Invariants
| Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $4900$ | ||
| Index: | $1680$ | $\PSL_2$-index: | $1680$ | ||||
| Genus: | $125 = 1 + \frac{ 1680 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $2$ are rational) | Cusp widths | $70^{24}$ | Cusp orbits | $1^{2}\cdot3^{2}\cdot4\cdot12$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $19$ | ||||||
| $\Q$-gonality: | $17 \le \gamma \le 56$ | ||||||
| $\overline{\Q}$-gonality: | $17 \le \gamma \le 56$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.1680.125.3 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}11&11\\65&48\end{bmatrix}$, $\begin{bmatrix}14&57\\45&63\end{bmatrix}$, $\begin{bmatrix}18&65\\15&38\end{bmatrix}$, $\begin{bmatrix}69&65\\15&14\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 70-isogeny field degree: | $6$ |
| Cyclic 70-torsion field degree: | $144$ |
| Full 70-torsion field degree: | $3456$ |
Jacobian
| Conductor: | $2^{136}\cdot5^{208}\cdot7^{211}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{29}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| 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, 245.2.a.e$^{2}$, 245.2.a.f$^{2}$, 245.2.a.h$^{2}$, 700.2.e.a, 700.2.e.b, 700.2.e.c, 700.2.e.d, 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 2 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $60$ | $60$ | $0$ | $0$ | full Jacobian |
| 10.60.2.c.1 | $10$ | $28$ | $28$ | $2$ | $0$ | $1^{7}\cdot2^{28}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.60.2.c.1 | $10$ | $28$ | $28$ | $2$ | $0$ | $1^{7}\cdot2^{28}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| 35.840.57.d.1 | $35$ | $2$ | $2$ | $57$ | $19$ | $2^{13}\cdot4^{7}\cdot6\cdot8$ |
| 70.336.21.f.1 | $70$ | $5$ | $5$ | $21$ | $2$ | $1^{6}\cdot2^{23}\cdot3^{2}\cdot4^{8}\cdot6\cdot8$ |
| 70.336.21.f.2 | $70$ | $5$ | $5$ | $21$ | $2$ | $1^{6}\cdot2^{23}\cdot3^{2}\cdot4^{8}\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 70.3360.249.bk.1 | $70$ | $2$ | $2$ | $249$ | $48$ | $1^{54}\cdot2^{13}\cdot3^{2}\cdot4^{8}\cdot6$ |
| 70.3360.249.bl.1 | $70$ | $2$ | $2$ | $249$ | $44$ | $1^{54}\cdot2^{13}\cdot3^{2}\cdot4^{8}\cdot6$ |
| 70.3360.249.bs.1 | $70$ | $2$ | $2$ | $249$ | $52$ | $1^{54}\cdot2^{13}\cdot3^{2}\cdot4^{8}\cdot6$ |
| 70.3360.249.bt.1 | $70$ | $2$ | $2$ | $249$ | $46$ | $1^{54}\cdot2^{13}\cdot3^{2}\cdot4^{8}\cdot6$ |
| 70.3360.257.m.1 | $70$ | $2$ | $2$ | $257$ | $43$ | $1^{30}\cdot2^{15}\cdot3^{2}\cdot4^{13}\cdot6\cdot8$ |
| 70.3360.257.cs.1 | $70$ | $2$ | $2$ | $257$ | $42$ | $1^{30}\cdot2^{15}\cdot3^{2}\cdot4^{13}\cdot6\cdot8$ |
| 70.3360.257.jg.1 | $70$ | $2$ | $2$ | $257$ | $48$ | $1^{30}\cdot2^{15}\cdot3^{2}\cdot4^{13}\cdot6\cdot8$ |
| 70.3360.257.jh.1 | $70$ | $2$ | $2$ | $257$ | $51$ | $1^{30}\cdot2^{15}\cdot3^{2}\cdot4^{13}\cdot6\cdot8$ |
| 70.3360.257.ky.1 | $70$ | $2$ | $2$ | $257$ | $42$ | $1^{22}\cdot2^{41}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.3360.257.kz.1 | $70$ | $2$ | $2$ | $257$ | $41$ | $1^{22}\cdot2^{41}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.3360.257.lg.1 | $70$ | $2$ | $2$ | $257$ | $40$ | $1^{22}\cdot2^{41}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.3360.257.lh.1 | $70$ | $2$ | $2$ | $257$ | $41$ | $1^{22}\cdot2^{41}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.5040.381.bn.1 | $70$ | $3$ | $3$ | $381$ | $64$ | $1^{26}\cdot2^{61}\cdot3^{6}\cdot4^{16}\cdot6^{3}\cdot8$ |
| 70.5040.381.by.1 | $70$ | $3$ | $3$ | $381$ | $56$ | $1^{42}\cdot2^{51}\cdot3^{2}\cdot4^{21}\cdot6\cdot8^{2}$ |