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Properties

Label 312.168.11.cl.1
Level $312$
Index $168$
Genus $11$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $312$ $\SL_2$-level: $104$ Newform level: $1$
Index: $168$ $\PSL_2$-index:$168$
Genus: $11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $1^{2}\cdot2\cdot8\cdot13^{2}\cdot26\cdot104$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $4 \le \gamma \le 11$
$\overline{\Q}$-gonality: $4 \le \gamma \le 11$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 104D11

Level structure

$\GL_2(\Z/312\Z)$-generators: $\begin{bmatrix}42&187\\203&234\end{bmatrix}$, $\begin{bmatrix}80&253\\143&294\end{bmatrix}$, $\begin{bmatrix}95&54\\126&23\end{bmatrix}$, $\begin{bmatrix}121&210\\22&205\end{bmatrix}$, $\begin{bmatrix}161&292\\162&239\end{bmatrix}$, $\begin{bmatrix}217&112\\106&15\end{bmatrix}$, $\begin{bmatrix}290&181\\9&46\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 312.336.11-312.cl.1.1, 312.336.11-312.cl.1.2, 312.336.11-312.cl.1.3, 312.336.11-312.cl.1.4, 312.336.11-312.cl.1.5, 312.336.11-312.cl.1.6, 312.336.11-312.cl.1.7, 312.336.11-312.cl.1.8, 312.336.11-312.cl.1.9, 312.336.11-312.cl.1.10, 312.336.11-312.cl.1.11, 312.336.11-312.cl.1.12, 312.336.11-312.cl.1.13, 312.336.11-312.cl.1.14, 312.336.11-312.cl.1.15, 312.336.11-312.cl.1.16, 312.336.11-312.cl.1.17, 312.336.11-312.cl.1.18, 312.336.11-312.cl.1.19, 312.336.11-312.cl.1.20, 312.336.11-312.cl.1.21, 312.336.11-312.cl.1.22, 312.336.11-312.cl.1.23, 312.336.11-312.cl.1.24, 312.336.11-312.cl.1.25, 312.336.11-312.cl.1.26, 312.336.11-312.cl.1.27, 312.336.11-312.cl.1.28, 312.336.11-312.cl.1.29, 312.336.11-312.cl.1.30, 312.336.11-312.cl.1.31, 312.336.11-312.cl.1.32, 312.336.11-312.cl.1.33, 312.336.11-312.cl.1.34, 312.336.11-312.cl.1.35, 312.336.11-312.cl.1.36, 312.336.11-312.cl.1.37, 312.336.11-312.cl.1.38, 312.336.11-312.cl.1.39, 312.336.11-312.cl.1.40, 312.336.11-312.cl.1.41, 312.336.11-312.cl.1.42, 312.336.11-312.cl.1.43, 312.336.11-312.cl.1.44, 312.336.11-312.cl.1.45, 312.336.11-312.cl.1.46, 312.336.11-312.cl.1.47, 312.336.11-312.cl.1.48, 312.336.11-312.cl.1.49, 312.336.11-312.cl.1.50, 312.336.11-312.cl.1.51, 312.336.11-312.cl.1.52, 312.336.11-312.cl.1.53, 312.336.11-312.cl.1.54, 312.336.11-312.cl.1.55, 312.336.11-312.cl.1.56, 312.336.11-312.cl.1.57, 312.336.11-312.cl.1.58, 312.336.11-312.cl.1.59, 312.336.11-312.cl.1.60, 312.336.11-312.cl.1.61, 312.336.11-312.cl.1.62, 312.336.11-312.cl.1.63, 312.336.11-312.cl.1.64
Cyclic 312-isogeny field degree: $8$
Cyclic 312-torsion field degree: $768$
Full 312-torsion field degree: $11501568$

Rational points

This modular curve has 4 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
$X_0(52)$ $52$ $2$ $2$ $5$ $0$
312.12.0.z.1 $312$ $14$ $14$ $0$ $?$

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

Covering curve Level Index Degree Genus
312.336.21.yb.1 $312$ $2$ $2$ $21$
312.336.21.yb.2 $312$ $2$ $2$ $21$
312.336.21.yd.1 $312$ $2$ $2$ $21$
312.336.21.yd.2 $312$ $2$ $2$ $21$
312.336.21.yf.1 $312$ $2$ $2$ $21$
312.336.21.yf.2 $312$ $2$ $2$ $21$
312.336.21.yh.1 $312$ $2$ $2$ $21$
312.336.21.yh.2 $312$ $2$ $2$ $21$
312.336.21.zx.1 $312$ $2$ $2$ $21$
312.336.21.zx.2 $312$ $2$ $2$ $21$
312.336.21.zz.1 $312$ $2$ $2$ $21$
312.336.21.zz.2 $312$ $2$ $2$ $21$
312.336.21.bab.1 $312$ $2$ $2$ $21$
312.336.21.bab.2 $312$ $2$ $2$ $21$
312.336.21.bad.1 $312$ $2$ $2$ $21$
312.336.21.bad.2 $312$ $2$ $2$ $21$
312.336.23.db.1 $312$ $2$ $2$ $23$
312.336.23.dk.1 $312$ $2$ $2$ $23$
312.336.23.eh.1 $312$ $2$ $2$ $23$
312.336.23.ei.1 $312$ $2$ $2$ $23$
312.336.23.gl.1 $312$ $2$ $2$ $23$
312.336.23.go.1 $312$ $2$ $2$ $23$
312.336.23.gu.1 $312$ $2$ $2$ $23$
312.336.23.gv.1 $312$ $2$ $2$ $23$
312.336.23.hy.1 $312$ $2$ $2$ $23$
312.336.23.hz.1 $312$ $2$ $2$ $23$
312.336.23.if.1 $312$ $2$ $2$ $23$
312.336.23.ii.1 $312$ $2$ $2$ $23$
312.336.23.iy.1 $312$ $2$ $2$ $23$
312.336.23.iz.1 $312$ $2$ $2$ $23$
312.336.23.jv.1 $312$ $2$ $2$ $23$
312.336.23.jy.1 $312$ $2$ $2$ $23$
312.336.23.mc.1 $312$ $2$ $2$ $23$
312.336.23.mc.2 $312$ $2$ $2$ $23$
312.336.23.me.1 $312$ $2$ $2$ $23$
312.336.23.me.2 $312$ $2$ $2$ $23$
312.336.23.mg.1 $312$ $2$ $2$ $23$
312.336.23.mg.2 $312$ $2$ $2$ $23$
312.336.23.mi.1 $312$ $2$ $2$ $23$
312.336.23.mi.2 $312$ $2$ $2$ $23$
312.336.23.ny.1 $312$ $2$ $2$ $23$
312.336.23.ny.2 $312$ $2$ $2$ $23$
312.336.23.oa.1 $312$ $2$ $2$ $23$
312.336.23.oa.2 $312$ $2$ $2$ $23$
312.336.23.oc.1 $312$ $2$ $2$ $23$
312.336.23.oc.2 $312$ $2$ $2$ $23$
312.336.23.oe.1 $312$ $2$ $2$ $23$
312.336.23.oe.2 $312$ $2$ $2$ $23$