Invariants
| Level: | $76$ | $\SL_2$-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
| $\GL_2(\Z/76\Z)$-generators: | $\begin{bmatrix}9&46\\58&59\end{bmatrix}$, $\begin{bmatrix}19&18\\18&31\end{bmatrix}$, $\begin{bmatrix}19&54\\44&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 76.12.0.b.1 for the level structure with $-I$) |
| Cyclic 76-isogeny field degree: | $40$ |
| Cyclic 76-torsion field degree: | $720$ |
| Full 76-torsion field degree: | $492480$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.12.0-2.a.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 76.12.0-2.a.1.1 | $76$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 76.48.0-76.b.1.1 | $76$ | $2$ | $2$ | $0$ |
| 76.48.0-76.b.1.2 | $76$ | $2$ | $2$ | $0$ |
| 76.48.0-76.c.1.2 | $76$ | $2$ | $2$ | $0$ |
| 76.48.0-76.c.1.3 | $76$ | $2$ | $2$ | $0$ |
| 76.480.17-76.d.1.4 | $76$ | $20$ | $20$ | $17$ |
| 152.48.0-152.d.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.d.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.g.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.g.1.4 | $152$ | $2$ | $2$ | $0$ |
| 228.48.0-228.e.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.e.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.g.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.g.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.72.2-228.b.1.1 | $228$ | $3$ | $3$ | $2$ |
| 228.96.1-228.b.1.2 | $228$ | $4$ | $4$ | $1$ |