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Properties

Label 156.8.0.a.1
Level $156$
Index $8$
Genus $0$
Cusps $2$
$\Q$-cusps $2$

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Invariants

Level: $156$ $\SL_2$-level: $6$
Index: $8$ $\PSL_2$-index:$8$
Genus: $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$
Cusps: $2$ (all of which are rational) Cusp widths $2\cdot6$ Cusp orbits $1^{2}$
Elliptic points: $0$ of order $2$ and $2$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 6C0

Level structure

$\GL_2(\Z/156\Z)$-generators: $\begin{bmatrix}15&23\\55&92\end{bmatrix}$, $\begin{bmatrix}59&48\\101&7\end{bmatrix}$, $\begin{bmatrix}104&69\\21&35\end{bmatrix}$, $\begin{bmatrix}112&27\\135&70\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 156.16.0-156.a.1.1, 156.16.0-156.a.1.2, 156.16.0-156.a.1.3, 156.16.0-156.a.1.4, 156.16.0-156.a.1.5, 156.16.0-156.a.1.6, 156.16.0-156.a.1.7, 156.16.0-156.a.1.8, 312.16.0-156.a.1.1, 312.16.0-156.a.1.2, 312.16.0-156.a.1.3, 312.16.0-156.a.1.4, 312.16.0-156.a.1.5, 312.16.0-156.a.1.6, 312.16.0-156.a.1.7, 312.16.0-156.a.1.8
Cyclic 156-isogeny field degree: $84$
Cyclic 156-torsion field degree: $4032$
Full 156-torsion field degree: $15095808$

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

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
$X_0(3)$ $3$ $2$ $2$ $0$ $0$
52.2.0.a.1 $52$ $4$ $4$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(3)$ $3$ $2$ $2$ $0$ $0$
52.2.0.a.1 $52$ $4$ $4$ $0$ $0$

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

Covering curve Level Index Degree Genus
156.24.0.o.1 $156$ $3$ $3$ $0$
156.24.1.d.1 $156$ $3$ $3$ $1$
156.32.1.b.1 $156$ $4$ $4$ $1$
156.112.7.e.1 $156$ $14$ $14$ $7$