Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}79&212\\186&227\end{bmatrix}$, $\begin{bmatrix}145&306\\150&229\end{bmatrix}$, $\begin{bmatrix}179&102\\38&253\end{bmatrix}$, $\begin{bmatrix}191&136\\32&129\end{bmatrix}$, $\begin{bmatrix}229&156\\150&61\end{bmatrix}$, $\begin{bmatrix}231&82\\124&63\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.1.dh.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $20127744$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | not computed |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 104.12.0.b.1 | $104$ | $8$ | $4$ | $0$ | $?$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 312.48.0-6.a.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.0-312.fp.1.2 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.0-312.fp.1.31 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.1-312.hm.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1-312.hm.1.24 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 312.192.1-312.lp.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lp.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lp.3.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lp.4.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lr.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lr.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lr.3.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lr.4.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.dq.1.19 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.dr.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ds.1.36 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.dt.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ef.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.eh.1.16 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.el.1.16 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.en.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fn.1.20 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fn.2.24 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fp.1.4 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fp.2.8 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.gi.1.20 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.gi.2.24 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.gj.1.4 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.gj.2.8 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.5-312.d.1.2 | $312$ | $3$ | $3$ | $5$ | $?$ | not computed |