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Properties

Label 312.36.2.cx.1
Level $312$
Index $36$
Genus $2$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $312$ $\SL_2$-level: $24$ Newform level: $1$
Index: $36$ $\PSL_2$-index:$36$
Genus: $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $3^{2}\cdot6\cdot24$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24B2

Level structure

$\GL_2(\Z/312\Z)$-generators: $\begin{bmatrix}47&128\\278&73\end{bmatrix}$, $\begin{bmatrix}54&151\\173&228\end{bmatrix}$, $\begin{bmatrix}57&80\\122&207\end{bmatrix}$, $\begin{bmatrix}113&262\\82&121\end{bmatrix}$, $\begin{bmatrix}121&114\\246&97\end{bmatrix}$, $\begin{bmatrix}230&235\\71&86\end{bmatrix}$, $\begin{bmatrix}252&103\\281&30\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 312.72.2-312.cx.1.1, 312.72.2-312.cx.1.2, 312.72.2-312.cx.1.3, 312.72.2-312.cx.1.4, 312.72.2-312.cx.1.5, 312.72.2-312.cx.1.6, 312.72.2-312.cx.1.7, 312.72.2-312.cx.1.8, 312.72.2-312.cx.1.9, 312.72.2-312.cx.1.10, 312.72.2-312.cx.1.11, 312.72.2-312.cx.1.12, 312.72.2-312.cx.1.13, 312.72.2-312.cx.1.14, 312.72.2-312.cx.1.15, 312.72.2-312.cx.1.16, 312.72.2-312.cx.1.17, 312.72.2-312.cx.1.18, 312.72.2-312.cx.1.19, 312.72.2-312.cx.1.20, 312.72.2-312.cx.1.21, 312.72.2-312.cx.1.22, 312.72.2-312.cx.1.23, 312.72.2-312.cx.1.24, 312.72.2-312.cx.1.25, 312.72.2-312.cx.1.26, 312.72.2-312.cx.1.27, 312.72.2-312.cx.1.28, 312.72.2-312.cx.1.29, 312.72.2-312.cx.1.30, 312.72.2-312.cx.1.31, 312.72.2-312.cx.1.32, 312.72.2-312.cx.1.33, 312.72.2-312.cx.1.34, 312.72.2-312.cx.1.35, 312.72.2-312.cx.1.36, 312.72.2-312.cx.1.37, 312.72.2-312.cx.1.38, 312.72.2-312.cx.1.39, 312.72.2-312.cx.1.40, 312.72.2-312.cx.1.41, 312.72.2-312.cx.1.42, 312.72.2-312.cx.1.43, 312.72.2-312.cx.1.44, 312.72.2-312.cx.1.45, 312.72.2-312.cx.1.46, 312.72.2-312.cx.1.47, 312.72.2-312.cx.1.48, 312.72.2-312.cx.1.49, 312.72.2-312.cx.1.50, 312.72.2-312.cx.1.51, 312.72.2-312.cx.1.52, 312.72.2-312.cx.1.53, 312.72.2-312.cx.1.54, 312.72.2-312.cx.1.55, 312.72.2-312.cx.1.56, 312.72.2-312.cx.1.57, 312.72.2-312.cx.1.58, 312.72.2-312.cx.1.59, 312.72.2-312.cx.1.60, 312.72.2-312.cx.1.61, 312.72.2-312.cx.1.62, 312.72.2-312.cx.1.63, 312.72.2-312.cx.1.64
Cyclic 312-isogeny field degree: $112$
Cyclic 312-torsion field degree: $10752$
Full 312-torsion field degree: $53673984$

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
12.18.1.c.1 $12$ $2$ $2$ $1$ $0$
312.12.0.z.1 $312$ $3$ $3$ $0$ $?$

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

Covering curve Level Index Degree Genus
312.72.3.bwb.1 $312$ $2$ $2$ $3$
312.72.3.bwd.1 $312$ $2$ $2$ $3$
312.72.3.bwj.1 $312$ $2$ $2$ $3$
312.72.3.bwl.1 $312$ $2$ $2$ $3$
312.72.3.bxj.1 $312$ $2$ $2$ $3$
312.72.3.bxl.1 $312$ $2$ $2$ $3$
312.72.3.bxr.1 $312$ $2$ $2$ $3$
312.72.3.bxt.1 $312$ $2$ $2$ $3$
312.72.3.bzh.1 $312$ $2$ $2$ $3$
312.72.3.bzj.1 $312$ $2$ $2$ $3$
312.72.3.bzl.1 $312$ $2$ $2$ $3$
312.72.3.bzn.1 $312$ $2$ $2$ $3$
312.72.3.cbp.1 $312$ $2$ $2$ $3$
312.72.3.cbr.1 $312$ $2$ $2$ $3$
312.72.3.ccb.1 $312$ $2$ $2$ $3$
312.72.3.ccd.1 $312$ $2$ $2$ $3$
312.72.4.ep.1 $312$ $2$ $2$ $4$
312.72.4.fa.1 $312$ $2$ $2$ $4$
312.72.4.gb.1 $312$ $2$ $2$ $4$
312.72.4.gc.1 $312$ $2$ $2$ $4$
312.72.4.im.1 $312$ $2$ $2$ $4$
312.72.4.ip.1 $312$ $2$ $2$ $4$
312.72.4.ir.1 $312$ $2$ $2$ $4$
312.72.4.is.1 $312$ $2$ $2$ $4$
312.72.4.kb.1 $312$ $2$ $2$ $4$
312.72.4.kc.1 $312$ $2$ $2$ $4$
312.72.4.ku.1 $312$ $2$ $2$ $4$
312.72.4.kx.1 $312$ $2$ $2$ $4$
312.72.4.mf.1 $312$ $2$ $2$ $4$
312.72.4.mg.1 $312$ $2$ $2$ $4$
312.72.4.nk.1 $312$ $2$ $2$ $4$
312.72.4.nn.1 $312$ $2$ $2$ $4$
312.72.4.pl.1 $312$ $2$ $2$ $4$
312.72.4.pn.1 $312$ $2$ $2$ $4$
312.72.4.px.1 $312$ $2$ $2$ $4$
312.72.4.pz.1 $312$ $2$ $2$ $4$
312.72.4.sb.1 $312$ $2$ $2$ $4$
312.72.4.sd.1 $312$ $2$ $2$ $4$
312.72.4.sf.1 $312$ $2$ $2$ $4$
312.72.4.sh.1 $312$ $2$ $2$ $4$
312.72.4.tx.1 $312$ $2$ $2$ $4$
312.72.4.tz.1 $312$ $2$ $2$ $4$
312.72.4.uf.1 $312$ $2$ $2$ $4$
312.72.4.uh.1 $312$ $2$ $2$ $4$
312.72.4.vd.1 $312$ $2$ $2$ $4$
312.72.4.vf.1 $312$ $2$ $2$ $4$
312.72.4.vl.1 $312$ $2$ $2$ $4$
312.72.4.vn.1 $312$ $2$ $2$ $4$