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: | $\begin{bmatrix}41&232\\238&289\end{bmatrix}$, $\begin{bmatrix}77&106\\122&57\end{bmatrix}$, $\begin{bmatrix}191&0\\310&89\end{bmatrix}$, $\begin{bmatrix}211&224\\90&289\end{bmatrix}$, $\begin{bmatrix}283&310\\196&231\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.12.0.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 | $12$ | $2$ | $2$ | $0$ | $0$ |
104.12.0-2.a.1.1 | $104$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.48.0-312.c.1.2 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.c.1.5 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.d.1.4 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.d.1.8 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.e.1.9 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.e.1.17 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.f.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.f.1.10 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.k.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.k.1.10 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.l.1.5 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.l.1.9 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.n.1.2 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.n.1.4 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.o.1.11 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.o.1.16 | $312$ | $2$ | $2$ | $0$ |
312.72.2-312.d.1.30 | $312$ | $3$ | $3$ | $2$ |
312.96.1-312.dj.1.37 | $312$ | $4$ | $4$ | $1$ |
312.336.11-312.d.1.10 | $312$ | $14$ | $14$ | $11$ |