Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $70$ | ||
| Index: | $288$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $8$ are rational) | Cusp widths | $1^{2}\cdot2^{2}\cdot5^{2}\cdot7^{2}\cdot10^{2}\cdot14^{2}\cdot35^{2}\cdot70^{2}$ | Cusp orbits | $1^{8}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 70J17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.288.17.2 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}3&6\\0&61\end{bmatrix}$, $\begin{bmatrix}11&12\\0&41\end{bmatrix}$, $\begin{bmatrix}33&3\\0&11\end{bmatrix}$, $\begin{bmatrix}67&30\\0&61\end{bmatrix}$, $\begin{bmatrix}67&68\\0&69\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 70.576.17-70.a.2.1, 70.576.17-70.a.2.2, 70.576.17-70.a.2.3, 70.576.17-70.a.2.4, 70.576.17-70.a.2.5, 70.576.17-70.a.2.6, 70.576.17-70.a.2.7, 70.576.17-70.a.2.8, 70.576.17-70.a.2.9, 70.576.17-70.a.2.10, 70.576.17-70.a.2.11, 70.576.17-70.a.2.12, 70.576.17-70.a.2.13, 70.576.17-70.a.2.14, 70.576.17-70.a.2.15, 70.576.17-70.a.2.16 |
| Cyclic 70-isogeny field degree: | $1$ |
| Cyclic 70-torsion field degree: | $24$ |
| Full 70-torsion field degree: | $20160$ |
Jacobian
| Conductor: | $2^{7}\cdot5^{15}\cdot7^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{4}\cdot4$ |
| 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 |
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 |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $96$ | $96$ | $0$ | $0$ | full Jacobian |
| 5.12.0.a.2 | $5$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
| $X_0(7)$ | $7$ | $36$ | $36$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.36.0.a.1 | $10$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 35.96.5.a.1 | $35$ | $3$ | $3$ | $5$ | $0$ | $1^{4}\cdot2^{2}\cdot4$ |
| $X_0(70)$ | $70$ | $2$ | $2$ | $9$ | $0$ | $2^{2}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.576.33.a.3 | $70$ | $2$ | $2$ | $33$ | $0$ | $4^{2}\cdot8$ |
| 70.576.33.a.4 | $70$ | $2$ | $2$ | $33$ | $0$ | $4^{2}\cdot8$ |
| 70.576.33.b.1 | $70$ | $2$ | $2$ | $33$ | $0$ | $4^{2}\cdot8$ |
| 70.576.33.b.2 | $70$ | $2$ | $2$ | $33$ | $0$ | $4^{2}\cdot8$ |
| 70.576.37.a.1 | $70$ | $2$ | $2$ | $37$ | $0$ | $1^{8}\cdot2^{4}\cdot4$ |
| 70.576.37.q.1 | $70$ | $2$ | $2$ | $37$ | $2$ | $1^{8}\cdot2^{4}\cdot4$ |
| 70.576.37.u.1 | $70$ | $2$ | $2$ | $37$ | $2$ | $1^{8}\cdot2^{4}\cdot4$ |
| 70.576.37.v.2 | $70$ | $2$ | $2$ | $37$ | $6$ | $1^{8}\cdot2^{4}\cdot4$ |
| 70.576.37.be.1 | $70$ | $2$ | $2$ | $37$ | $0$ | $4\cdot8^{2}$ |
| 70.576.37.be.2 | $70$ | $2$ | $2$ | $37$ | $0$ | $4\cdot8^{2}$ |
| 70.576.37.bf.3 | $70$ | $2$ | $2$ | $37$ | $0$ | $4\cdot8^{2}$ |
| 70.576.37.bf.4 | $70$ | $2$ | $2$ | $37$ | $0$ | $4\cdot8^{2}$ |
| 70.864.49.a.2 | $70$ | $3$ | $3$ | $49$ | $0$ | $2^{4}\cdot4^{6}$ |
| 70.864.49.a.4 | $70$ | $3$ | $3$ | $49$ | $0$ | $2^{4}\cdot4^{6}$ |
| 70.864.49.b.2 | $70$ | $3$ | $3$ | $49$ | $2$ | $1^{8}\cdot2^{12}$ |
| 70.1440.97.a.1 | $70$ | $5$ | $5$ | $97$ | $6$ | $1^{20}\cdot2^{20}\cdot4^{5}$ |
| 70.2016.137.a.2 | $70$ | $7$ | $7$ | $137$ | $13$ | $1^{28}\cdot2^{30}\cdot4^{8}$ |