Invariants
| Level: | $312$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{4}$ | ||
| 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: | 8B1 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}11&98\\240&217\end{bmatrix}$, $\begin{bmatrix}63&268\\104&61\end{bmatrix}$, $\begin{bmatrix}83&186\\152&173\end{bmatrix}$, $\begin{bmatrix}171&178\\4&149\end{bmatrix}$, $\begin{bmatrix}243&16\\280&173\end{bmatrix}$, $\begin{bmatrix}257&106\\200&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.d.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $40255488$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 36x $ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{3888x^{2}y^{4}z^{2}+36xy^{6}z+5038848xy^{2}z^{5}+y^{8}+2176782336z^{8}}{z^{2}y^{4}x^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 104.24.0-4.b.1.7 | $104$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.24.0-4.b.1.3 | $312$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 312.96.1-24.n.2.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.n.2.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bb.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bb.1.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bg.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bg.1.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bg.2.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bg.2.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bh.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bh.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bh.2.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bh.2.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bi.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bi.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bi.2.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bi.2.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bj.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bj.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bj.2.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bj.2.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bs.1.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bs.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bv.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-24.bv.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bw.1.18 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bw.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.by.1.18 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.by.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ce.1.18 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ce.1.25 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ce.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ce.2.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cf.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cf.1.31 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cf.2.26 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cf.2.29 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cg.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cg.1.31 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cg.2.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cg.2.29 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ch.1.17 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ch.1.29 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ch.2.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ch.2.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.dc.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.dc.1.22 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.de.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.de.1.22 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.144.5-24.h.1.29 | $312$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 312.192.5-24.h.1.28 | $312$ | $4$ | $4$ | $5$ | $?$ | not computed |