Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $300$ | ||
| Index: | $960$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $31 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $10^{8}\cdot20^{2}\cdot30^{8}\cdot60^{2}$ | Cusp orbits | $2^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.960.31.52 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}23&24\\6&7\end{bmatrix}$, $\begin{bmatrix}31&10\\36&59\end{bmatrix}$, $\begin{bmatrix}31&18\\12&29\end{bmatrix}$, $\begin{bmatrix}35&16\\54&1\end{bmatrix}$, $\begin{bmatrix}55&4\\24&35\end{bmatrix}$, $\begin{bmatrix}59&34\\18&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.480.31.a.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $96$ |
| Full 60-torsion field degree: | $2304$ |
Jacobian
| Conductor: | $2^{44}\cdot3^{25}\cdot5^{62}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot8^{2}$ |
| Newforms: | 50.2.a.b$^{4}$, 75.2.a.a$^{3}$, 75.2.a.b$^{3}$, 100.2.a.a$^{2}$, 150.2.a.b$^{2}$, 300.2.a.b, 300.2.e.c, 300.2.e.e |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.9 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.9 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 60.480.15-30.a.1.2 | $60$ | $2$ | $2$ | $15$ | $0$ | $8^{2}$ |
| 60.480.15-30.a.1.10 | $60$ | $2$ | $2$ | $15$ | $0$ | $8^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.1920.61-60.a.1.3 | $60$ | $2$ | $2$ | $61$ | $1$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.b.2.9 | $60$ | $2$ | $2$ | $61$ | $8$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.c.2.1 | $60$ | $2$ | $2$ | $61$ | $5$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.d.1.11 | $60$ | $2$ | $2$ | $61$ | $0$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.i.1.9 | $60$ | $2$ | $2$ | $61$ | $1$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.j.1.2 | $60$ | $2$ | $2$ | $61$ | $4$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.k.1.10 | $60$ | $2$ | $2$ | $61$ | $3$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.61-60.l.2.1 | $60$ | $2$ | $2$ | $61$ | $8$ | $1^{14}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.a.1.1 | $60$ | $2$ | $2$ | $65$ | $7$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.a.1.24 | $60$ | $2$ | $2$ | $65$ | $7$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.b.1.16 | $60$ | $2$ | $2$ | $65$ | $2$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.b.4.2 | $60$ | $2$ | $2$ | $65$ | $2$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.c.1.24 | $60$ | $2$ | $2$ | $65$ | $8$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.c.2.2 | $60$ | $2$ | $2$ | $65$ | $8$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.d.1.21 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.d.2.4 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.i.1.21 | $60$ | $2$ | $2$ | $65$ | $5$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.i.3.4 | $60$ | $2$ | $2$ | $65$ | $5$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.j.1.2 | $60$ | $2$ | $2$ | $65$ | $1$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.j.3.21 | $60$ | $2$ | $2$ | $65$ | $1$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.k.2.19 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.k.3.2 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.l.2.1 | $60$ | $2$ | $2$ | $65$ | $5$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.l.3.23 | $60$ | $2$ | $2$ | $65$ | $5$ | $1^{18}\cdot8^{2}$ |
| 60.1920.65-60.dc.2.8 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dc.3.10 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dd.2.15 | $60$ | $2$ | $2$ | $65$ | $10$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dd.3.7 | $60$ | $2$ | $2$ | $65$ | $10$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.de.2.7 | $60$ | $2$ | $2$ | $65$ | $4$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.de.4.16 | $60$ | $2$ | $2$ | $65$ | $4$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.df.2.14 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.df.4.3 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dk.2.12 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dk.4.3 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dl.2.3 | $60$ | $2$ | $2$ | $65$ | $10$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dl.4.10 | $60$ | $2$ | $2$ | $65$ | $10$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dm.1.11 | $60$ | $2$ | $2$ | $65$ | $4$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dm.4.4 | $60$ | $2$ | $2$ | $65$ | $4$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dn.1.4 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.65-60.dn.4.11 | $60$ | $2$ | $2$ | $65$ | $6$ | $1^{18}\cdot4^{2}\cdot8$ |
| 60.1920.69-60.bx.1.23 | $60$ | $2$ | $2$ | $69$ | $6$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.by.1.20 | $60$ | $2$ | $2$ | $69$ | $10$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.bz.1.23 | $60$ | $2$ | $2$ | $69$ | $4$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.ca.1.17 | $60$ | $2$ | $2$ | $69$ | $6$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.cn.1.18 | $60$ | $2$ | $2$ | $69$ | $6$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.co.1.21 | $60$ | $2$ | $2$ | $69$ | $10$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.cp.1.19 | $60$ | $2$ | $2$ | $69$ | $4$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.cq.1.23 | $60$ | $2$ | $2$ | $69$ | $6$ | $1^{18}\cdot2^{4}\cdot4^{3}$ |
| 60.1920.69-60.fd.2.13 | $60$ | $2$ | $2$ | $69$ | $5$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.fe.2.7 | $60$ | $2$ | $2$ | $69$ | $1$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.ff.1.11 | $60$ | $2$ | $2$ | $69$ | $6$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.fg.1.13 | $60$ | $2$ | $2$ | $69$ | $5$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.fh.2.2 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.fi.2.7 | $60$ | $2$ | $2$ | $69$ | $2$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.fj.1.7 | $60$ | $2$ | $2$ | $69$ | $8$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.1920.69-60.fk.1.6 | $60$ | $2$ | $2$ | $69$ | $6$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.2880.91-60.i.1.16 | $60$ | $3$ | $3$ | $91$ | $1$ | $1^{28}\cdot4^{2}\cdot8^{3}$ |
| 60.2880.101-60.d.1.24 | $60$ | $3$ | $3$ | $101$ | $7$ | $1^{34}\cdot2^{2}\cdot8^{4}$ |