Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $350$ | ||
| Index: | $1152$ | $\PSL_2$-index: | $576$ | ||||
| Genus: | $33 = 1 + \frac{ 576 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{4}\cdot5^{4}\cdot7^{4}\cdot10^{4}\cdot14^{4}\cdot35^{4}\cdot70^{4}$ | Cusp orbits | $2^{8}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.1152.33.45 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}11&14\\0&31\end{bmatrix}$, $\begin{bmatrix}19&27\\0&61\end{bmatrix}$, $\begin{bmatrix}29&3\\0&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 70.576.33.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: | $5040$ |
Jacobian
| Conductor: | $2^{15}\cdot5^{47}\cdot7^{33}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{4}\cdot4^{3}\cdot8$ |
| Newforms: | 14.2.a.a$^{2}$, 35.2.a.a$^{2}$, 35.2.a.b$^{2}$, 35.2.b.a$^{2}$, 70.2.a.a, 70.2.c.a, 175.2.f.c$^{2}$, 350.2.g.a |
Rational points
This modular curve has no $\Q_p$ points for $p=37,53$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.384.9-35.a.4.2 | $70$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}\cdot4^{2}\cdot8$ |
| 70.576.17-70.a.1.10 | $70$ | $2$ | $2$ | $17$ | $0$ | $4^{2}\cdot8$ |
| 70.576.17-70.a.1.12 | $70$ | $2$ | $2$ | $17$ | $0$ | $4^{2}\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.2304.73-70.a.4.3 | $70$ | $2$ | $2$ | $73$ | $0$ | $1^{8}\cdot2^{4}\cdot4^{2}\cdot8^{2}$ |
| 70.2304.73-70.t.4.1 | $70$ | $2$ | $2$ | $73$ | $2$ | $1^{8}\cdot2^{4}\cdot4^{2}\cdot8^{2}$ |
| 70.2304.73-70.bg.4.4 | $70$ | $2$ | $2$ | $73$ | $2$ | $1^{8}\cdot2^{4}\cdot4^{2}\cdot8^{2}$ |
| 70.2304.73-70.bi.4.3 | $70$ | $2$ | $2$ | $73$ | $6$ | $1^{8}\cdot2^{4}\cdot4^{2}\cdot8^{2}$ |
| 70.3456.97-70.b.1.7 | $70$ | $3$ | $3$ | $97$ | $0$ | $2^{4}\cdot4^{10}\cdot16$ |
| 70.3456.97-70.d.3.7 | $70$ | $3$ | $3$ | $97$ | $0$ | $2^{4}\cdot4^{10}\cdot16$ |
| 70.3456.97-70.f.1.6 | $70$ | $3$ | $3$ | $97$ | $2$ | $1^{8}\cdot2^{12}\cdot4^{4}\cdot16$ |
| 70.5760.193-70.a.2.7 | $70$ | $5$ | $5$ | $193$ | $6$ | $1^{20}\cdot2^{20}\cdot4^{13}\cdot8^{4}\cdot16$ |
| 70.8064.273-70.a.1.6 | $70$ | $7$ | $7$ | $273$ | $13$ | $1^{28}\cdot2^{30}\cdot4^{14}\cdot8^{2}\cdot16^{2}\cdot24^{2}$ |