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Properties

Label 156.24.0.s.1
Level $156$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $156$ $\SL_2$-level: $12$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $1^{2}\cdot3^{2}\cdot4\cdot12$ Cusp orbits $1^{2}\cdot2^{2}$
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: 12E0

Level structure

$\GL_2(\Z/156\Z)$-generators: $\begin{bmatrix}10&83\\145&150\end{bmatrix}$, $\begin{bmatrix}44&151\\105&124\end{bmatrix}$, $\begin{bmatrix}64&11\\93&50\end{bmatrix}$, $\begin{bmatrix}109&44\\100&3\end{bmatrix}$, $\begin{bmatrix}120&47\\127&74\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 156.48.0-156.s.1.1, 156.48.0-156.s.1.2, 156.48.0-156.s.1.3, 156.48.0-156.s.1.4, 156.48.0-156.s.1.5, 156.48.0-156.s.1.6, 156.48.0-156.s.1.7, 156.48.0-156.s.1.8, 156.48.0-156.s.1.9, 156.48.0-156.s.1.10, 156.48.0-156.s.1.11, 156.48.0-156.s.1.12, 156.48.0-156.s.1.13, 156.48.0-156.s.1.14, 156.48.0-156.s.1.15, 156.48.0-156.s.1.16, 312.48.0-156.s.1.1, 312.48.0-156.s.1.2, 312.48.0-156.s.1.3, 312.48.0-156.s.1.4, 312.48.0-156.s.1.5, 312.48.0-156.s.1.6, 312.48.0-156.s.1.7, 312.48.0-156.s.1.8, 312.48.0-156.s.1.9, 312.48.0-156.s.1.10, 312.48.0-156.s.1.11, 312.48.0-156.s.1.12, 312.48.0-156.s.1.13, 312.48.0-156.s.1.14, 312.48.0-156.s.1.15, 312.48.0-156.s.1.16
Cyclic 156-isogeny field degree: $28$
Cyclic 156-torsion field degree: $1344$
Full 156-torsion field degree: $5031936$

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
$X_0(6)$ $6$ $2$ $2$ $0$ $0$
156.6.0.b.1 $156$ $4$ $4$ $0$ $?$

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

Covering curve Level Index Degree Genus
156.48.1.c.1 $156$ $2$ $2$ $1$
156.48.1.f.1 $156$ $2$ $2$ $1$
156.48.1.z.1 $156$ $2$ $2$ $1$
156.48.1.ba.1 $156$ $2$ $2$ $1$
156.48.1.bh.1 $156$ $2$ $2$ $1$
156.48.1.bi.1 $156$ $2$ $2$ $1$
156.48.1.bt.1 $156$ $2$ $2$ $1$
156.48.1.bu.1 $156$ $2$ $2$ $1$
156.72.1.r.1 $156$ $3$ $3$ $1$
156.336.23.df.1 $156$ $14$ $14$ $23$
312.48.1.gh.1 $312$ $2$ $2$ $1$
312.48.1.ka.1 $312$ $2$ $2$ $1$
312.48.1.bla.1 $312$ $2$ $2$ $1$
312.48.1.bld.1 $312$ $2$ $2$ $1$
312.48.1.byo.1 $312$ $2$ $2$ $1$
312.48.1.byr.1 $312$ $2$ $2$ $1$
312.48.1.bzy.1 $312$ $2$ $2$ $1$
312.48.1.cab.1 $312$ $2$ $2$ $1$