Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot5^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 5D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.24.0.4 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}6&25\\5&22\end{bmatrix}$, $\begin{bmatrix}11&5\\17&12\end{bmatrix}$, $\begin{bmatrix}19&0\\2&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 5.12.0.a.2 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $12$ |
| Cyclic 30-torsion field degree: | $96$ |
| Full 30-torsion field degree: | $5760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1545 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^2\cdot7}\cdot\frac{(105x+11y)^{12}(388962000x^{4}+157437000x^{3}y+23549400x^{2}y^{2}+1543920xy^{3}+37469y^{4})^{3}}{(15x+2y)(35x+3y)(105x+11y)^{12}(2205x^{2}+525xy+29y^{2})^{5}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 30.120.0-5.a.1.2 | $30$ | $5$ | $5$ | $0$ |
| 30.48.1-10.a.2.1 | $30$ | $2$ | $2$ | $1$ |
| 30.48.1-10.b.2.2 | $30$ | $2$ | $2$ | $1$ |
| 30.72.0-10.a.1.3 | $30$ | $3$ | $3$ | $0$ |
| 30.72.2-15.a.2.5 | $30$ | $3$ | $3$ | $2$ |
| 30.96.1-15.a.2.5 | $30$ | $4$ | $4$ | $1$ |
| 60.48.1-20.b.2.4 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-20.e.2.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.3-20.i.2.7 | $60$ | $4$ | $4$ | $3$ |
| 150.120.0-25.a.2.1 | $150$ | $5$ | $5$ | $0$ |
| 30.48.1-30.d.2.3 | $30$ | $2$ | $2$ | $1$ |
| 30.48.1-30.i.2.4 | $30$ | $2$ | $2$ | $1$ |
| 210.192.5-35.a.1.8 | $210$ | $8$ | $8$ | $5$ |
| 210.504.16-35.a.1.3 | $210$ | $21$ | $21$ | $16$ |
| 120.48.1-40.bx.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.cd.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.cj.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.cp.2.5 | $120$ | $2$ | $2$ | $1$ |
| 330.288.9-55.a.1.5 | $330$ | $12$ | $12$ | $9$ |
| 60.48.1-60.k.2.4 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-60.bd.2.4 | $60$ | $2$ | $2$ | $1$ |
| 210.48.1-70.c.2.3 | $210$ | $2$ | $2$ | $1$ |
| 210.48.1-70.d.2.4 | $210$ | $2$ | $2$ | $1$ |
| 330.48.1-110.c.2.2 | $330$ | $2$ | $2$ | $1$ |
| 330.48.1-110.d.2.1 | $330$ | $2$ | $2$ | $1$ |
| 120.48.1-120.en.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.et.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.jl.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.jr.2.8 | $120$ | $2$ | $2$ | $1$ |
| 210.48.1-210.s.2.2 | $210$ | $2$ | $2$ | $1$ |
| 210.48.1-210.v.2.4 | $210$ | $2$ | $2$ | $1$ |
| 330.48.1-330.s.2.3 | $330$ | $2$ | $2$ | $1$ |
| 330.48.1-330.v.2.7 | $330$ | $2$ | $2$ | $1$ |