Invariants
| Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
| Index: | $720$ | $\PSL_2$-index: | $360$ | ||||
| Genus: | $25 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $30^{12}$ | Cusp orbits | $1^{2}\cdot2\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $6$ | ||||||
| $\overline{\Q}$-gonality: | $6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.720.25.105 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&10\\25&9\end{bmatrix}$, $\begin{bmatrix}26&5\\15&23\end{bmatrix}$ |
| $\GL_2(\Z/30\Z)$-subgroup: | $C_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 30.360.25.bx.1 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $12$ |
| Cyclic 30-torsion field degree: | $24$ |
| Full 30-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{50}\cdot5^{40}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{7}$ |
| Newforms: | 45.2.b.a$^{2}$, 90.2.a.a$^{2}$, 90.2.a.b$^{2}$, 180.2.a.a$^{2}$, 225.2.a.c, 225.2.a.d, 225.2.a.f, 225.2.b.c, 450.2.a.a, 450.2.a.e, 450.2.c.a, 450.2.c.d, 900.2.a.b, 900.2.d.c |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 6.6.0.b.1 | $6$ | $120$ | $60$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.144.5-30.k.1.2 | $30$ | $5$ | $5$ | $5$ | $0$ | $1^{8}\cdot2^{6}$ |
| 30.144.5-30.k.2.1 | $30$ | $5$ | $5$ | $5$ | $0$ | $1^{8}\cdot2^{6}$ |
| 30.240.5-30.e.1.4 | $30$ | $3$ | $3$ | $5$ | $1$ | $1^{8}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.2160.73-30.bg.1.6 | $30$ | $3$ | $3$ | $73$ | $7$ | $1^{20}\cdot2^{14}$ |
| 30.2880.97-30.ba.1.8 | $30$ | $4$ | $4$ | $97$ | $7$ | $1^{36}\cdot2^{18}$ |
| 60.2880.109-60.dhv.1.8 | $60$ | $4$ | $4$ | $109$ | $19$ | $1^{42}\cdot2^{21}$ |
| 60.2880.109-60.dhx.1.8 | $60$ | $4$ | $4$ | $109$ | $25$ | $1^{40}\cdot2^{22}$ |