Invariants
Level: | $88$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-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: | $\begin{bmatrix}9&6\\42&15\end{bmatrix}$, $\begin{bmatrix}9&16\\26&39\end{bmatrix}$, $\begin{bmatrix}19&34\\2&15\end{bmatrix}$, $\begin{bmatrix}35&10\\86&65\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 88.12.0.a.1 for the level structure with $-I$) |
Cyclic 88-isogeny field degree: | $48$ |
Cyclic 88-torsion field degree: | $1920$ |
Full 88-torsion field degree: | $844800$ |
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 |
---|---|---|---|---|---|
8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
44.12.0-2.a.1.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
88.48.0-88.a.1.2 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.a.1.3 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.b.1.4 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.b.1.6 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.e.1.7 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.e.1.11 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.g.1.1 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.g.1.6 | $88$ | $2$ | $2$ | $0$ |
88.288.9-88.c.1.1 | $88$ | $12$ | $12$ | $9$ |
264.48.0-264.h.1.2 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.h.1.16 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.j.1.2 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.j.1.16 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.n.1.5 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.n.1.13 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.p.1.5 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.p.1.13 | $264$ | $2$ | $2$ | $0$ |
264.72.2-264.a.1.13 | $264$ | $3$ | $3$ | $2$ |
264.96.1-264.dg.1.3 | $264$ | $4$ | $4$ | $1$ |