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: | $\begin{bmatrix}1&28\\1&55\end{bmatrix}$, $\begin{bmatrix}15&52\\47&67\end{bmatrix}$, $\begin{bmatrix}35&64\\71&85\end{bmatrix}$, $\begin{bmatrix}49&52\\53&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 44.12.0.h.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 347 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{11}\cdot\frac{(11x-2y)^{12}(121x^{4}+154x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(11x-2y)^{12}(11x^{2}-y^{2})^{4}}$ |
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.1.2 | $88$ | $2$ | $2$ | $0$ | $?$ |
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.bi.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bi.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bj.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bj.1.10 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bq.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bq.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.br.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.br.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-44.l.1.4 | $88$ | $12$ | $12$ | $9$ |
| 264.48.0-264.ck.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ck.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cl.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cl.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cs.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cs.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ct.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ct.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-132.t.1.1 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-132.l.1.2 | $264$ | $4$ | $4$ | $1$ |