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 | $4^{6}$ | 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: | 4G0 |
Level structure
| $\GL_2(\Z/88\Z)$-generators: | $\begin{bmatrix}33&12\\68&81\end{bmatrix}$, $\begin{bmatrix}35&56\\72&43\end{bmatrix}$, $\begin{bmatrix}37&26\\56&73\end{bmatrix}$, $\begin{bmatrix}67&18\\52&41\end{bmatrix}$, $\begin{bmatrix}77&32\\36&87\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 44.24.0.c.1 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $24$ |
| Cyclic 88-torsion field degree: | $960$ |
| 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, including 29 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^8\cdot11^2}\cdot\frac{(3x-y)^{24}(6698781x^{8}+41290560x^{7}y+126044100x^{6}y^{2}+252360360x^{5}y^{3}+348054084x^{4}y^{4}+318139920x^{3}y^{5}+180860400x^{2}y^{6}+57996480xy^{7}+8221456y^{8})^{3}}{(3x-y)^{24}(3x+2y)^{4}(3x+8y)^{4}(3x^{2}+5xy+3y^{2})^{4}(45x^{2}+42xy-10y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 88.24.0-4.b.1.8 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.