Invariants
| Level: | $260$ | $\SL_2$-level: | $65$ | Newform level: | $65$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot5^{2}\cdot13^{2}\cdot65^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 65A11 |
Level structure
| $\GL_2(\Z/260\Z)$-generators: | $\begin{bmatrix}134&89\\45&103\end{bmatrix}$, $\begin{bmatrix}184&139\\105&98\end{bmatrix}$, $\begin{bmatrix}189&220\\205&239\end{bmatrix}$, $\begin{bmatrix}197&256\\181&187\end{bmatrix}$, $\begin{bmatrix}212&257\\133&36\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 65.168.11.a.2 for the level structure with $-I$) |
| Cyclic 260-isogeny field degree: | $6$ |
| Cyclic 260-torsion field degree: | $576$ |
| Full 260-torsion field degree: | $3594240$ |
Models
Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ x^{2} - x w + x t - x u - x v - x r + x s - x a - x b + v s + v a + s a + a^{2} $ |
| $=$ | $x^{2} - x z + x v + x s + x a + y^{2} + 2 y t - z t + 2 v s + 2 v a + s a + s b + a^{2} + a b$ | |
| $=$ | $x^{2} - x y - x w + x t + x u + x a + y z + 2 y t + u s + u a + r s + r a - s^{2} + a^{2}$ | |
| $=$ | $x^{2} - x y + 2 x z + x w + 2 x t - 2 x v + x r + x a - x b - y^{2} + y z - y w - y s + z t + s^{2} + \cdots - a b$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|
| 20.24.0-5.a.2.1 | $20$ | $14$ | $14$ | $0$ | $0$ |
| $X_0(13)$ | $13$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.24.0-5.a.2.1 | $20$ | $14$ | $14$ | $0$ | $0$ |