Properties

Label 312.96.0-312.dg.1.3
Level $312$
Index $96$
Genus $0$
Cusps $10$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $312$ $\SL_2$-level: $8$
Index: $96$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot8^{4}$ Cusp orbits $2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1 \le \gamma \le 2$
$\overline{\Q}$-gonality: $1$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8O0

Level structure

$\GL_2(\Z/312\Z)$-generators: $\begin{bmatrix}237&280\\13&113\end{bmatrix}$, $\begin{bmatrix}245&192\\203&283\end{bmatrix}$, $\begin{bmatrix}269&272\\24&287\end{bmatrix}$, $\begin{bmatrix}289&176\\307&1\end{bmatrix}$
Contains $-I$: no $\quad$ (see 312.48.0.dg.1 for the level structure with $-I$)
Cyclic 312-isogeny field degree: $56$
Cyclic 312-torsion field degree: $5376$
Full 312-torsion field degree: $20127744$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
24.48.0-24.bh.1.1 $24$ $2$ $2$ $0$ $0$
104.48.0-104.ca.1.6 $104$ $2$ $2$ $0$ $?$
312.48.0-24.bh.1.8 $312$ $2$ $2$ $0$ $?$
312.48.0-104.ca.1.4 $312$ $2$ $2$ $0$ $?$
312.48.0-312.ei.1.19 $312$ $2$ $2$ $0$ $?$
312.48.0-312.ei.1.20 $312$ $2$ $2$ $0$ $?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
312.288.8-312.qk.2.2 $312$ $3$ $3$ $8$
312.384.7-312.kr.1.10 $312$ $4$ $4$ $7$