Invariants
Level: | $78$ | $\SL_2$-level: | $78$ | Newform level: | $1$ | ||
Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
Genus: | $11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6\cdot13\cdot26\cdot39\cdot78$ | Cusp orbits | $1^{8}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 11$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 11$ | ||||||
Rational cusps: | $8$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 78C11 |
Level structure
$\GL_2(\Z/78\Z)$-generators: | $\begin{bmatrix}7&21\\0&59\end{bmatrix}$, $\begin{bmatrix}29&58\\0&67\end{bmatrix}$, $\begin{bmatrix}61&53\\0&71\end{bmatrix}$, $\begin{bmatrix}67&73\\0&35\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 78.168.11.a.1 for the level structure with $-I$) |
Cyclic 78-isogeny field degree: | $1$ |
Cyclic 78-torsion field degree: | $24$ |
Full 78-torsion field degree: | $22464$ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
6.24.0-6.a.1.3 | $6$ | $14$ | $14$ | $0$ | $0$ |
$X_0(13)$ | $13$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
6.24.0-6.a.1.3 | $6$ | $14$ | $14$ | $0$ | $0$ |
78.112.3-39.a.1.6 | $78$ | $3$ | $3$ | $3$ | $?$ |