Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $3600$ | ||
| Index: | $1920$ | $\PSL_2$-index: | $960$ | ||||
| Genus: | $69 = 1 + \frac{ 960 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $20^{12}\cdot60^{12}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 20$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 20$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.1920.69.77 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&34\\18&59\end{bmatrix}$, $\begin{bmatrix}19&24\\54&37\end{bmatrix}$, $\begin{bmatrix}23&22\\54&37\end{bmatrix}$, $\begin{bmatrix}47&18\\18&23\end{bmatrix}$, $\begin{bmatrix}59&38\\18&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.960.69.fk.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $96$ |
| Full 60-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{163}\cdot3^{81}\cdot5^{132}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{33}\cdot2^{6}\cdot4^{2}\cdot8^{2}$ |
| Newforms: | 48.2.c.a, 50.2.a.b$^{4}$, 72.2.a.a, 75.2.a.a$^{3}$, 75.2.a.b$^{3}$, 100.2.a.a$^{2}$, 150.2.a.b$^{2}$, 225.2.a.b, 225.2.a.e, 300.2.a.b, 300.2.e.c, 300.2.e.e, 450.2.a.c$^{2}$, 450.2.a.d, 900.2.a.b$^{2}$, 900.2.a.e, 1200.2.h.a, 1200.2.h.c, 1200.2.h.e, 1200.2.h.f, 1200.2.h.i, 1200.2.h.k, 1200.2.h.m, 1800.2.a.c, 1800.2.a.e, 1800.2.a.h$^{2}$, 1800.2.a.m, 1800.2.a.n, 1800.2.a.v$^{2}$, 1800.2.a.x |
Rational points
This modular curve has no $\Q_p$ points for $p=7,43,53$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $192$ | $96$ | $0$ | $0$ | full Jacobian |
| 12.192.3-12.i.2.6 | $12$ | $10$ | $10$ | $3$ | $0$ | $1^{32}\cdot2^{5}\cdot4^{2}\cdot8^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.192.3-12.i.2.6 | $12$ | $10$ | $10$ | $3$ | $0$ | $1^{32}\cdot2^{5}\cdot4^{2}\cdot8^{2}$ |
| 60.960.31-60.a.2.7 | $60$ | $2$ | $2$ | $31$ | $0$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.960.31-60.a.2.21 | $60$ | $2$ | $2$ | $31$ | $0$ | $1^{18}\cdot2^{6}\cdot4^{2}$ |
| 60.960.33-60.p.1.5 | $60$ | $2$ | $2$ | $33$ | $6$ | $2^{6}\cdot4^{2}\cdot8^{2}$ |
| 60.960.33-60.p.1.15 | $60$ | $2$ | $2$ | $33$ | $6$ | $2^{6}\cdot4^{2}\cdot8^{2}$ |
| 60.960.35-60.bi.1.2 | $60$ | $2$ | $2$ | $35$ | $0$ | $1^{18}\cdot8^{2}$ |
| 60.960.35-60.bi.1.6 | $60$ | $2$ | $2$ | $35$ | $0$ | $1^{18}\cdot8^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.3840.137-60.bo.2.2 | $60$ | $2$ | $2$ | $137$ | $19$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.bo.2.8 | $60$ | $2$ | $2$ | $137$ | $19$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.bs.2.3 | $60$ | $2$ | $2$ | $137$ | $16$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.bs.3.1 | $60$ | $2$ | $2$ | $137$ | $16$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.jq.2.3 | $60$ | $2$ | $2$ | $137$ | $16$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.jq.3.1 | $60$ | $2$ | $2$ | $137$ | $16$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.jr.1.4 | $60$ | $2$ | $2$ | $137$ | $13$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.jr.4.3 | $60$ | $2$ | $2$ | $137$ | $13$ | $1^{36}\cdot8^{4}$ |
| 60.3840.137-60.mj.2.3 | $60$ | $2$ | $2$ | $137$ | $13$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.ml.2.5 | $60$ | $2$ | $2$ | $137$ | $24$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.mr.2.1 | $60$ | $2$ | $2$ | $137$ | $15$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.mt.2.5 | $60$ | $2$ | $2$ | $137$ | $12$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.np.1.5 | $60$ | $2$ | $2$ | $137$ | $13$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.nr.2.1 | $60$ | $2$ | $2$ | $137$ | $20$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.nx.2.4 | $60$ | $2$ | $2$ | $137$ | $13$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.nz.2.1 | $60$ | $2$ | $2$ | $137$ | $20$ | $1^{32}\cdot2^{4}\cdot4^{5}\cdot8$ |
| 60.3840.137-60.qi.1.5 | $60$ | $2$ | $2$ | $137$ | $20$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qi.3.7 | $60$ | $2$ | $2$ | $137$ | $20$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qj.1.6 | $60$ | $2$ | $2$ | $137$ | $18$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qj.4.1 | $60$ | $2$ | $2$ | $137$ | $18$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qs.1.3 | $60$ | $2$ | $2$ | $137$ | $20$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qs.3.2 | $60$ | $2$ | $2$ | $137$ | $20$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qt.2.4 | $60$ | $2$ | $2$ | $137$ | $18$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.3840.137-60.qt.4.2 | $60$ | $2$ | $2$ | $137$ | $18$ | $1^{36}\cdot4^{4}\cdot8^{2}$ |
| 60.5760.205-60.ri.2.7 | $60$ | $3$ | $3$ | $205$ | $19$ | $1^{64}\cdot2^{6}\cdot4^{9}\cdot8^{3}$ |
| 60.5760.217-60.sg.1.5 | $60$ | $3$ | $3$ | $217$ | $24$ | $1^{72}\cdot2^{10}\cdot4^{4}\cdot8^{5}$ |