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: | $\begin{bmatrix}83&0\\252&229\end{bmatrix}$, $\begin{bmatrix}213&256\\35&295\end{bmatrix}$, $\begin{bmatrix}223&132\\284&293\end{bmatrix}$, $\begin{bmatrix}241&56\\80&173\end{bmatrix}$, $\begin{bmatrix}251&196\\231&13\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 104.12.0.v.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $112$ |
Cyclic 312-torsion field degree: | $10752$ |
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 |
---|---|---|---|---|---|
12.12.0-4.c.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
312.12.0-4.c.1.2 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.bo.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.bo.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.bp.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.bp.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.by.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.by.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.bz.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-104.bz.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.cs.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.cs.1.16 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.ct.1.3 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.ct.1.14 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.da.1.3 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.da.1.14 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.db.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.db.1.16 | $312$ | $2$ | $2$ | $0$ |
312.72.2-312.cf.1.8 | $312$ | $3$ | $3$ | $2$ |
312.96.1-312.zf.1.7 | $312$ | $4$ | $4$ | $1$ |
312.336.11-104.bt.1.4 | $312$ | $14$ | $14$ | $11$ |