Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $1800$ | ||
| Index: | $960$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $33 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}\cdot30^{4}\cdot60^{4}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-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: | 60.960.33.165 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}25&12\\56&49\end{bmatrix}$, $\begin{bmatrix}25&18\\14&41\end{bmatrix}$, $\begin{bmatrix}47&54\\52&7\end{bmatrix}$, $\begin{bmatrix}47&54\\52&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.480.33.d.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: | $2^{57}\cdot3^{38}\cdot5^{64}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot8^{2}$ |
| Newforms: | 50.2.a.b$^{4}$, 72.2.a.a, 75.2.a.a$^{3}$, 225.2.a.e, 300.2.a.b, 300.2.e.e$^{2}$, 450.2.a.c$^{2}$, 900.2.a.e, 1800.2.a.e, 1800.2.a.h$^{2}$, 1800.2.a.x |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $192$ | $96$ | $0$ | $0$ | full Jacobian |
| 12.192.1-12.d.1.8 | $12$ | $5$ | $5$ | $1$ | $0$ | $1^{16}\cdot8^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.192.1-12.d.1.8 | $12$ | $5$ | $5$ | $1$ | $0$ | $1^{16}\cdot8^{2}$ |
| 60.480.16-60.a.1.5 | $60$ | $2$ | $2$ | $16$ | $0$ | $1^{9}\cdot8$ |
| 60.480.16-60.a.1.26 | $60$ | $2$ | $2$ | $16$ | $0$ | $1^{9}\cdot8$ |
| 60.480.16-60.a.2.3 | $60$ | $2$ | $2$ | $16$ | $0$ | $1^{9}\cdot8$ |
| 60.480.16-60.a.2.32 | $60$ | $2$ | $2$ | $16$ | $0$ | $1^{9}\cdot8$ |
| 60.480.17-60.d.1.1 | $60$ | $2$ | $2$ | $17$ | $2$ | $8^{2}$ |
| 60.480.17-60.d.1.15 | $60$ | $2$ | $2$ | $17$ | $2$ | $8^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.1920.69-60.i.1.4 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.i.2.2 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.k.2.1 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.k.4.4 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.x.2.6 | $60$ | $2$ | $2$ | $69$ | $9$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.x.3.8 | $60$ | $2$ | $2$ | $69$ | $9$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.y.1.5 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.1920.69-60.y.2.8 | $60$ | $2$ | $2$ | $69$ | $7$ | $1^{18}\cdot2^{5}\cdot8$ |
| 60.2880.97-60.p.2.24 | $60$ | $3$ | $3$ | $97$ | $8$ | $1^{32}\cdot8^{4}$ |
| 60.2880.105-60.ba.2.8 | $60$ | $3$ | $3$ | $105$ | $11$ | $1^{36}\cdot2^{2}\cdot8^{4}$ |
| 60.3840.129-60.v.2.15 | $60$ | $4$ | $4$ | $129$ | $13$ | $1^{48}\cdot4^{4}\cdot8^{4}$ |