Invariants
Level: | $312$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 4E0 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}35&288\\70&101\end{bmatrix}$, $\begin{bmatrix}73&42\\172&113\end{bmatrix}$, $\begin{bmatrix}241&74\\0&245\end{bmatrix}$, $\begin{bmatrix}269&256\\190&207\end{bmatrix}$, $\begin{bmatrix}311&292\\188&285\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 104.12.0.b.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $224$ |
Cyclic 312-torsion field degree: | $21504$ |
Full 312-torsion field degree: | $80510976$ |
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 |
---|---|---|---|---|---|
12.12.0-2.a.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
312.12.0-2.a.1.1 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.48.0-104.c.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.c.1.4 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.d.1.6 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.d.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.e.1.9 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.e.1.11 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.f.1.6 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.f.1.7 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.k.1.2 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.k.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.m.1.14 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.m.1.15 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.q.1.15 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.q.1.16 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.s.1.4 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.s.1.8 | $312$ | $2$ | $2$ | $0$ |
312.72.2-312.b.1.30 | $312$ | $3$ | $3$ | $2$ |
312.96.1-312.dh.1.31 | $312$ | $4$ | $4$ | $1$ |
312.336.11-104.d.1.4 | $312$ | $14$ | $14$ | $11$ |