Invariants
Level: | $70$ | $\SL_2$-level: | $70$ | Newform level: | $4900$ | ||
Index: | $420$ | $\PSL_2$-index: | $420$ | ||||
Genus: | $28 = 1 + \frac{ 420 }{12} - \frac{ 20 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $70^{6}$ | Cusp orbits | $6$ | ||
Elliptic points: | $20$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $21$ | ||||||
$\Q$-gonality: | $9 \le \gamma \le 20$ | ||||||
$\overline{\Q}$-gonality: | $9 \le \gamma \le 20$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.420.28.4 |
Level structure
$\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}24&13\\9&22\end{bmatrix}$, $\begin{bmatrix}24&35\\69&11\end{bmatrix}$, $\begin{bmatrix}48&45\\43&47\end{bmatrix}$, $\begin{bmatrix}67&21\\63&18\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 70-isogeny field degree: | $144$ |
Cyclic 70-torsion field degree: | $3456$ |
Full 70-torsion field degree: | $13824$ |
Jacobian
Conductor: | $2^{30}\cdot5^{56}\cdot7^{56}$ |
Simple: | no |
Squarefree: | yes |
Decomposition: | $2^{3}\cdot3^{2}\cdot4^{4}$ |
Newforms: | 1225.2.a.ba, 1225.2.a.k, 1225.2.a.p, 1225.2.a.r, 1225.2.a.y, 4900.2.a.bd, 4900.2.a.bf, 4900.2.a.bg, 4900.2.a.bh |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
35.210.13.a.1 | $35$ | $2$ | $2$ | $13$ | $13$ | $3\cdot4^{3}$ |
70.20.0.b.1 | $70$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
70.840.57.b.1 | $70$ | $2$ | $2$ | $57$ | $32$ | $1^{17}\cdot2^{3}\cdot3^{2}$ |
70.840.57.d.1 | $70$ | $2$ | $2$ | $57$ | $40$ | $1^{17}\cdot2^{3}\cdot3^{2}$ |
70.840.57.f.1 | $70$ | $2$ | $2$ | $57$ | $31$ | $1^{17}\cdot2^{3}\cdot3^{2}$ |
70.840.57.h.1 | $70$ | $2$ | $2$ | $57$ | $42$ | $1^{17}\cdot2^{3}\cdot3^{2}$ |
70.840.63.o.1 | $70$ | $2$ | $2$ | $63$ | $31$ | $1^{13}\cdot2^{4}\cdot3^{2}\cdot4^{2}$ |
70.840.63.q.1 | $70$ | $2$ | $2$ | $63$ | $39$ | $1^{13}\cdot2^{4}\cdot3^{2}\cdot4^{2}$ |
70.840.63.s.1 | $70$ | $2$ | $2$ | $63$ | $27$ | $1^{13}\cdot2^{4}\cdot3^{2}\cdot4^{2}$ |
70.840.63.u.1 | $70$ | $2$ | $2$ | $63$ | $31$ | $1^{13}\cdot2^{4}\cdot3^{2}\cdot4^{2}$ |
70.840.65.d.1 | $70$ | $2$ | $2$ | $65$ | $26$ | $1^{7}\cdot2^{6}\cdot3^{2}\cdot4^{3}$ |
70.840.65.bd.1 | $70$ | $2$ | $2$ | $65$ | $36$ | $1^{7}\cdot2^{6}\cdot3^{2}\cdot4^{3}$ |
70.840.65.bh.1 | $70$ | $2$ | $2$ | $65$ | $29$ | $1^{7}\cdot2^{6}\cdot3^{2}\cdot4^{3}$ |
70.840.65.bk.1 | $70$ | $2$ | $2$ | $65$ | $36$ | $1^{7}\cdot2^{6}\cdot3^{2}\cdot4^{3}$ |
70.1260.92.b.1 | $70$ | $3$ | $3$ | $92$ | $50$ | $1^{10}\cdot2^{14}\cdot3^{2}\cdot4^{5}$ |
70.1260.92.d.1 | $70$ | $3$ | $3$ | $92$ | $50$ | $1^{16}\cdot2^{11}\cdot3^{2}\cdot4^{5}$ |