Invariants
| Level: | $88$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
Level structure
| $\GL_2(\Z/88\Z)$-generators: | $\begin{bmatrix}14&15\\33&68\end{bmatrix}$, $\begin{bmatrix}17&12\\10&59\end{bmatrix}$, $\begin{bmatrix}26&1\\59&24\end{bmatrix}$, $\begin{bmatrix}39&2\\52&61\end{bmatrix}$, $\begin{bmatrix}63&56\\32&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $12$ |
| Cyclic 88-torsion field degree: | $480$ |
| 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, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 44.12.0-4.c.1.2 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 88.12.0-4.c.1.5 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 88.48.0-8.i.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.k.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.q.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.r.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.ba.1.8 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.ba.2.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.bb.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.bb.2.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bf.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bh.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bj.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bl.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bu.1.13 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bu.2.9 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bv.1.13 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bv.2.9 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-88.bl.1.45 | $88$ | $12$ | $12$ | $9$ |
| 176.48.0-16.e.1.3 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.e.2.2 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.e.1.25 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.e.2.17 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.f.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.f.2.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.f.1.25 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.f.2.17 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.g.1.5 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.g.1.31 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.h.1.5 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.h.1.29 | $176$ | $2$ | $2$ | $0$ |
| 176.48.1-16.a.1.12 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.a.1.4 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-16.b.1.12 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.b.1.2 | $176$ | $2$ | $2$ | $1$ |
| 264.48.0-24.bh.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bj.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bl.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bn.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.by.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.by.2.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bz.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bz.2.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cx.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cz.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.db.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dd.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ec.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ec.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ed.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ed.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-24.cj.1.7 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-24.ir.1.2 | $264$ | $4$ | $4$ | $1$ |