Invariants
| Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
| Index: | $1440$ | $\PSL_2$-index: | $720$ | ||||
| Genus: | $49 = 1 + \frac{ 720 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $4$ are rational) | Cusp widths | $30^{24}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $8 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $8 \le \gamma \le 12$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.1440.49.26 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&15\\15&26\end{bmatrix}$, $\begin{bmatrix}13&0\\15&11\end{bmatrix}$, $\begin{bmatrix}18&25\\25&6\end{bmatrix}$ |
| $\GL_2(\Z/30\Z)$-subgroup: | $C_2^3.D_6$ |
| Contains $-I$: | no $\quad$ (see 30.720.49.fh.1 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $6$ |
| Cyclic 30-torsion field degree: | $12$ |
| Full 30-torsion field degree: | $96$ |
Jacobian
| Conductor: | $2^{60}\cdot3^{59}\cdot5^{84}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{23}\cdot2^{13}$ |
| Newforms: | 15.2.a.a$^{2}$, 20.2.a.a$^{2}$, 36.2.a.a, 45.2.b.a$^{2}$, 60.2.d.a$^{2}$, 75.2.a.a, 75.2.a.b, 75.2.a.c, 75.2.b.a, 75.2.b.b, 100.2.a.a$^{4}$, 100.2.c.a$^{2}$, 225.2.a.c, 225.2.a.d, 225.2.a.f, 225.2.b.c, 300.2.a.a, 300.2.a.b, 300.2.a.c, 300.2.a.d, 300.2.d.a, 900.2.a.d, 900.2.a.f, 900.2.a.g$^{3}$, 900.2.d.b, 900.2.d.d |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $240$ | $120$ | $0$ | $0$ | full Jacobian |
| 10.240.5-10.e.1.2 | $10$ | $6$ | $6$ | $5$ | $0$ | $1^{20}\cdot2^{12}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.720.19-15.g.1.6 | $15$ | $2$ | $2$ | $19$ | $1$ | $1^{16}\cdot2^{7}$ |
| 30.288.9-30.bf.1.4 | $30$ | $5$ | $5$ | $9$ | $0$ | $1^{18}\cdot2^{11}$ |
| 30.288.9-30.bf.2.1 | $30$ | $5$ | $5$ | $9$ | $0$ | $1^{18}\cdot2^{11}$ |
| 30.720.19-15.g.1.1 | $30$ | $2$ | $2$ | $19$ | $1$ | $1^{16}\cdot2^{7}$ |
| 30.720.25-30.ea.1.2 | $30$ | $2$ | $2$ | $25$ | $2$ | $1^{14}\cdot2^{5}$ |
| 30.720.25-30.ea.1.7 | $30$ | $2$ | $2$ | $25$ | $2$ | $1^{14}\cdot2^{5}$ |
| 30.720.25-30.ew.1.1 | $30$ | $2$ | $2$ | $25$ | $2$ | $1^{12}\cdot2^{6}$ |
| 30.720.25-30.ew.1.8 | $30$ | $2$ | $2$ | $25$ | $2$ | $1^{12}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.2880.97-30.x.1.6 | $30$ | $2$ | $2$ | $97$ | $8$ | $1^{24}\cdot2^{12}$ |
| 30.2880.97-30.cn.1.2 | $30$ | $2$ | $2$ | $97$ | $7$ | $1^{24}\cdot2^{12}$ |
| 30.4320.145-30.da.1.6 | $30$ | $3$ | $3$ | $145$ | $8$ | $1^{44}\cdot2^{26}$ |
| 60.2880.97-60.ur.1.8 | $60$ | $2$ | $2$ | $97$ | $13$ | $1^{24}\cdot2^{12}$ |
| 60.2880.97-60.ccw.1.2 | $60$ | $2$ | $2$ | $97$ | $12$ | $1^{24}\cdot2^{12}$ |
| 60.5760.217-60.gkw.1.12 | $60$ | $4$ | $4$ | $217$ | $38$ | $1^{82}\cdot2^{43}$ |