Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F1 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}48&197\\35&270\end{bmatrix}$, $\begin{bmatrix}58&63\\195&88\end{bmatrix}$, $\begin{bmatrix}197&222\\282&47\end{bmatrix}$, $\begin{bmatrix}273&86\\86&273\end{bmatrix}$, $\begin{bmatrix}304&231\\267&256\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.1.bk.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $13418496$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 216 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3\cdot3^3}\cdot\frac{(y^{2}-648z^{2})^{3}(y^{6}-48600y^{4}z^{2}-18895680y^{2}z^{4}-2448880128z^{6})^{3}}{z^{2}y^{6}(y^{2}+216z^{2})^{2}(y^{2}+1944z^{2})^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 156.72.0-6.a.1.5 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.0-24.cb.1.9 | $312$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 312.48.0-24.cb.1.11 | $312$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 312.72.0-6.a.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 312.288.5-24.fo.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.fr.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.gq.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.gt.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.hs.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.hw.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.iu.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.iy.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bhw.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bhx.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bik.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bil.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bsq.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bsr.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bte.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.btf.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |