Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $4900$ | ||
| Index: | $1260$ | $\PSL_2$-index: | $1260$ | ||||
| Genus: | $92 = 1 + \frac{ 1260 }{12} - \frac{ 20 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (none of which are rational) | Cusp widths | $70^{18}$ | Cusp orbits | $3^{2}\cdot12$ | ||
| Elliptic points: | $20$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $17$ | ||||||
| $\Q$-gonality: | $13 \le \gamma \le 42$ | ||||||
| $\overline{\Q}$-gonality: | $13 \le \gamma \le 42$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.1260.92.2 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}13&15\\10&43\end{bmatrix}$, $\begin{bmatrix}19&15\\35&44\end{bmatrix}$, $\begin{bmatrix}38&23\\35&46\end{bmatrix}$, $\begin{bmatrix}61&47\\5&48\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 70-isogeny field degree: | $24$ |
| Cyclic 70-torsion field degree: | $576$ |
| Full 70-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{100}\cdot5^{156}\cdot7^{180}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2^{15}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| Newforms: | 100.2.c.a, 245.2.a.e$^{2}$, 245.2.a.f$^{2}$, 245.2.a.h$^{2}$, 980.2.e.e$^{2}$, 980.2.e.f$^{2}$, 1225.2.a.ba, 1225.2.a.bb, 1225.2.a.bc, 1225.2.a.d, 1225.2.a.f, 1225.2.a.k, 1225.2.a.l, 1225.2.a.p, 1225.2.a.r, 1225.2.a.v, 1225.2.a.x, 1225.2.a.y, 4900.2.e.c, 4900.2.e.i, 4900.2.e.m, 4900.2.e.p, 4900.2.e.q, 4900.2.e.r, 4900.2.e.t, 4900.2.e.u |
Rational points
This modular curve has real points and $\Q_p$ points for good $p < 8192$, but no known 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $60$ | $60$ | $0$ | $0$ | full Jacobian |
| 10.60.2.c.1 | $10$ | $21$ | $21$ | $2$ | $0$ | $1^{2}\cdot2^{14}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.60.2.c.1 | $10$ | $21$ | $21$ | $2$ | $0$ | $1^{2}\cdot2^{14}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| 35.630.42.a.1 | $35$ | $2$ | $2$ | $42$ | $17$ | $2^{4}\cdot4^{7}\cdot6\cdot8$ |
| 70.252.14.b.1 | $70$ | $5$ | $5$ | $14$ | $2$ | $1^{2}\cdot2^{12}\cdot3^{2}\cdot4^{8}\cdot6\cdot8$ |
| 70.252.14.b.2 | $70$ | $5$ | $5$ | $14$ | $2$ | $1^{2}\cdot2^{12}\cdot3^{2}\cdot4^{8}\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.2520.185.w.1 | $70$ | $2$ | $2$ | $185$ | $37$ | $1^{37}\cdot2^{10}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.2520.185.x.1 | $70$ | $2$ | $2$ | $185$ | $35$ | $1^{37}\cdot2^{10}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.2520.185.bg.1 | $70$ | $2$ | $2$ | $185$ | $44$ | $1^{37}\cdot2^{10}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.2520.185.bh.1 | $70$ | $2$ | $2$ | $185$ | $39$ | $1^{37}\cdot2^{10}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.2520.191.ba.1 | $70$ | $2$ | $2$ | $191$ | $38$ | $1^{17}\cdot2^{27}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.2520.191.bb.1 | $70$ | $2$ | $2$ | $191$ | $36$ | $1^{17}\cdot2^{27}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.2520.191.bk.1 | $70$ | $2$ | $2$ | $191$ | $30$ | $1^{17}\cdot2^{27}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.2520.191.bl.1 | $70$ | $2$ | $2$ | $191$ | $32$ | $1^{17}\cdot2^{27}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.2520.193.p.1 | $70$ | $2$ | $2$ | $193$ | $36$ | $1^{13}\cdot2^{12}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.2520.193.cr.1 | $70$ | $2$ | $2$ | $193$ | $34$ | $1^{13}\cdot2^{12}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.2520.193.ia.1 | $70$ | $2$ | $2$ | $193$ | $39$ | $1^{13}\cdot2^{12}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.2520.193.ib.1 | $70$ | $2$ | $2$ | $193$ | $40$ | $1^{13}\cdot2^{12}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.3780.284.c.1 | $70$ | $3$ | $3$ | $284$ | $50$ | $1^{24}\cdot2^{36}\cdot3^{2}\cdot4^{17}\cdot6\cdot8^{2}$ |