Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $980$ | ||
| Index: | $3024$ | $\PSL_2$-index: | $1512$ | ||||
| Genus: | $109 = 1 + \frac{ 1512 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (none of which are rational) | Cusp widths | $14^{18}\cdot70^{18}$ | Cusp orbits | $3^{6}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $14$ | ||||||
| $\Q$-gonality: | $15 \le \gamma \le 42$ | ||||||
| $\overline{\Q}$-gonality: | $15 \le \gamma \le 42$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.3024.109.1 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}11&40\\0&31\end{bmatrix}$, $\begin{bmatrix}25&56\\68&3\end{bmatrix}$, $\begin{bmatrix}33&38\\40&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 70.1512.109.a.1 for the level structure with $-I$) |
| Cyclic 70-isogeny field degree: | $8$ |
| Cyclic 70-torsion field degree: | $48$ |
| Full 70-torsion field degree: | $1920$ |
Jacobian
| Conductor: | $2^{82}\cdot5^{95}\cdot7^{216}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{28}\cdot4^{10}$ |
| Newforms: | 20.2.a.a, 98.2.a.b$^{4}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 245.2.a.e$^{3}$, 245.2.a.f$^{3}$, 245.2.a.h$^{3}$, 245.2.b.c$^{3}$, 245.2.b.e$^{3}$, 245.2.b.f$^{3}$, 490.2.a.a$^{2}$, 490.2.a.c$^{2}$, 490.2.a.g$^{2}$, 490.2.a.k$^{2}$, 490.2.a.l$^{2}$, 490.2.a.m$^{2}$, 490.2.c.a$^{2}$, 490.2.c.c$^{2}$, 490.2.c.g$^{2}$, 980.2.a.a, 980.2.a.g, 980.2.a.j, 980.2.a.k, 980.2.e.e, 980.2.e.f |
Rational points
This modular curve has no $\Q_p$ points for $p=3,11,23,109,149,263,389$, and therefore no 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 |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $504$ | $252$ | $0$ | $0$ | full Jacobian |
| $X_1(5)$ | $5$ | $126$ | $126$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $144$ | $72$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $21$ | $21$ | $1$ | $0$ | $1^{12}\cdot2^{28}\cdot4^{10}$ |
| 70.1008.37-70.a.1.2 | $70$ | $3$ | $3$ | $37$ | $4$ | $1^{8}\cdot2^{20}\cdot4^{6}$ |
| 70.1512.52-70.a.2.2 | $70$ | $2$ | $2$ | $52$ | $7$ | $1^{9}\cdot2^{14}\cdot4^{5}$ |
| 70.1512.52-70.a.2.15 | $70$ | $2$ | $2$ | $52$ | $7$ | $1^{9}\cdot2^{14}\cdot4^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.6048.217-70.a.1.4 | $70$ | $2$ | $2$ | $217$ | $28$ | $1^{42}\cdot2^{21}\cdot4^{6}$ |
| 70.6048.217-70.b.2.4 | $70$ | $2$ | $2$ | $217$ | $31$ | $1^{42}\cdot2^{21}\cdot4^{6}$ |
| 70.6048.217-70.c.2.1 | $70$ | $2$ | $2$ | $217$ | $22$ | $1^{42}\cdot2^{21}\cdot4^{6}$ |
| 70.6048.217-70.d.1.3 | $70$ | $2$ | $2$ | $217$ | $30$ | $1^{42}\cdot2^{21}\cdot4^{6}$ |
| 70.15120.577-70.a.1.4 | $70$ | $5$ | $5$ | $577$ | $102$ | $1^{50}\cdot2^{71}\cdot3^{8}\cdot4^{45}\cdot6^{4}\cdot8^{6}$ |