Invariants
| Level: | $88$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-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
| $\GL_2(\Z/88\Z)$-generators: | $\begin{bmatrix}15&4\\54&57\end{bmatrix}$, $\begin{bmatrix}17&20\\48&87\end{bmatrix}$, $\begin{bmatrix}63&4\\78&75\end{bmatrix}$, $\begin{bmatrix}65&44\\26&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 88.48.0.i.2 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $24$ |
| Cyclic 88-torsion field degree: | $960$ |
| Full 88-torsion field degree: | $211200$ |
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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 88.48.0-44.c.1.5 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.48.0-44.c.1.11 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.48.0-8.e.1.12 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.48.0-88.h.1.8 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.48.0-88.h.1.18 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 88.192.1-88.i.1.3 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.y.1.5 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.bc.1.6 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.bg.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.bu.1.2 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.by.1.8 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.cb.1.6 | $88$ | $2$ | $2$ | $1$ |
| 88.192.1-88.cd.1.4 | $88$ | $2$ | $2$ | $1$ |
| 264.192.1-264.fx.1.11 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.gd.2.5 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.hc.2.6 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.hi.1.15 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.me.2.12 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.mk.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.nk.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.nq.2.10 | $264$ | $2$ | $2$ | $1$ |
| 264.288.8-264.bz.2.59 | $264$ | $3$ | $3$ | $8$ |
| 264.384.7-264.cf.2.50 | $264$ | $4$ | $4$ | $7$ |