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Properties

Label 156.12.0.a.1
Level $156$
Index $12$
Genus $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $156$ $\SL_2$-level: $4$
Index: $12$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $2^{2}\cdot4^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 4E0

Level structure

$\GL_2(\Z/156\Z)$-generators: $\begin{bmatrix}9&124\\128&151\end{bmatrix}$, $\begin{bmatrix}31&136\\116&145\end{bmatrix}$, $\begin{bmatrix}35&50\\36&49\end{bmatrix}$, $\begin{bmatrix}91&32\\132&79\end{bmatrix}$, $\begin{bmatrix}117&64\\46&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 156.24.0-156.a.1.1, 156.24.0-156.a.1.2, 156.24.0-156.a.1.3, 156.24.0-156.a.1.4, 156.24.0-156.a.1.5, 156.24.0-156.a.1.6, 156.24.0-156.a.1.7, 156.24.0-156.a.1.8, 312.24.0-156.a.1.1, 312.24.0-156.a.1.2, 312.24.0-156.a.1.3, 312.24.0-156.a.1.4, 312.24.0-156.a.1.5, 312.24.0-156.a.1.6, 312.24.0-156.a.1.7, 312.24.0-156.a.1.8
Cyclic 156-isogeny field degree: $112$
Cyclic 156-torsion field degree: $5376$
Full 156-torsion field degree: $10063872$

Models

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

Rational points

This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X(2)$ $2$ $2$ $2$ $0$ $0$
156.6.0.b.1 $156$ $2$ $2$ $0$ $?$
156.6.0.e.1 $156$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
156.24.0.a.1 $156$ $2$ $2$ $0$
156.24.0.c.1 $156$ $2$ $2$ $0$
156.24.0.d.1 $156$ $2$ $2$ $0$
156.24.0.g.1 $156$ $2$ $2$ $0$
156.36.2.c.1 $156$ $3$ $3$ $2$
156.48.1.c.1 $156$ $4$ $4$ $1$
156.168.11.c.1 $156$ $14$ $14$ $11$
312.24.0.b.1 $312$ $2$ $2$ $0$
312.24.0.f.1 $312$ $2$ $2$ $0$
312.24.0.j.1 $312$ $2$ $2$ $0$
312.24.0.s.1 $312$ $2$ $2$ $0$