Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $3600$ | ||
| 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 | $5^{4}\cdot15^{4}\cdot20^{2}\cdot60^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 60V15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.480.15.55 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}14&17\\21&56\end{bmatrix}$, $\begin{bmatrix}20&13\\33&26\end{bmatrix}$, $\begin{bmatrix}25&56\\54&35\end{bmatrix}$, $\begin{bmatrix}32&19\\15&28\end{bmatrix}$, $\begin{bmatrix}37&52\\42&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.240.15.di.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: | $2^{35}\cdot3^{21}\cdot5^{30}$ |
| 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, 3600.2.a.bc$^{2}$, 3600.2.a.be$^{2}$, 3600.2.a.f, 3600.2.a.j, 3600.2.a.s, 3600.2.a.u |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.i.1.1 | $12$ | $10$ | $10$ | $0$ | $0$ | 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.i.1.1 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 30.240.7-30.h.1.3 | $30$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 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.if.1.7 | $60$ | $2$ | $2$ | $29$ | $8$ | $1^{14}$ |
| 60.960.29-60.ig.1.5 | $60$ | $2$ | $2$ | $29$ | $13$ | $1^{14}$ |
| 60.960.29-60.jc.1.3 | $60$ | $2$ | $2$ | $29$ | $12$ | $1^{14}$ |
| 60.960.29-60.jd.1.5 | $60$ | $2$ | $2$ | $29$ | $11$ | $1^{14}$ |
| 60.960.29-60.kg.1.5 | $60$ | $2$ | $2$ | $29$ | $13$ | $1^{14}$ |
| 60.960.29-60.kh.1.7 | $60$ | $2$ | $2$ | $29$ | $12$ | $1^{14}$ |
| 60.960.29-60.ld.1.5 | $60$ | $2$ | $2$ | $29$ | $9$ | $1^{14}$ |
| 60.960.29-60.le.1.1 | $60$ | $2$ | $2$ | $29$ | $16$ | $1^{14}$ |
| 60.960.33-60.o.1.39 | $60$ | $2$ | $2$ | $33$ | $8$ | $1^{18}$ |
| 60.960.33-60.bd.1.19 | $60$ | $2$ | $2$ | $33$ | $13$ | $1^{18}$ |
| 60.960.33-60.go.1.9 | $60$ | $2$ | $2$ | $33$ | $13$ | $1^{18}$ |
| 60.960.33-60.gp.1.11 | $60$ | $2$ | $2$ | $33$ | $15$ | $1^{18}$ |
| 60.960.33-60.hk.1.1 | $60$ | $2$ | $2$ | $33$ | $11$ | $1^{18}$ |
| 60.960.33-60.hl.1.2 | $60$ | $2$ | $2$ | $33$ | $11$ | $1^{18}$ |
| 60.960.33-60.ig.1.3 | $60$ | $2$ | $2$ | $33$ | $13$ | $1^{18}$ |
| 60.960.33-60.ih.1.2 | $60$ | $2$ | $2$ | $33$ | $15$ | $1^{18}$ |
| 60.960.33-60.jq.1.2 | $60$ | $2$ | $2$ | $33$ | $15$ | $1^{18}$ |
| 60.960.33-60.jr.1.1 | $60$ | $2$ | $2$ | $33$ | $15$ | $1^{18}$ |
| 60.960.33-60.km.1.2 | $60$ | $2$ | $2$ | $33$ | $17$ | $1^{18}$ |
| 60.960.33-60.kn.1.6 | $60$ | $2$ | $2$ | $33$ | $19$ | $1^{18}$ |
| 60.960.33-60.li.1.11 | $60$ | $2$ | $2$ | $33$ | $10$ | $1^{18}$ |
| 60.960.33-60.lj.1.9 | $60$ | $2$ | $2$ | $33$ | $11$ | $1^{18}$ |
| 60.960.33-60.ln.1.11 | $60$ | $2$ | $2$ | $33$ | $18$ | $1^{18}$ |
| 60.960.33-60.lo.1.23 | $60$ | $2$ | $2$ | $33$ | $16$ | $1^{18}$ |
| 60.1440.43-60.jy.1.6 | $60$ | $3$ | $3$ | $43$ | $17$ | $1^{28}$ |
| 60.1440.49-60.bkm.1.14 | $60$ | $3$ | $3$ | $49$ | $19$ | $1^{34}$ |