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: | $4$ | ||||||
| $\Q$-gonality: | $6$ | ||||||
| $\overline{\Q}$-gonality: | $6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.720.25.7 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&0\\25&29\end{bmatrix}$, $\begin{bmatrix}11&5\\15&22\end{bmatrix}$, $\begin{bmatrix}16&5\\5&11\end{bmatrix}$ |
| $\GL_2(\Z/30\Z)$-subgroup: | $Q_8:(C_4\times S_3)$ |
| Contains $-I$: | no $\quad$ (see 30.360.25.eu.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: | $2^{30}\cdot3^{50}\cdot5^{43}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{7}$ |
| Newforms: | 36.2.a.a, 45.2.b.a$^{2}$, 180.2.a.a, 225.2.a.c, 225.2.a.d, 225.2.a.f, 225.2.b.c, 900.2.a.b$^{2}$, 900.2.a.d, 900.2.a.f, 900.2.a.g$^{3}$, 900.2.d.b, 900.2.d.c, 900.2.d.d |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.360.10-15.b.1.5 | $15$ | $2$ | $2$ | $10$ | $1$ | $1^{9}\cdot2^{3}$ |
| 30.360.10-15.b.1.3 | $30$ | $2$ | $2$ | $10$ | $1$ | $1^{9}\cdot2^{3}$ |
| 30.144.5-30.ba.1.2 | $30$ | $5$ | $5$ | $5$ | $1$ | $1^{8}\cdot2^{6}$ |
| 30.144.5-30.ba.2.1 | $30$ | $5$ | $5$ | $5$ | $1$ | $1^{8}\cdot2^{6}$ |
| 30.240.5-30.m.1.4 | $30$ | $3$ | $3$ | $5$ | $2$ | $1^{8}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.1440.49-30.fb.1.3 | $30$ | $2$ | $2$ | $49$ | $6$ | $1^{12}\cdot2^{6}$ |
| 30.1440.49-30.ff.1.6 | $30$ | $2$ | $2$ | $49$ | $7$ | $1^{12}\cdot2^{6}$ |
| 30.1440.49-30.fp.1.2 | $30$ | $2$ | $2$ | $49$ | $6$ | $1^{12}\cdot2^{6}$ |
| 30.1440.49-30.fu.1.4 | $30$ | $2$ | $2$ | $49$ | $6$ | $1^{12}\cdot2^{6}$ |
| 30.2160.73-30.dg.1.7 | $30$ | $3$ | $3$ | $73$ | $9$ | $1^{20}\cdot2^{14}$ |
| 60.1440.49-60.cba.1.6 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{12}\cdot2^{6}$ |
| 60.1440.49-60.ccc.1.6 | $60$ | $2$ | $2$ | $49$ | $7$ | $1^{12}\cdot2^{6}$ |
| 60.1440.49-60.cdv.1.6 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{12}\cdot2^{6}$ |
| 60.1440.49-60.cfa.1.6 | $60$ | $2$ | $2$ | $49$ | $6$ | $1^{12}\cdot2^{6}$ |
| 60.2880.109-60.ern.1.10 | $60$ | $4$ | $4$ | $109$ | $26$ | $1^{40}\cdot2^{22}$ |