Invariants
Level: | $312$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C0 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}59&174\\156&1\end{bmatrix}$, $\begin{bmatrix}82&33\\61&38\end{bmatrix}$, $\begin{bmatrix}84&205\\29&208\end{bmatrix}$, $\begin{bmatrix}116&17\\97&188\end{bmatrix}$, $\begin{bmatrix}281&40\\38&135\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 24.12.0.y.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $112$ |
Cyclic 312-torsion field degree: | $10752$ |
Full 312-torsion field degree: | $80510976$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1284 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3}\cdot\frac{x^{12}(3x^{2}-24xy+32y^{2})^{3}(3x^{2}+24xy+32y^{2})^{3}}{y^{8}x^{14}(3x^{2}-128y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
52.12.0-4.c.1.1 | $52$ | $2$ | $2$ | $0$ | $0$ |
312.12.0-4.c.1.6 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.48.0-24.l.1.3 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.n.1.7 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.u.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.w.1.7 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.bk.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.bn.1.7 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.bp.1.2 | $312$ | $2$ | $2$ | $0$ |
312.48.0-24.bq.1.7 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.cm.1.2 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.co.1.6 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.cq.1.5 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.cs.1.4 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.dk.1.9 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.dm.1.10 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.do.1.1 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.dq.1.10 | $312$ | $2$ | $2$ | $0$ |
312.72.2-24.ck.1.7 | $312$ | $3$ | $3$ | $2$ |
312.96.1-24.is.1.23 | $312$ | $4$ | $4$ | $1$ |
312.336.11-312.ci.1.41 | $312$ | $14$ | $14$ | $11$ |