Invariants
| Level: | $78$ | $\SL_2$-level: | $6$ | ||||
| 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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
| $\GL_2(\Z/78\Z)$-generators: | $\begin{bmatrix}34&21\\61&26\end{bmatrix}$, $\begin{bmatrix}39&28\\46&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 78.24.0.b.1 for the level structure with $-I$) |
| Cyclic 78-isogeny field degree: | $14$ |
| Cyclic 78-torsion field degree: | $336$ |
| Full 78-torsion field degree: | $157248$ |
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 |
|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 78.16.0-78.a.1.2 | $78$ | $3$ | $3$ | $0$ | $?$ |
| 78.24.0-6.a.1.4 | $78$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 78.144.1-78.d.1.1 | $78$ | $3$ | $3$ | $1$ |
| 156.96.1-156.i.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.k.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.u.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.w.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bg.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bi.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bo.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bq.1.2 | $156$ | $2$ | $2$ | $1$ |
| 234.144.1-234.i.1.2 | $234$ | $3$ | $3$ | $1$ |
| 234.144.4-234.x.1.2 | $234$ | $3$ | $3$ | $4$ |
| 234.144.4-234.bd.1.3 | $234$ | $3$ | $3$ | $4$ |
| 312.96.1-312.yx.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zd.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bap.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bav.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byk.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byq.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzi.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzo.1.1 | $312$ | $2$ | $2$ | $1$ |