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Properties

Label 312.96.3.su.3
Level $312$
Index $96$
Genus $3$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $312$ $\SL_2$-level: $24$ Newform level: $1$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{5}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 3$
$\overline{\Q}$-gonality: $2 \le \gamma \le 3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24W3

Level structure

$\GL_2(\Z/312\Z)$-generators: $\begin{bmatrix}31&44\\120&179\end{bmatrix}$, $\begin{bmatrix}34&303\\195&154\end{bmatrix}$, $\begin{bmatrix}82&231\\127&182\end{bmatrix}$, $\begin{bmatrix}207&88\\28&63\end{bmatrix}$, $\begin{bmatrix}237&194\\152&303\end{bmatrix}$, $\begin{bmatrix}290&187\\311&126\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 312.192.3-312.su.3.1, 312.192.3-312.su.3.2, 312.192.3-312.su.3.3, 312.192.3-312.su.3.4, 312.192.3-312.su.3.5, 312.192.3-312.su.3.6, 312.192.3-312.su.3.7, 312.192.3-312.su.3.8, 312.192.3-312.su.3.9, 312.192.3-312.su.3.10, 312.192.3-312.su.3.11, 312.192.3-312.su.3.12, 312.192.3-312.su.3.13, 312.192.3-312.su.3.14, 312.192.3-312.su.3.15, 312.192.3-312.su.3.16, 312.192.3-312.su.3.17, 312.192.3-312.su.3.18, 312.192.3-312.su.3.19, 312.192.3-312.su.3.20, 312.192.3-312.su.3.21, 312.192.3-312.su.3.22, 312.192.3-312.su.3.23, 312.192.3-312.su.3.24, 312.192.3-312.su.3.25, 312.192.3-312.su.3.26, 312.192.3-312.su.3.27, 312.192.3-312.su.3.28, 312.192.3-312.su.3.29, 312.192.3-312.su.3.30, 312.192.3-312.su.3.31, 312.192.3-312.su.3.32
Cyclic 312-isogeny field degree: $28$
Cyclic 312-torsion field degree: $2688$
Full 312-torsion field degree: $20127744$

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
12.48.0.c.3 $12$ $2$ $2$ $0$ $0$
312.48.1.baa.1 $312$ $2$ $2$ $1$ $?$
312.48.2.h.1 $312$ $2$ $2$ $2$ $?$

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

Covering curve Level Index Degree Genus
312.192.5.mo.4 $312$ $2$ $2$ $5$
312.192.5.rf.4 $312$ $2$ $2$ $5$
312.192.5.um.1 $312$ $2$ $2$ $5$
312.192.5.uu.4 $312$ $2$ $2$ $5$
312.192.5.wq.1 $312$ $2$ $2$ $5$
312.192.5.wz.1 $312$ $2$ $2$ $5$
312.192.5.ye.1 $312$ $2$ $2$ $5$
312.192.5.yn.1 $312$ $2$ $2$ $5$
312.192.5.zd.4 $312$ $2$ $2$ $5$
312.192.5.zg.4 $312$ $2$ $2$ $5$
312.192.5.bbi.4 $312$ $2$ $2$ $5$
312.192.5.bbj.4 $312$ $2$ $2$ $5$
312.192.5.bcn.1 $312$ $2$ $2$ $5$
312.192.5.bcq.1 $312$ $2$ $2$ $5$
312.192.5.bdm.1 $312$ $2$ $2$ $5$
312.192.5.bdn.1 $312$ $2$ $2$ $5$
312.288.13.bvq.2 $312$ $3$ $3$ $13$