of the three local models the fibre over $u$ is the normalization of $\mathcal B_u$ in the field (3.1), by the same computations as in Proposition 3.1. Hence the fibre of $\mathcal C_m$ over $t$ is $C_{m,t}$. $\square$

By Proposition 3.1 the fibres are geometrically connected of genus $m$. The open set $U$ may be shrunk by finitely many points whenever this is convenient; this changes none of the statements of the course.

5. The involution and the rotations

Proposition 5.1. For every $t\in U(k)$ and every divisor $m>1$ of $n$:

there is an involution $w$ of $C_{m,t}$, defined over $k$, given by $w(Z,Y)=(-Z,\,Y^{-1});\quad(5.1)$
if $k$ contains the group $\mu_m$ of $m$-th roots of unity, the maps $u_\zeta(Z,Y)=(Z,\zeta Y)$, $\zeta\in\mu_m$, are automorphisms of $C_{m,t}$ over $B_t$, they form the full group of deck transformations, and $w\,u_\zeta\,w=u_{\zeta^{-1}}; \tag{5.2}$
for $d\mid m$, the inclusion $k(B_t)(Y^{m/d})\subset k(C_{m,t})$ defines a map $C_{m,t}\to C_{d,t}$, $Y\mapsto Y^{m/d}$, which is the quotient by the subgroup $\mu_{m/d}$ of rotations; for $d=1$ it is $C_{m,t}\to B_t$.

All these maps extend to the family $\mathcal C_m\to U$.

Proof. (1) On function fields, define $w^*$ by $w^*(g)=g\circ[-1]$ for $g\in k(B_t)$ and $w^*(Y)=Y^{-1}$. This is compatible with the relation $Y^m=f_t$, because $(Y^{-1})^m=f_t^{-1}=f_t\circ[-1]$ by (2.2). So $w^*$ is a field automorphism of $k(C_{m,t})$ over $k$ with $(w^*)^2=1$, and it induces an involution of the curve.

(2) If $\mu_m\subset k$, the extension (3.1) is a Kummer extension of degree $m$ with Galois group $\{Y\mapsto\zeta Y\}\cong\mu_m$. The automorphisms of the curve over $B_t$ are the automorphisms of its function field over $k(B_t)$, so they are the $u_\zeta$. For (5.2), compute on points: $\begin{aligned} w\,u_\zeta\,w(Z,Y)&=w\,u_\zeta(-Z,Y^{-1})\\ &=w(-Z,\zeta Y^{-1})\\ &=(Z,\zeta^{-1}Y). \end{aligned}$

(3) The element $Y^{m/d}$ satisfies $(Y^{m/d})^d=f_t$, so $k(B_t)(Y^{m/d})$ is the function field of $C_{d,t}$, and it is the fixed field of the rotations by $\zeta$ with $\zeta^{m/d}=1$.

The same formulas define automorphisms and maps of the generic fibres of the families, and they extend to the normal schemes $\mathcal C_m$, $\mathcal C_d$ and $\mathcal B$, which are the normalizations of $\mathcal B$ in the respective fields. $\square$

The involution has a small fixed-point set, which already shows how it pairs the rotation characters.

Proposition 5.2. The involution $w$ of $C_{m,t}$ has exactly four fixed points over $\bar k$, one above each point of $B_t[2](\bar k)$. The quotient $C_{m,t}/\langle w\rangle$ has genus $(m-1)/2$.

Proof. A fixed point of $w$ lies over a point $Z$ with $-Z=Z$, that is, over $Z\in B_t[2]$, which consists of $O$ and the three points with $y=0$. None of these is in the support of $\operatorname{div}(f_t)$, because $\pm R_t$ and $\pm S_t$ are not $2$-torsion: $R_t\ne-R_t$, and $S_t\ne-S_t$ by the definition of $U$. So $f_t$ is a unit at $Z$, and (2.2) at $Z=-Z$ gives $f_t(Z)^2=1$. The $m$ points above $Z$ are the solutions of $Y^m=f_t(Z)$, and such a point is fixed by $w$ exactly when $Y=Y^{-1}$, that is, $Y=\pm1$. Since $m$ is odd, $(\pm1)^m=\pm1$, so exactly one of $Y=1$, $Y=-1$ satisfies $Y^m=f_t(Z)$, namely $Y=f_t(Z)$. This gives exactly one fixed point above each of the four points of $B_t[2]$.

The quotient map $C_{m,t}\to C_{m,t}/\langle w\rangle$ has degree two and is ramified exactly at the four fixed points, each with index two. Riemann–Hurwitz gives $2m-2=2(2g'-2)+4$, so $g'=(m-1)/2$. $\square$

For $m=3$ the quotient $D_t=C_{3,t}/\langle w\rangle$ has genus one. Its Jacobian is the elliptic curve $A_t$ that starts the chain of congruences; Lesson 5 identifies its cohomology with a character piece of $H^1(C_{n,t})$.

6. The moving-point family

For the geometry of Lessons 3 and 4 it is convenient to keep the elliptic curve fixed and move the point. Let $\begin{aligned} &E^\circ=\{R\in E:\ O,\,R,\,-R,\,[n]R,\,-[n]R\\ &\qquad\qquad\text{pairwise distinct}\}, \end{aligned}$ an open subset of $E$ whose complement is finite. On the constant surface $E\times E^\circ\to E^\circ$, with the parameter $R$ in the second factor, the graphs of $Z=\pm R$ and $Z=\pm[n]R$ are disjoint sections. Lemma 4.1 and Proposition 4.2 apply verbatim: there is a rational function $f_R(Z)$, normalized by $f_R(O)=1$, with divisor $([n]R)-(-[n]R)-n(R)+n(-R)$ on every fibre, and the normalization of $E\times E^\circ$ in $Y^m=f_R(Z)$ is a smooth proper family $\mathcal C^E_m\to E^\circ$ with fibres $C_{m,R}$.

Proposition 6.1. The map $x\colon E^\circ\to\mathbb A^1$ takes values in $U$ and is finite étale of degree two onto $U$, and $E^\circ=x^{-1}(U)$. For $R\in E^\circ$ with $t=x(R)$, the map $\begin{gathered} \phi_R\colon B_t\longrightarrow E,\\ (x,y)\longmapsto\bigl(x,\,y(R)\,y\bigr),\quad(6.1) \end{gathered}$ is an isomorphism of elliptic curves with $\phi_R(R_t)=R$, and it identifies $f_t$ with $f_R$ and $C_{m,t}$ with $C_{m,R}$. These identifications form an isomorphism over $E^\circ$ between $\mathcal C^E_m$ and the pullback of $\mathcal C_m$ along $x$.

Proof. A point $R=(x_0,y_0)\in E^\circ$ is not $2$-torsion, because $R\ne-R$, so $y_0\ne0$ and $h(x_0)=y_0^2\ne0$. For $(x,y)\in B_t$, $(y_0y)^2=h(t)y^2=h(x)$, so $\phi_R$ maps $B_t$ to $E$; it is an isomorphism with inverse $(x,y)\mapsto(x,y/y_0)$, it fixes the point at infinity, and $\phi_R(t,1)=(x_0,y_0)=R$. An isomorphism of elliptic curves preserving the origins is a group isomorphism, so $\phi_R([m]R_t)=[m]R$ for every $m$. Hence the five points $O,\pm R_t,\pm S_t$ are pairwise distinct exactly when $O,\pm R,\pm[n]R$ are, which gives $E^\circ=x^{-1}(U)$. The map $x\colon E\to\mathbb P^1$ has degree two and is ramified exactly at the points of $E[2]$, none of which lies in $E^\circ$; so it is finite étale of degree two over $U$.

The function $f_R\circ\phi_R$ on $B_t$ has the divisor (2.1) and value $1$ at $O$, so it equals $f_t$ by Lemma 2.1. Therefore $\phi_R$ extends to an isomorphism $C_{m,t}\to C_{m,R}$ with $Y\mapsto Y$. The coefficient $y(R)$ is a regular invertible function on $E^\circ$, so the same formula defines the isomorphism of families. $\square$

The normalization $f(O)=1$ is what makes this identification exact: no constant appears in the equation $Y^m=f$, so no twist of the cover appears either (Remark 2.2).

7. Exercises

7.1. Let $E\colon y^2=x^3-x$ over $\mathbb Q$ and $n=3$. Show that $t=2$ lies in $U$, and write down the five points $O,\pm R_2,\pm S_2$ of $B_2\colon6y^2=x^3-x$.

7.2. Show that the function $f_t$ of Lemma 2.1 satisfies $f_t(Q)\in\{1,-1\}$ for every $Q\in B_t[2](\bar k)$, and that $f_t(O)=1$ is consistent with this.

7.3. Let $m\mid n$, $m>1$, and suppose $\mu_m\subset k$. Show that the fixed points of $u_\zeta$ for $\zeta\ne1$ are exactly the two points above $S_t$ and $-S_t$, and deduce the genus formula of Proposition 3.1 from the Lefschetz-free count: the cover $C_{m,t}\setminus\{\text{2 points}\}\to B_t\setminus\{\pm S_t\}$ is a connected étale cover of degree $m$.

7.4. Show that $w\,u_\zeta$ is an involution for every $\zeta\in\mu_m$, and count its fixed points.

8. Solutions

7.1. Here $h(2)=6\ne0$, and $R_2=(2,1)$ lies on $6y^2=x^3-x$ since $6=8-2$. On $B_2$ the tangent line at $R_2$ has slope $\lambda$ with $12y\lambda=3x^2-1$, so $\lambda=11/12$. For $6y^2=x^3-x$, the third intersection of a line $y=\lambda x+c$ with the curve satisfies $x_1+x_2+x_3=6\lambda^2$. So $x([2]R_2)=6\cdot\frac{121}{144}-4=\frac{121}{24}-4=\frac{25}{24}$, and $y([2]R_2)$$=-\bigl(1+\frac{11}{12}(\frac{25}{24}-2)\bigr)$$=-\bigl(1-\frac{253}{288}\bigr)$$=-\frac{35}{288}$. Then $S_2=[3]R_2=[2]R_2+R_2$: the chord through $(2,1)$ and $(\frac{25}{24},-\frac{35}{288})$ has slope $\lambda'=\frac{1+35/288}{2-25/24}=\frac{323/288}{23/24}=\frac{323}{276}$. Hence $\begin{gathered} x(S_2)=6\lambda'^2-2-\tfrac{25}{24}=\tfrac{2738}{529},\\ y(S_2)=-\tfrac{57387}{12167}, \end{gathered}$ with $529=23^2$ and $12167=23^3$. These points are affine with $y\ne0$ (the $2$-torsion points of $B_2$ are $O$ and $(0,0),(\pm1,0)$), so $\pm R_2$ and $\pm S_2$ are not $2$-torsion and $R_2$, $S_2$ are different from $O$. Since $x(S_2)\ne2$, also $S_2\ne\pm R_2$. Hence the five points $O$, $(2,\pm1)$ and $\bigl(\frac{2738}{529},\pm\frac{57387}{12167}\bigr)$ are distinct and $2\in U$.

7.2. A point $Q\in B_t[2]$ satisfies $Q=-Q$ and is not in the support of $\operatorname{div}(f_t)$ (see the proof of Proposition 5.2), so (2.2) gives $f_t(Q)=f_t(-Q)=f_t(Q)^{-1}$, that is, $f_t(Q)^2=1$. For $Q=O$ the normalization chooses the value $+1$.

7.3. A point $(Z,Y)$ is fixed by $u_\zeta$, $\zeta\ne1$, exactly when $\zeta Y=Y$ in the local description, which happens only where the fibre of the cover over $Z$ has fewer than $m$ points, that is, over the totally ramified points $\pm S_t$ (Proposition 3.1(2)); the unique point above each is fixed by every rotation. Removing the two branch points, the cover is a connected étale cover of degree $m$ of the twice-punctured torus $B_t\setminus\{\pm S_t\}$, whose Euler characteristic over $\mathbb C$ is $0-2=-2$. The cover has Euler characteristic $-2m$, and adding back the two points gives $2-2g=-2m+2$, so $g=m$. (This is the topological form of the same Riemann–Hurwitz count, valid after a complex embedding of $k$.)

7.4. By (5.2), $(wu_\zeta)^2=w u_\zeta w u_\zeta=u_{\zeta^{-1}}u_\zeta=1$. The involution $wu_\zeta$ acts by $(Z,Y)\mapsto(-Z,\zeta^{-1}Y^{-1})$, so its fixed points lie over $B_t[2]$ and satisfy $Y^2=\zeta^{-1}$, together with $Y^m=f_t(Z)=\pm1$. Since $m$ is odd, $\zeta^{-1}$ has a unique square root $\eta_0$ in $\mu_m$, and the solutions are $Y=\pm\eta_0$. Exactly one of them satisfies $Y^m=f_t(Z)$, because $(\pm\eta_0)^m=\pm1$. So $wu_\zeta$ also has exactly four fixed points, one above each point of $B_t[2]$. The $m$ involutions $wu_\zeta$ are the reflections of the dihedral group generated by $w$ and the rotations.

Sources

[OpenAI-030] OpenAI, Modularity of elliptic curves over imaginary quadratic fields, OpenAI Math Release preprint, 4 October 2026, Section 3.1 (the family and its dihedral action). https://github.com/openai/math/tree/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026