Invariants
| Level: | $70$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.72.1.12 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}9&36\\58&21\end{bmatrix}$, $\begin{bmatrix}29&66\\44&59\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 140.144.1-70.a.2.1, 140.144.1-70.a.2.2, 140.144.1-70.a.2.3, 140.144.1-70.a.2.4, 140.144.1-70.a.2.5, 140.144.1-70.a.2.6, 140.144.1-70.a.2.7, 140.144.1-70.a.2.8, 280.144.1-70.a.2.1, 280.144.1-70.a.2.2, 280.144.1-70.a.2.3, 280.144.1-70.a.2.4, 280.144.1-70.a.2.5, 280.144.1-70.a.2.6, 280.144.1-70.a.2.7, 280.144.1-70.a.2.8 |
| Cyclic 70-isogeny field degree: | $8$ |
| Cyclic 70-torsion field degree: | $192$ |
| Full 70-torsion field degree: | $80640$ |
Jacobian
| Conductor: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 35.12.0.a.2 | $35$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.36.1.a.1 | $10$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 70.24.1.a.2 | $70$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 70.36.0.a.1 | $70$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 70.36.0.b.1 | $70$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 70.360.13.b.1 | $70$ | $5$ | $5$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 70.576.37.b.1 | $70$ | $8$ | $8$ | $37$ | $0$ | $1^{12}\cdot2^{8}\cdot4^{2}$ |
| 70.1512.109.b.1 | $70$ | $21$ | $21$ | $109$ | $14$ | $1^{12}\cdot2^{28}\cdot4^{10}$ |
| 70.2016.145.b.2 | $70$ | $28$ | $28$ | $145$ | $14$ | $1^{24}\cdot2^{36}\cdot4^{12}$ |
| 140.144.5.i.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.j.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.m.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.n.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.q.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.r.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.u.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.v.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.216.13.a.1 | $210$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 210.288.13.a.1 | $210$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 280.144.5.y.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.bb.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.bk.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.bn.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.bw.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.bz.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.ci.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.cl.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |