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: | $\begin{bmatrix}23&0\\30&37\end{bmatrix}$, $\begin{bmatrix}27&42\\40&47\end{bmatrix}$, $\begin{bmatrix}37&80\\12&43\end{bmatrix}$, $\begin{bmatrix}81&28\\74&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 44.12.0.b.1 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $48$ |
| Cyclic 88-torsion field degree: | $1920$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^4\cdot11^2}\cdot\frac{x^{12}(1936x^{4}+1100x^{2}y^{2}+625y^{4})^{3}}{y^{4}x^{16}(44x^{2}+25y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 88.12.0-2.a.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 |
|---|---|---|---|---|
| 88.48.0-44.b.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-44.b.1.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-44.c.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-44.c.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.d.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.d.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.g.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.g.1.8 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-44.d.1.1 | $88$ | $12$ | $12$ | $9$ |
| 264.48.0-132.e.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.e.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.g.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.g.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.l.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.l.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.r.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.r.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-132.b.1.1 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-132.b.1.4 | $264$ | $4$ | $4$ | $1$ |