Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $70$ | ||
| Index: | $1728$ | $\PSL_2$-index: | $864$ | ||||
| Genus: | $49 = 1 + \frac{ 864 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$ | ||||||
| Cusps: | $48$ (of which $12$ are rational) | Cusp widths | $1^{6}\cdot2^{6}\cdot5^{6}\cdot7^{6}\cdot10^{6}\cdot14^{6}\cdot35^{6}\cdot70^{6}$ | Cusp orbits | $1^{12}\cdot2^{6}\cdot3^{4}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 24$ | ||||||
| Rational cusps: | $12$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.1728.49.81 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}19&63\\0&27\end{bmatrix}$, $\begin{bmatrix}29&65\\0&41\end{bmatrix}$, $\begin{bmatrix}41&12\\0&27\end{bmatrix}$, $\begin{bmatrix}61&26\\0&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 70.864.49.a.3 for the level structure with $-I$) |
| Cyclic 70-isogeny field degree: | $1$ |
| Cyclic 70-torsion field degree: | $12$ |
| Full 70-torsion field degree: | $3360$ |
Jacobian
| Conductor: | $2^{23}\cdot5^{47}\cdot7^{49}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{8}\cdot4^{7}$ |
| Newforms: | 14.2.a.a$^{2}$, 35.2.a.a$^{2}$, 35.2.a.b$^{2}$, 35.2.b.a$^{2}$, 35.2.e.a$^{2}$, 35.2.j.a$^{2}$, 70.2.a.a, 70.2.c.a, 70.2.e.a, 70.2.e.b, 70.2.e.c, 70.2.e.d, 70.2.i.a, 70.2.i.b |
Rational points
This modular curve has 12 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 |
|---|---|---|---|---|---|---|
| 7.24.0.a.2 | $7$ | $72$ | $36$ | $0$ | $0$ | full Jacobian |
| 10.72.0-10.a.2.4 | $10$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.576.13-35.a.4.6 | $70$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{6}\cdot4^{5}$ |
| 70.576.17-70.a.1.10 | $70$ | $3$ | $3$ | $17$ | $0$ | $2^{4}\cdot4^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.3456.97-70.a.4.8 | $70$ | $2$ | $2$ | $97$ | $0$ | $4^{6}\cdot8\cdot16$ |
| 70.3456.97-70.b.3.6 | $70$ | $2$ | $2$ | $97$ | $0$ | $4^{6}\cdot8\cdot16$ |
| 70.3456.97-70.c.4.8 | $70$ | $2$ | $2$ | $97$ | $0$ | $4^{6}\cdot8\cdot16$ |
| 70.3456.97-70.d.3.7 | $70$ | $2$ | $2$ | $97$ | $0$ | $4^{6}\cdot8\cdot16$ |
| 70.3456.109-70.a.4.2 | $70$ | $2$ | $2$ | $109$ | $0$ | $1^{8}\cdot2^{12}\cdot4^{7}$ |
| 70.3456.109-70.i.4.1 | $70$ | $2$ | $2$ | $109$ | $2$ | $1^{8}\cdot2^{12}\cdot4^{7}$ |
| 70.3456.109-70.m.4.4 | $70$ | $2$ | $2$ | $109$ | $4$ | $1^{8}\cdot2^{12}\cdot4^{7}$ |
| 70.3456.109-70.n.4.3 | $70$ | $2$ | $2$ | $109$ | $6$ | $1^{8}\cdot2^{12}\cdot4^{7}$ |
| 70.3456.109-70.s.3.7 | $70$ | $2$ | $2$ | $109$ | $0$ | $4^{3}\cdot8^{2}\cdot16^{2}$ |
| 70.3456.109-70.t.4.8 | $70$ | $2$ | $2$ | $109$ | $0$ | $4^{3}\cdot8^{2}\cdot16^{2}$ |
| 70.3456.109-70.u.3.6 | $70$ | $2$ | $2$ | $109$ | $0$ | $4^{3}\cdot8^{2}\cdot16^{2}$ |
| 70.3456.109-70.v.4.8 | $70$ | $2$ | $2$ | $109$ | $0$ | $4^{3}\cdot8^{2}\cdot16^{2}$ |
| 70.8640.289-70.a.1.10 | $70$ | $5$ | $5$ | $289$ | $8$ | $1^{20}\cdot2^{40}\cdot4^{19}\cdot6^{4}\cdot8^{2}\cdot12^{2}$ |
| 70.12096.409-70.c.2.13 | $70$ | $7$ | $7$ | $409$ | $15$ | $1^{28}\cdot2^{58}\cdot4^{42}\cdot8^{6}$ |