Invariants
Level: | $264$ | $\SL_2$-level: | $12$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E0 |
Level structure
$\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}69&248\\158&33\end{bmatrix}$, $\begin{bmatrix}74&3\\41&238\end{bmatrix}$, $\begin{bmatrix}144&191\\23&42\end{bmatrix}$, $\begin{bmatrix}182&241\\183&148\end{bmatrix}$, $\begin{bmatrix}237&76\\218&55\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 264.24.0.fj.1 for the level structure with $-I$) |
Cyclic 264-isogeny field degree: | $48$ |
Cyclic 264-torsion field degree: | $3840$ |
Full 264-torsion field degree: | $20275200$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
264.24.0-6.a.1.5 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
264.96.1-264.dh.1.17 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.gl.1.2 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.kf.1.3 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.kh.1.12 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bad.1.13 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.baf.1.10 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.baj.1.10 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bal.1.12 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.byy.1.9 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.byz.1.2 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bze.1.2 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bzf.1.4 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bzg.1.7 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bzi.1.6 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bzp.1.6 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.bzr.1.24 | $264$ | $2$ | $2$ | $1$ |
264.144.1-264.bn.1.15 | $264$ | $3$ | $3$ | $1$ |