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Properties

Label 312.48.2.i.1
Level $312$
Index $48$
Genus $2$
Cusps $6$
$\Q$-cusps $6$

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Invariants

Level: $312$ $\SL_2$-level: $24$ Newform level: $1$
Index: $48$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $2^{2}\cdot6^{2}\cdot8\cdot24$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24F2

Level structure

$\GL_2(\Z/312\Z)$-generators: $\begin{bmatrix}44&63\\243&224\end{bmatrix}$, $\begin{bmatrix}71&168\\144&203\end{bmatrix}$, $\begin{bmatrix}92&15\\59&280\end{bmatrix}$, $\begin{bmatrix}133&128\\16&165\end{bmatrix}$, $\begin{bmatrix}215&100\\120&127\end{bmatrix}$, $\begin{bmatrix}229&86\\304&99\end{bmatrix}$, $\begin{bmatrix}266&219\\245&292\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 312.96.2-312.i.1.1, 312.96.2-312.i.1.2, 312.96.2-312.i.1.3, 312.96.2-312.i.1.4, 312.96.2-312.i.1.5, 312.96.2-312.i.1.6, 312.96.2-312.i.1.7, 312.96.2-312.i.1.8, 312.96.2-312.i.1.9, 312.96.2-312.i.1.10, 312.96.2-312.i.1.11, 312.96.2-312.i.1.12, 312.96.2-312.i.1.13, 312.96.2-312.i.1.14, 312.96.2-312.i.1.15, 312.96.2-312.i.1.16, 312.96.2-312.i.1.17, 312.96.2-312.i.1.18, 312.96.2-312.i.1.19, 312.96.2-312.i.1.20, 312.96.2-312.i.1.21, 312.96.2-312.i.1.22, 312.96.2-312.i.1.23, 312.96.2-312.i.1.24, 312.96.2-312.i.1.25, 312.96.2-312.i.1.26, 312.96.2-312.i.1.27, 312.96.2-312.i.1.28, 312.96.2-312.i.1.29, 312.96.2-312.i.1.30, 312.96.2-312.i.1.31, 312.96.2-312.i.1.32, 312.96.2-312.i.1.33, 312.96.2-312.i.1.34, 312.96.2-312.i.1.35, 312.96.2-312.i.1.36, 312.96.2-312.i.1.37, 312.96.2-312.i.1.38, 312.96.2-312.i.1.39, 312.96.2-312.i.1.40, 312.96.2-312.i.1.41, 312.96.2-312.i.1.42, 312.96.2-312.i.1.43, 312.96.2-312.i.1.44, 312.96.2-312.i.1.45, 312.96.2-312.i.1.46, 312.96.2-312.i.1.47, 312.96.2-312.i.1.48, 312.96.2-312.i.1.49, 312.96.2-312.i.1.50, 312.96.2-312.i.1.51, 312.96.2-312.i.1.52, 312.96.2-312.i.1.53, 312.96.2-312.i.1.54, 312.96.2-312.i.1.55, 312.96.2-312.i.1.56, 312.96.2-312.i.1.57, 312.96.2-312.i.1.58, 312.96.2-312.i.1.59, 312.96.2-312.i.1.60, 312.96.2-312.i.1.61, 312.96.2-312.i.1.62, 312.96.2-312.i.1.63, 312.96.2-312.i.1.64
Cyclic 312-isogeny field degree: $28$
Cyclic 312-torsion field degree: $2688$
Full 312-torsion field degree: $40255488$

Rational points

This modular curve has 6 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
$X_0(12)$ $12$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
312.96.3.er.1 $312$ $2$ $2$ $3$
312.96.3.hv.1 $312$ $2$ $2$ $3$
312.96.3.ki.1 $312$ $2$ $2$ $3$
312.96.3.kl.1 $312$ $2$ $2$ $3$
312.96.3.lz.1 $312$ $2$ $2$ $3$
312.96.3.mb.2 $312$ $2$ $2$ $3$
312.96.3.ml.1 $312$ $2$ $2$ $3$
312.96.3.mn.2 $312$ $2$ $2$ $3$
312.96.3.na.2 $312$ $2$ $2$ $3$
312.96.3.nd.1 $312$ $2$ $2$ $3$
312.96.3.ne.2 $312$ $2$ $2$ $3$
312.96.3.nh.1 $312$ $2$ $2$ $3$
312.96.3.nq.1 $312$ $2$ $2$ $3$
312.96.3.nt.2 $312$ $2$ $2$ $3$
312.96.3.nu.1 $312$ $2$ $2$ $3$
312.96.3.nx.2 $312$ $2$ $2$ $3$
312.96.3.sc.1 $312$ $2$ $2$ $3$
312.96.3.sc.3 $312$ $2$ $2$ $3$
312.96.3.sd.1 $312$ $2$ $2$ $3$
312.96.3.sd.3 $312$ $2$ $2$ $3$
312.96.3.sg.1 $312$ $2$ $2$ $3$
312.96.3.sg.3 $312$ $2$ $2$ $3$
312.96.3.sh.1 $312$ $2$ $2$ $3$
312.96.3.sh.3 $312$ $2$ $2$ $3$
312.96.3.sk.1 $312$ $2$ $2$ $3$
312.96.3.sk.3 $312$ $2$ $2$ $3$
312.96.3.sl.1 $312$ $2$ $2$ $3$
312.96.3.sl.3 $312$ $2$ $2$ $3$
312.96.3.so.2 $312$ $2$ $2$ $3$
312.96.3.so.4 $312$ $2$ $2$ $3$
312.96.3.sp.2 $312$ $2$ $2$ $3$
312.96.3.sp.4 $312$ $2$ $2$ $3$
312.96.3.ss.2 $312$ $2$ $2$ $3$
312.96.3.ss.4 $312$ $2$ $2$ $3$
312.96.3.st.2 $312$ $2$ $2$ $3$
312.96.3.st.4 $312$ $2$ $2$ $3$
312.96.3.sw.1 $312$ $2$ $2$ $3$
312.96.3.sw.3 $312$ $2$ $2$ $3$
312.96.3.sx.1 $312$ $2$ $2$ $3$
312.96.3.sx.3 $312$ $2$ $2$ $3$
312.96.3.ta.1 $312$ $2$ $2$ $3$
312.96.3.ta.3 $312$ $2$ $2$ $3$
312.96.3.tb.1 $312$ $2$ $2$ $3$
312.96.3.tb.3 $312$ $2$ $2$ $3$
312.96.3.te.1 $312$ $2$ $2$ $3$
312.96.3.te.3 $312$ $2$ $2$ $3$
312.96.3.tf.1 $312$ $2$ $2$ $3$
312.96.3.tf.3 $312$ $2$ $2$ $3$
312.144.7.des.1 $312$ $3$ $3$ $7$