Invariants
Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12L3 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}9&44\\76&293\end{bmatrix}$, $\begin{bmatrix}49&26\\268&189\end{bmatrix}$, $\begin{bmatrix}99&82\\230&301\end{bmatrix}$, $\begin{bmatrix}135&272\\86&147\end{bmatrix}$, $\begin{bmatrix}205&254\\198&35\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.96.3.ft.2 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $56$ |
Cyclic 312-torsion field degree: | $2688$ |
Full 312-torsion field degree: | $10063872$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
312.96.0-12.a.2.11 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.96.1-312.dj.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.96.1-312.dj.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.96.2-312.b.1.11 | $312$ | $2$ | $2$ | $2$ | $?$ |
312.96.2-312.b.1.15 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.384.5-312.jj.2.5 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jj.2.18 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jl.1.15 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jl.3.5 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jo.1.24 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jo.3.9 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jq.1.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.jq.3.10 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kc.2.7 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kc.4.15 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ke.3.16 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ke.4.5 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kj.2.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kj.4.14 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kl.2.7 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kl.4.16 | $312$ | $2$ | $2$ | $5$ |