Invariants
| Level: | $130$ | $\SL_2$-level: | $10$ | ||||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $0 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $5^{12}$ | Cusp orbits | $1^{2}\cdot2\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 5H0 |
Level structure
| $\GL_2(\Z/130\Z)$-generators: | $\begin{bmatrix}87&73\\59&58\end{bmatrix}$, $\begin{bmatrix}109&113\\9&0\end{bmatrix}$, $\begin{bmatrix}121&85\\110&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 5.60.0.a.1 for the level structure with $-I$) |
| Cyclic 130-isogeny field degree: | $42$ |
| Cyclic 130-torsion field degree: | $1008$ |
| Full 130-torsion field degree: | $628992$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 60 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{60}(x^{4}+3x^{3}y-x^{2}y^{2}-3xy^{3}+y^{4})^{3}(x^{8}-4x^{7}y+7x^{6}y^{2}-2x^{5}y^{3}+15x^{4}y^{4}+2x^{3}y^{5}+7x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+x^{7}y+7x^{6}y^{2}-7x^{5}y^{3}+7x^{3}y^{5}+7x^{2}y^{6}-xy^{7}+y^{8})^{3}}{y^{5}x^{65}(x^{2}-xy-y^{2})^{5}(x^{4}-2x^{3}y+4x^{2}y^{2}-3xy^{3}+y^{4})^{5}(x^{4}+3x^{3}y+4x^{2}y^{2}+2xy^{3}+y^{4})^{5}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 130.24.0-5.a.1.1 | $130$ | $5$ | $5$ | $0$ | $?$ |
| 130.24.0-5.a.2.1 | $130$ | $5$ | $5$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 130.240.5-10.c.1.2 | $130$ | $2$ | $2$ | $5$ |
| 130.240.5-10.e.1.1 | $130$ | $2$ | $2$ | $5$ |
| 130.240.5-130.j.1.4 | $130$ | $2$ | $2$ | $5$ |
| 130.240.5-130.k.1.1 | $130$ | $2$ | $2$ | $5$ |
| 130.360.4-10.a.1.3 | $130$ | $3$ | $3$ | $4$ |
| 260.240.5-20.p.1.1 | $260$ | $2$ | $2$ | $5$ |
| 260.240.5-20.u.1.1 | $260$ | $2$ | $2$ | $5$ |
| 260.240.5-260.z.1.6 | $260$ | $2$ | $2$ | $5$ |
| 260.240.5-260.bc.1.3 | $260$ | $2$ | $2$ | $5$ |
| 260.480.15-20.bj.1.7 | $260$ | $4$ | $4$ | $15$ |