Invariants
| Level: | $88$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
Level structure
| $\GL_2(\Z/88\Z)$-generators: | $\begin{bmatrix}3&56\\50&1\end{bmatrix}$, $\begin{bmatrix}35&64\\15&25\end{bmatrix}$, $\begin{bmatrix}47&24\\30&67\end{bmatrix}$, $\begin{bmatrix}67&56\\60&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 88.24.0.bj.1 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $12$ |
| Cyclic 88-torsion field degree: | $480$ |
| Full 88-torsion field degree: | $422400$ |
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 |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 88.24.0-44.h.1.2 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.24.0-44.h.1.3 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.24.0-8.n.1.6 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.24.0-88.z.1.5 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 88.24.0-88.z.1.15 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 88.96.0-88.bk.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bk.2.7 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bl.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bl.2.6 | $88$ | $2$ | $2$ | $0$ |
| 176.96.0-176.y.1.12 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.y.2.5 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.z.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.z.2.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-176.u.1.14 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.w.1.16 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.ci.1.12 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.ck.1.6 | $176$ | $2$ | $2$ | $1$ |
| 264.96.0-264.dv.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dv.2.10 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dw.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dw.2.10 | $264$ | $2$ | $2$ | $0$ |
| 264.144.4-264.iv.1.17 | $264$ | $3$ | $3$ | $4$ |
| 264.192.3-264.lt.1.25 | $264$ | $4$ | $4$ | $3$ |