Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $1200$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $15 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $10^{6}\cdot30^{6}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30A15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.480.15.75 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&49\\24&55\end{bmatrix}$, $\begin{bmatrix}25&13\\42&5\end{bmatrix}$, $\begin{bmatrix}31&14\\54&23\end{bmatrix}$, $\begin{bmatrix}47&30\\30&47\end{bmatrix}$, $\begin{bmatrix}49&41\\36&7\end{bmatrix}$, $\begin{bmatrix}53&23\\36&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.240.15.bk.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{35}\cdot3^{9}\cdot5^{30}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}$ |
| Newforms: | 50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 75.2.a.b$^{2}$, 150.2.a.b, 400.2.a.c$^{2}$, 400.2.a.f$^{2}$, 1200.2.a.c, 1200.2.a.e, 1200.2.a.k, 1200.2.a.n |
Rational points
This modular curve has no $\Q_p$ points for $p=53$, and therefore no 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.d.1.10 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.d.1.10 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 60.240.7-30.h.1.13 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 60.240.7-30.h.1.14 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.960.29-60.ea.1.20 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
| 60.960.29-60.eb.1.23 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 60.960.29-60.ec.1.16 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 60.960.29-60.ed.1.20 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
| 60.960.29-60.ee.1.15 | $60$ | $2$ | $2$ | $29$ | $8$ | $1^{14}$ |
| 60.960.29-60.ef.1.16 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{14}$ |
| 60.960.29-60.eg.1.20 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{14}$ |
| 60.960.29-60.eh.1.13 | $60$ | $2$ | $2$ | $29$ | $6$ | $1^{14}$ |
| 60.960.31-60.e.1.21 | $60$ | $2$ | $2$ | $31$ | $2$ | $8^{2}$ |
| 60.960.31-60.e.2.22 | $60$ | $2$ | $2$ | $31$ | $2$ | $8^{2}$ |
| 60.960.31-60.f.1.22 | $60$ | $2$ | $2$ | $31$ | $2$ | $8^{2}$ |
| 60.960.31-60.f.2.21 | $60$ | $2$ | $2$ | $31$ | $2$ | $8^{2}$ |
| 60.960.31-60.g.1.21 | $60$ | $2$ | $2$ | $31$ | $2$ | $4^{2}\cdot8$ |
| 60.960.31-60.g.2.23 | $60$ | $2$ | $2$ | $31$ | $2$ | $4^{2}\cdot8$ |
| 60.960.31-60.h.1.23 | $60$ | $2$ | $2$ | $31$ | $2$ | $4^{2}\cdot8$ |
| 60.960.31-60.h.2.21 | $60$ | $2$ | $2$ | $31$ | $2$ | $4^{2}\cdot8$ |
| 60.960.33-60.bc.1.20 | $60$ | $2$ | $2$ | $33$ | $6$ | $1^{18}$ |
| 60.960.33-60.bd.1.19 | $60$ | $2$ | $2$ | $33$ | $13$ | $1^{18}$ |
| 60.960.33-60.be.1.21 | $60$ | $2$ | $2$ | $33$ | $6$ | $1^{18}$ |
| 60.960.33-60.bf.1.22 | $60$ | $2$ | $2$ | $33$ | $7$ | $1^{18}$ |
| 60.960.33-60.bg.1.18 | $60$ | $2$ | $2$ | $33$ | $4$ | $1^{18}$ |
| 60.960.33-60.bh.1.18 | $60$ | $2$ | $2$ | $33$ | $14$ | $1^{18}$ |
| 60.960.33-60.bi.1.20 | $60$ | $2$ | $2$ | $33$ | $6$ | $1^{18}$ |
| 60.960.33-60.bj.1.19 | $60$ | $2$ | $2$ | $33$ | $10$ | $1^{18}$ |
| 60.960.33-60.bm.1.20 | $60$ | $2$ | $2$ | $33$ | $10$ | $1^{18}$ |
| 60.960.33-60.bn.1.18 | $60$ | $2$ | $2$ | $33$ | $8$ | $1^{18}$ |
| 60.960.33-60.bo.1.17 | $60$ | $2$ | $2$ | $33$ | $6$ | $1^{18}$ |
| 60.960.33-60.bp.1.18 | $60$ | $2$ | $2$ | $33$ | $10$ | $1^{18}$ |
| 60.960.33-60.bq.1.17 | $60$ | $2$ | $2$ | $33$ | $8$ | $1^{18}$ |
| 60.960.33-60.br.1.17 | $60$ | $2$ | $2$ | $33$ | $4$ | $1^{18}$ |
| 60.960.33-60.bs.1.23 | $60$ | $2$ | $2$ | $33$ | $5$ | $1^{18}$ |
| 60.960.33-60.bt.1.39 | $60$ | $2$ | $2$ | $33$ | $7$ | $1^{18}$ |
| 60.960.35-60.br.1.18 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{4}\cdot4^{3}$ |
| 60.960.35-60.br.2.19 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{4}\cdot4^{3}$ |
| 60.960.35-60.bs.1.18 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{4}\cdot4^{3}$ |
| 60.960.35-60.bs.2.19 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{4}\cdot4^{3}$ |
| 60.960.35-60.bt.1.21 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{6}\cdot4^{2}$ |
| 60.960.35-60.bt.2.22 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{6}\cdot4^{2}$ |
| 60.960.35-60.bu.1.22 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{6}\cdot4^{2}$ |
| 60.960.35-60.bu.2.21 | $60$ | $2$ | $2$ | $35$ | $2$ | $2^{6}\cdot4^{2}$ |
| 60.1440.43-60.fg.1.23 | $60$ | $3$ | $3$ | $43$ | $6$ | $1^{28}$ |
| 60.1440.49-60.cr.1.18 | $60$ | $3$ | $3$ | $49$ | $13$ | $1^{34}$ |