Invariants
Level: | $312$ | $\SL_2$-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
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}71&130\\20&225\end{bmatrix}$, $\begin{bmatrix}107&204\\252&53\end{bmatrix}$, $\begin{bmatrix}137&244\\60&133\end{bmatrix}$, $\begin{bmatrix}174&49\\311&214\end{bmatrix}$, $\begin{bmatrix}251&232\\122&81\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.24.0.fp.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $56$ |
Cyclic 312-torsion field degree: | $2688$ |
Full 312-torsion field degree: | $40255488$ |
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.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
312.24.0-6.a.1.6 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.96.1-312.dh.1.30 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.gl.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.kf.1.21 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.kh.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.baf.1.15 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bah.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bal.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.ban.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bza.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzb.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzg.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzh.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzi.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzk.1.15 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzr.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzt.1.17 | $312$ | $2$ | $2$ | $1$ |
312.144.1-312.br.1.2 | $312$ | $3$ | $3$ | $1$ |