Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $700$ | ||
| Index: | $480$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $35 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $10^{6}\cdot70^{6}$ | Cusp orbits | $1^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 16$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.480.35.1 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}3&35\\40&13\end{bmatrix}$, $\begin{bmatrix}3&63\\50&67\end{bmatrix}$, $\begin{bmatrix}33&42\\30&19\end{bmatrix}$, $\begin{bmatrix}58&7\\15&69\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 70.960.35-70.bp.1.1, 70.960.35-70.bp.1.2, 70.960.35-70.bp.1.3, 70.960.35-70.bp.1.4, 70.960.35-70.bp.1.5, 70.960.35-70.bp.1.6, 70.960.35-70.bp.1.7, 70.960.35-70.bp.1.8 |
| Cyclic 70-isogeny field degree: | $3$ |
| Cyclic 70-torsion field degree: | $72$ |
| Full 70-torsion field degree: | $12096$ |
Jacobian
| Conductor: | $2^{40}\cdot5^{56}\cdot7^{31}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{15}$ |
| Newforms: | 35.2.a.a$^{2}$, 35.2.a.b$^{2}$, 100.2.c.a$^{2}$, 140.2.e.a$^{2}$, 140.2.e.b$^{2}$, 175.2.a.a, 175.2.a.b, 175.2.a.c, 175.2.a.d, 175.2.a.e, 175.2.a.f, 700.2.e.a, 700.2.e.b, 700.2.e.c, 700.2.e.d |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $60$ | $60$ | $0$ | $0$ | full Jacobian |
| 10.60.2.c.1 | $10$ | $8$ | $8$ | $2$ | $0$ | $1^{5}\cdot2^{14}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.60.2.c.1 | $10$ | $8$ | $8$ | $2$ | $0$ | $1^{5}\cdot2^{14}$ |
| 35.240.15.c.1 | $35$ | $2$ | $2$ | $15$ | $2$ | $2^{10}$ |
| 70.96.7.f.1 | $70$ | $5$ | $5$ | $7$ | $0$ | $1^{4}\cdot2^{12}$ |
| 70.96.7.f.2 | $70$ | $5$ | $5$ | $7$ | $0$ | $1^{4}\cdot2^{12}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.960.69.e.1 | $70$ | $2$ | $2$ | $69$ | $7$ | $1^{20}\cdot2^{3}\cdot4^{2}$ |
| 70.960.69.z.1 | $70$ | $2$ | $2$ | $69$ | $8$ | $1^{20}\cdot2^{3}\cdot4^{2}$ |
| 70.960.69.ck.1 | $70$ | $2$ | $2$ | $69$ | $10$ | $1^{20}\cdot2^{3}\cdot4^{2}$ |
| 70.960.69.cl.1 | $70$ | $2$ | $2$ | $69$ | $13$ | $1^{20}\cdot2^{3}\cdot4^{2}$ |
| 70.1440.103.cw.1 | $70$ | $3$ | $3$ | $103$ | $2$ | $2^{2}\cdot4^{10}\cdot6^{2}\cdot12$ |
| 70.1440.103.cw.2 | $70$ | $3$ | $3$ | $103$ | $2$ | $2^{2}\cdot4^{10}\cdot6^{2}\cdot12$ |
| 70.1440.103.df.1 | $70$ | $3$ | $3$ | $103$ | $12$ | $1^{4}\cdot2^{12}\cdot3^{4}\cdot4^{4}\cdot6^{2}$ |
| 70.1440.103.dm.1 | $70$ | $3$ | $3$ | $103$ | $6$ | $1^{20}\cdot2^{16}\cdot4^{4}$ |
| 70.3360.257.ky.1 | $70$ | $7$ | $7$ | $257$ | $42$ | $1^{24}\cdot2^{55}\cdot3^{4}\cdot4^{14}\cdot6^{2}\cdot8$ |