Invariants
Level: | $312$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8K1 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}11&60\\20&215\end{bmatrix}$, $\begin{bmatrix}31&236\\160&295\end{bmatrix}$, $\begin{bmatrix}67&96\\200&287\end{bmatrix}$, $\begin{bmatrix}191&280\\116&189\end{bmatrix}$, $\begin{bmatrix}215&276\\52&7\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 24.96.1.a.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $112$ |
Cyclic 312-torsion field degree: | $10752$ |
Full 312-torsion field degree: | $10063872$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y^{2} + 2 z^{2} $ |
$=$ | $3 x^{2} - 3 y^{2} - w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{(36z^{4}-36z^{3}w+18z^{2}w^{2}-6zw^{3}+w^{4})^{3}(36z^{4}+36z^{3}w+18z^{2}w^{2}+6zw^{3}+w^{4})^{3}}{w^{8}z^{8}(6z^{2}-w^{2})^{2}(6z^{2}+w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
104.96.0-8.a.1.5 | $104$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
312.96.0-8.a.1.7 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
312.96.0-24.b.2.8 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
312.96.0-24.b.2.20 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
312.96.1-24.n.2.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.96.1-24.n.2.21 | $312$ | $2$ | $2$ | $1$ | $?$ | 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 |
---|---|---|---|---|---|---|
312.384.5-24.i.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.i.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.k.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.k.2.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.l.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.l.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.n.4.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-24.n.4.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.p.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.p.2.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.q.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.q.2.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.u.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.u.2.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.w.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.384.5-312.w.2.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |