Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $720$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot10^{2}\cdot12^{2}\cdot20^{2}\cdot30^{2}\cdot60^{2}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 60T17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.576.17.78 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&35\\12&19\end{bmatrix}$, $\begin{bmatrix}19&0\\36&31\end{bmatrix}$, $\begin{bmatrix}29&40\\30&1\end{bmatrix}$, $\begin{bmatrix}47&20\\18&7\end{bmatrix}$, $\begin{bmatrix}47&40\\42&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.288.17.n.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $2$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $3840$ |
Jacobian
| Conductor: | $2^{51}\cdot3^{25}\cdot5^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}$ |
| Newforms: | 15.2.a.a$^{2}$, 30.2.a.a, 72.2.a.a$^{2}$, 80.2.a.b$^{2}$, 240.2.a.b, 240.2.a.d, 360.2.a.a$^{2}$, 360.2.a.b, 360.2.a.e, 720.2.a.c, 720.2.a.h$^{2}$, 720.2.a.j |
Rational points
This modular curve has no $\Q_p$ points for $p=19,43$, 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 |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 12.96.1-12.f.1.6 | $12$ | $6$ | $6$ | $1$ | $0$ | $1^{16}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.1-12.f.1.6 | $12$ | $6$ | $6$ | $1$ | $0$ | $1^{16}$ |
| 60.288.7-60.fm.1.23 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{10}$ |
| 60.288.7-60.fm.1.41 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{10}$ |
| 60.288.7-60.jz.1.16 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{10}$ |
| 60.288.7-60.jz.1.25 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{10}$ |
| 60.288.9-60.fu.1.16 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 60.288.9-60.fu.1.25 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.1152.33-60.cq.1.24 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.cq.2.24 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.cr.1.12 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.cr.2.12 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.cs.1.10 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.cs.2.10 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.ct.1.7 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.33-60.ct.2.7 | $60$ | $2$ | $2$ | $33$ | $3$ | $2^{8}$ |
| 60.1152.37-60.dm.1.12 | $60$ | $2$ | $2$ | $37$ | $3$ | $4^{3}\cdot8$ |
| 60.1152.37-60.dm.2.12 | $60$ | $2$ | $2$ | $37$ | $3$ | $4^{3}\cdot8$ |
| 60.1152.37-60.dm.3.15 | $60$ | $2$ | $2$ | $37$ | $3$ | $4^{3}\cdot8$ |
| 60.1152.37-60.dm.4.14 | $60$ | $2$ | $2$ | $37$ | $3$ | $4^{3}\cdot8$ |
| 60.1152.37-60.dn.1.12 | $60$ | $2$ | $2$ | $37$ | $3$ | $2^{2}\cdot4^{2}\cdot8$ |
| 60.1152.37-60.dn.2.14 | $60$ | $2$ | $2$ | $37$ | $3$ | $2^{2}\cdot4^{2}\cdot8$ |
| 60.1152.37-60.do.1.12 | $60$ | $2$ | $2$ | $37$ | $3$ | $2^{2}\cdot4^{2}\cdot8$ |
| 60.1152.37-60.do.2.12 | $60$ | $2$ | $2$ | $37$ | $3$ | $2^{2}\cdot4^{2}\cdot8$ |
| 60.1728.57-60.tv.1.3 | $60$ | $3$ | $3$ | $57$ | $8$ | $1^{40}$ |
| 60.2880.97-60.fs.1.14 | $60$ | $5$ | $5$ | $97$ | $28$ | $1^{80}$ |