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: | $\begin{bmatrix}19&12\\29&85\end{bmatrix}$, $\begin{bmatrix}59&8\\76&25\end{bmatrix}$, $\begin{bmatrix}67&56\\53&63\end{bmatrix}$, $\begin{bmatrix}73&24\\65&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 44.12.0.g.1 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $24$ |
| Cyclic 88-torsion field degree: | $960$ |
| 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 323 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{11}\cdot\frac{(x+2y)^{12}(2x^{4}-30x^{3}y-65x^{2}y^{2}-30xy^{3}+2y^{4})^{3}}{(x-y)^{2}(x+y)^{2}(x+2y)^{12}(3x^{2}+5xy+3y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 88.12.0-4.c.1.1 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 88.48.0-88.be.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.be.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bf.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bf.1.10 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bm.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bm.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bn.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bn.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-44.k.1.1 | $88$ | $12$ | $12$ | $9$ |
| 264.48.0-264.cg.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cg.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ch.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ch.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.co.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.co.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cp.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cp.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-132.s.1.1 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-132.k.1.2 | $264$ | $4$ | $4$ | $1$ |