Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $350$ | ||
| Index: | $2880$ | $\PSL_2$-index: | $1440$ | ||||
| Genus: | $97 = 1 + \frac{ 1440 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$ | ||||||
| Cusps: | $48$ (of which $8$ are rational) | Cusp widths | $5^{12}\cdot10^{12}\cdot35^{12}\cdot70^{12}$ | Cusp orbits | $1^{8}\cdot2^{4}\cdot4^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $15 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $15 \le \gamma \le 24$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.2880.97.3 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}13&23\\0&9\end{bmatrix}$, $\begin{bmatrix}39&25\\0&59\end{bmatrix}$, $\begin{bmatrix}51&39\\0&39\end{bmatrix}$, $\begin{bmatrix}53&63\\0&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 70.1440.97.a.1 for the level structure with $-I$) |
| Cyclic 70-isogeny field degree: | $1$ |
| Cyclic 70-torsion field degree: | $12$ |
| Full 70-torsion field degree: | $2016$ |
Jacobian
| Conductor: | $2^{39}\cdot5^{158}\cdot7^{89}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{25}\cdot2^{24}\cdot4^{6}$ |
| Newforms: | 14.2.a.a$^{3}$, 35.2.a.a$^{4}$, 35.2.a.b$^{4}$, 35.2.b.a$^{4}$, 50.2.a.a$^{2}$, 50.2.a.b$^{2}$, 50.2.b.a$^{2}$, 70.2.a.a$^{2}$, 70.2.c.a$^{2}$, 175.2.a.a$^{2}$, 175.2.a.b$^{2}$, 175.2.a.c$^{2}$, 175.2.a.d$^{2}$, 175.2.a.e$^{2}$, 175.2.a.f$^{2}$, 175.2.b.a$^{2}$, 175.2.b.b$^{2}$, 175.2.b.c$^{2}$, 350.2.a.a, 350.2.a.b, 350.2.a.c, 350.2.a.d, 350.2.a.e, 350.2.a.f, 350.2.a.g, 350.2.a.h, 350.2.c.a, 350.2.c.b, 350.2.c.c, 350.2.c.d |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $360$ | $180$ | $0$ | $0$ | full Jacobian |
| 10.360.4-10.a.1.3 | $10$ | $8$ | $8$ | $4$ | $0$ | $1^{23}\cdot2^{23}\cdot4^{6}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.360.4-10.a.1.3 | $10$ | $8$ | $8$ | $4$ | $0$ | $1^{23}\cdot2^{23}\cdot4^{6}$ |
| 70.576.17-70.a.1.10 | $70$ | $5$ | $5$ | $17$ | $0$ | $1^{20}\cdot2^{20}\cdot4^{5}$ |
| 70.576.17-70.a.2.13 | $70$ | $5$ | $5$ | $17$ | $0$ | $1^{20}\cdot2^{20}\cdot4^{5}$ |
| 70.960.29-35.a.1.3 | $70$ | $3$ | $3$ | $29$ | $2$ | $1^{20}\cdot2^{16}\cdot4^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.5760.193-70.a.1.6 | $70$ | $2$ | $2$ | $193$ | $6$ | $4^{10}\cdot8^{5}\cdot16$ |
| 70.5760.193-70.a.2.7 | $70$ | $2$ | $2$ | $193$ | $6$ | $4^{10}\cdot8^{5}\cdot16$ |
| 70.5760.193-70.b.1.5 | $70$ | $2$ | $2$ | $193$ | $6$ | $4^{10}\cdot8^{5}\cdot16$ |
| 70.5760.193-70.b.2.5 | $70$ | $2$ | $2$ | $193$ | $6$ | $4^{10}\cdot8^{5}\cdot16$ |
| 70.5760.205-70.a.1.6 | $70$ | $2$ | $2$ | $205$ | $15$ | $1^{40}\cdot2^{26}\cdot4^{4}$ |
| 70.5760.205-70.cd.1.3 | $70$ | $2$ | $2$ | $205$ | $31$ | $1^{40}\cdot2^{26}\cdot4^{4}$ |
| 70.5760.205-70.ci.1.3 | $70$ | $2$ | $2$ | $205$ | $6$ | $4^{5}\cdot8^{9}\cdot16$ |
| 70.5760.205-70.ci.2.3 | $70$ | $2$ | $2$ | $205$ | $6$ | $4^{5}\cdot8^{9}\cdot16$ |
| 70.5760.205-70.cs.1.5 | $70$ | $2$ | $2$ | $205$ | $6$ | $4^{5}\cdot8^{9}\cdot16$ |
| 70.5760.205-70.cs.2.5 | $70$ | $2$ | $2$ | $205$ | $6$ | $4^{5}\cdot8^{9}\cdot16$ |
| 70.5760.205-70.ct.1.4 | $70$ | $2$ | $2$ | $205$ | $35$ | $1^{40}\cdot2^{26}\cdot4^{4}$ |
| 70.5760.205-70.cu.1.2 | $70$ | $2$ | $2$ | $205$ | $17$ | $1^{40}\cdot2^{26}\cdot4^{4}$ |
| 70.8640.289-70.a.1.10 | $70$ | $3$ | $3$ | $289$ | $8$ | $2^{24}\cdot4^{20}\cdot6^{4}\cdot8^{2}\cdot12^{2}$ |
| 70.8640.289-70.a.2.9 | $70$ | $3$ | $3$ | $289$ | $8$ | $2^{24}\cdot4^{20}\cdot6^{4}\cdot8^{2}\cdot12^{2}$ |
| 70.8640.289-70.b.1.11 | $70$ | $3$ | $3$ | $289$ | $42$ | $1^{48}\cdot2^{40}\cdot3^{8}\cdot4^{4}\cdot6^{4}$ |
| 70.20160.745-70.a.1.8 | $70$ | $7$ | $7$ | $745$ | $119$ | $1^{120}\cdot2^{136}\cdot3^{8}\cdot4^{46}\cdot6^{4}\cdot8^{3}$ |