Invariants
| Level: | $70$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 70.24.1.9 |
Level structure
| $\GL_2(\Z/70\Z)$-generators: | $\begin{bmatrix}26&31\\27&15\end{bmatrix}$, $\begin{bmatrix}58&33\\57&53\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 140.48.1-70.a.1.1, 140.48.1-70.a.1.2, 140.48.1-70.a.1.3, 140.48.1-70.a.1.4, 280.48.1-70.a.1.1, 280.48.1-70.a.1.2, 280.48.1-70.a.1.3, 280.48.1-70.a.1.4 |
| Cyclic 70-isogeny field degree: | $24$ |
| Cyclic 70-torsion field degree: | $576$ |
| Full 70-torsion field degree: | $241920$ |
Jacobian
| Conductor: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 7 x^{2} - 4 y^{2} - 3 y z + y w - 5 z^{2} + 4 z w - w^{2} $ |
| $=$ | $7 x^{2} + 6 y^{2} + 5 y z - 2 y w + 8 z^{2} - 6 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{4} + 13 x^{3} z - 420 x^{2} y^{2} + 18 x^{2} z^{2} - 406 x y^{2} z + 10 x z^{3} + 8281 y^{4} + \cdots + 5 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^5}\cdot\frac{3366528yz^{5}-5391140yz^{4}w-7735520yz^{3}w^{2}+4836120yz^{2}w^{3}-353760yzw^{4}-81603yw^{5}+3494068z^{6}+6094408z^{5}w-13164950z^{4}w^{2}+3760960z^{3}w^{3}+1642845z^{2}w^{4}-850218zw^{5}+94762w^{6}}{(z-w)^{5}(y+z+w)}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(2)$ | $2$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 35.12.0.a.1 | $35$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.12.1.a.1 | $10$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 35.12.0.a.1 | $35$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 70.12.0.a.2 | $70$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 70.72.1.a.1 | $70$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 70.72.1.c.2 | $70$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 70.120.5.f.1 | $70$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 70.192.13.b.2 | $70$ | $8$ | $8$ | $13$ | $0$ | $1^{4}\cdot2^{4}$ |
| 70.504.37.b.2 | $70$ | $21$ | $21$ | $37$ | $4$ | $1^{4}\cdot2^{8}\cdot4^{4}$ |
| 70.672.49.b.1 | $70$ | $28$ | $28$ | $49$ | $4$ | $1^{8}\cdot2^{12}\cdot4^{4}$ |
| 140.96.5.q.1 | $140$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 210.72.5.a.2 | $210$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 210.96.5.a.2 | $210$ | $4$ | $4$ | $5$ | $?$ | not computed |