Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $140$ | ||
| Index: | $1152$ | $\PSL_2$-index: | $576$ | ||||
| Genus: | $37 = 1 + \frac{ 576 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $8$ are rational) | Cusp widths | $2^{6}\cdot10^{6}\cdot14^{6}\cdot70^{6}$ | Cusp orbits | $1^{8}\cdot2^{4}\cdot4^{2}$ | ||
| 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: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.1152.37.633 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}9&41\\0&51\end{bmatrix}$, $\begin{bmatrix}41&40\\0&57\end{bmatrix}$, $\begin{bmatrix}59&23\\0&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 70.576.37.be.4 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: | $2^{31}\cdot5^{35}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{4}\cdot4^{2}\cdot8^{2}$ |
| Newforms: | 14.2.a.a$^{2}$, 35.2.a.a$^{2}$, 35.2.a.b$^{2}$, 35.2.b.a$^{2}$, 35.2.f.a, 70.2.a.a, 70.2.c.a, 70.2.g.a, 140.2.m.a |
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 |
|---|---|---|---|---|---|---|
| 7.16.0-7.a.1.1 | $7$ | $72$ | $72$ | $0$ | $0$ | full Jacobian |
| 10.72.0-10.a.2.4 | $10$ | $16$ | $16$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.384.13-70.g.4.7 | $70$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{2}\cdot4^{2}\cdot8$ |
| 70.576.17-70.a.1.2 | $70$ | $2$ | $2$ | $17$ | $0$ | $4\cdot8^{2}$ |
| 70.576.17-70.a.1.10 | $70$ | $2$ | $2$ | $17$ | $0$ | $4\cdot8^{2}$ |
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.b.1.3 | $70$ | $2$ | $2$ | $73$ | $0$ | $1^{8}\cdot2^{4}\cdot4^{3}\cdot8$ |
| 70.2304.73-70.t.3.1 | $70$ | $2$ | $2$ | $73$ | $2$ | $1^{8}\cdot2^{4}\cdot4^{3}\cdot8$ |
| 70.2304.73-70.bh.3.4 | $70$ | $2$ | $2$ | $73$ | $2$ | $1^{8}\cdot2^{4}\cdot4^{3}\cdot8$ |
| 70.2304.73-70.bi.4.3 | $70$ | $2$ | $2$ | $73$ | $6$ | $1^{8}\cdot2^{4}\cdot4^{3}\cdot8$ |
| 70.3456.109-70.r.4.6 | $70$ | $3$ | $3$ | $109$ | $2$ | $1^{8}\cdot2^{12}\cdot4^{2}\cdot16^{2}$ |
| 70.3456.109-70.t.2.7 | $70$ | $3$ | $3$ | $109$ | $0$ | $2^{4}\cdot4^{8}\cdot16^{2}$ |
| 70.3456.109-70.v.4.8 | $70$ | $3$ | $3$ | $109$ | $0$ | $2^{4}\cdot4^{8}\cdot16^{2}$ |
| 70.5760.205-70.ci.2.3 | $70$ | $5$ | $5$ | $205$ | $6$ | $1^{20}\cdot2^{20}\cdot4^{9}\cdot8^{7}\cdot16$ |
| 70.8064.289-70.cm.2.8 | $70$ | $7$ | $7$ | $289$ | $13$ | $1^{28}\cdot2^{30}\cdot4^{11}\cdot8^{3}\cdot16^{3}\cdot24^{2}$ |