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Properties

Label 232.48.0.o.2
Level $232$
Index $48$
Genus $0$
Cusps $10$
$\Q$-cusps $2$

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Invariants

Level: $232$ $\SL_2$-level: $8$
Index: $48$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $2$ are rational) Cusp widths $4^{8}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2^{2}\cdot4$
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: 8N0

Level structure

$\GL_2(\Z/232\Z)$-generators: $\begin{bmatrix}23&68\\200&169\end{bmatrix}$, $\begin{bmatrix}39&162\\4&133\end{bmatrix}$, $\begin{bmatrix}105&196\\84&33\end{bmatrix}$, $\begin{bmatrix}123&66\\228&133\end{bmatrix}$, $\begin{bmatrix}135&20\\16&103\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 232.96.0-232.o.2.1, 232.96.0-232.o.2.2, 232.96.0-232.o.2.3, 232.96.0-232.o.2.4, 232.96.0-232.o.2.5, 232.96.0-232.o.2.6, 232.96.0-232.o.2.7, 232.96.0-232.o.2.8, 232.96.0-232.o.2.9, 232.96.0-232.o.2.10, 232.96.0-232.o.2.11, 232.96.0-232.o.2.12, 232.96.0-232.o.2.13, 232.96.0-232.o.2.14, 232.96.0-232.o.2.15, 232.96.0-232.o.2.16
Cyclic 232-isogeny field degree: $60$
Cyclic 232-torsion field degree: $6720$
Full 232-torsion field degree: $21826560$

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
8.24.0.e.1 $8$ $2$ $2$ $0$ $0$
232.24.0.e.1 $232$ $2$ $2$ $0$ $?$
232.24.0.h.2 $232$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
232.96.1.j.2 $232$ $2$ $2$ $1$
232.96.1.z.1 $232$ $2$ $2$ $1$
232.96.1.bk.1 $232$ $2$ $2$ $1$
232.96.1.bo.2 $232$ $2$ $2$ $1$
232.96.1.bv.1 $232$ $2$ $2$ $1$
232.96.1.bz.2 $232$ $2$ $2$ $1$
232.96.1.cf.2 $232$ $2$ $2$ $1$
232.96.1.ch.1 $232$ $2$ $2$ $1$