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Properties

Label 88.12.0.a.1
Level $88$
Index $12$
Genus $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $88$ $\SL_2$-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

$\GL_2(\Z/88\Z)$-generators: $\begin{bmatrix}21&62\\38&1\end{bmatrix}$, $\begin{bmatrix}21&74\\36&15\end{bmatrix}$, $\begin{bmatrix}23&40\\24&81\end{bmatrix}$, $\begin{bmatrix}51&6\\66&29\end{bmatrix}$, $\begin{bmatrix}53&30\\64&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 88.24.0-88.a.1.1, 88.24.0-88.a.1.2, 88.24.0-88.a.1.3, 88.24.0-88.a.1.4, 88.24.0-88.a.1.5, 88.24.0-88.a.1.6, 88.24.0-88.a.1.7, 88.24.0-88.a.1.8, 264.24.0-88.a.1.1, 264.24.0-88.a.1.2, 264.24.0-88.a.1.3, 264.24.0-88.a.1.4, 264.24.0-88.a.1.5, 264.24.0-88.a.1.6, 264.24.0-88.a.1.7, 264.24.0-88.a.1.8
Cyclic 88-isogeny field degree: $48$
Cyclic 88-torsion field degree: $1920$
Full 88-torsion field degree: $1689600$

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
$X(2)$ $2$ $2$ $2$ $0$ $0$
88.6.0.c.1 $88$ $2$ $2$ $0$ $?$
88.6.0.f.1 $88$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
88.24.0.a.1 $88$ $2$ $2$ $0$
88.24.0.b.1 $88$ $2$ $2$ $0$
88.24.0.e.1 $88$ $2$ $2$ $0$
88.24.0.g.1 $88$ $2$ $2$ $0$
88.144.9.c.1 $88$ $12$ $12$ $9$
264.24.0.h.1 $264$ $2$ $2$ $0$
264.24.0.j.1 $264$ $2$ $2$ $0$
264.24.0.n.1 $264$ $2$ $2$ $0$
264.24.0.p.1 $264$ $2$ $2$ $0$
264.36.2.a.1 $264$ $3$ $3$ $2$
264.48.1.dg.1 $264$ $4$ $4$ $1$