Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $150$ | ||
| Index: | $720$ | $\PSL_2$-index: | $360$ | ||||
| Genus: | $19 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $8$ are rational) | Cusp widths | $5^{6}\cdot10^{6}\cdot15^{6}\cdot30^{6}$ | Cusp orbits | $1^{8}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30B19 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.720.19.1640 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&0\\30&19\end{bmatrix}$, $\begin{bmatrix}31&40\\0&43\end{bmatrix}$, $\begin{bmatrix}41&40\\0&37\end{bmatrix}$, $\begin{bmatrix}47&25\\0&37\end{bmatrix}$, $\begin{bmatrix}47&25\\30&11\end{bmatrix}$, $\begin{bmatrix}49&5\\30&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.360.19.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $2$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{9}\cdot3^{15}\cdot5^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{19}$ |
| Newforms: | 15.2.a.a$^{4}$, 30.2.a.a$^{2}$, 50.2.a.a$^{2}$, 50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 75.2.a.b$^{2}$, 75.2.a.c$^{2}$, 150.2.a.a, 150.2.a.b, 150.2.a.c |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 12.24.0-6.a.1.6 | $12$ | $30$ | $30$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.3-30.a.1.9 | $60$ | $5$ | $5$ | $3$ | $0$ | $1^{16}$ |
| 60.360.10-30.a.1.2 | $60$ | $2$ | $2$ | $10$ | $0$ | $1^{9}$ |
| 60.360.10-30.a.1.48 | $60$ | $2$ | $2$ | $10$ | $0$ | $1^{9}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.1440.37-30.a.1.19 | $60$ | $2$ | $2$ | $37$ | $0$ | $2^{9}$ |
| 60.1440.37-30.b.1.9 | $60$ | $2$ | $2$ | $37$ | $0$ | $2^{9}$ |
| 60.1440.37-60.nk.1.27 | $60$ | $2$ | $2$ | $37$ | $0$ | $2^{9}$ |
| 60.1440.37-60.nl.1.31 | $60$ | $2$ | $2$ | $37$ | $0$ | $2^{9}$ |
| 60.1440.43-30.a.1.13 | $60$ | $2$ | $2$ | $43$ | $1$ | $1^{24}$ |
| 60.1440.43-30.e.1.7 | $60$ | $2$ | $2$ | $43$ | $5$ | $1^{24}$ |
| 60.1440.43-30.i.1.13 | $60$ | $2$ | $2$ | $43$ | $1$ | $1^{24}$ |
| 60.1440.43-30.k.1.13 | $60$ | $2$ | $2$ | $43$ | $7$ | $1^{24}$ |
| 60.1440.43-30.q.1.6 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-30.r.1.4 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-60.fe.1.34 | $60$ | $2$ | $2$ | $43$ | $7$ | $1^{24}$ |
| 60.1440.43-60.gq.1.13 | $60$ | $2$ | $2$ | $43$ | $7$ | $1^{24}$ |
| 60.1440.43-60.gs.1.31 | $60$ | $2$ | $2$ | $43$ | $1$ | $1^{24}$ |
| 60.1440.43-60.iv.1.22 | $60$ | $2$ | $2$ | $43$ | $10$ | $1^{24}$ |
| 60.1440.43-60.ix.1.1 | $60$ | $2$ | $2$ | $43$ | $10$ | $1^{24}$ |
| 60.1440.43-60.iz.1.7 | $60$ | $2$ | $2$ | $43$ | $5$ | $1^{24}$ |
| 60.1440.43-60.nh.1.20 | $60$ | $2$ | $2$ | $43$ | $8$ | $1^{24}$ |
| 60.1440.43-60.nj.1.10 | $60$ | $2$ | $2$ | $43$ | $8$ | $1^{24}$ |
| 60.1440.43-60.nl.1.17 | $60$ | $2$ | $2$ | $43$ | $1$ | $1^{24}$ |
| 60.1440.43-60.nv.1.24 | $60$ | $2$ | $2$ | $43$ | $13$ | $1^{24}$ |
| 60.1440.43-60.nx.1.6 | $60$ | $2$ | $2$ | $43$ | $13$ | $1^{24}$ |
| 60.1440.43-60.nz.1.9 | $60$ | $2$ | $2$ | $43$ | $7$ | $1^{24}$ |
| 60.1440.43-60.pk.1.27 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-60.pl.1.29 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-60.pm.1.30 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-60.pn.1.15 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-60.po.1.15 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.43-60.pp.1.16 | $60$ | $2$ | $2$ | $43$ | $0$ | $2^{12}$ |
| 60.1440.49-60.bsd.1.11 | $60$ | $2$ | $2$ | $49$ | $2$ | $2^{15}$ |
| 60.1440.49-60.bse.1.21 | $60$ | $2$ | $2$ | $49$ | $0$ | $2^{15}$ |
| 60.1440.49-60.bsf.1.7 | $60$ | $2$ | $2$ | $49$ | $0$ | $2^{15}$ |
| 60.1440.49-60.bsg.1.11 | $60$ | $2$ | $2$ | $49$ | $4$ | $2^{15}$ |
| 60.1440.49-60.bvn.1.19 | $60$ | $2$ | $2$ | $49$ | $12$ | $1^{30}$ |
| 60.1440.49-60.bvp.1.21 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{30}$ |
| 60.1440.49-60.bwd.1.11 | $60$ | $2$ | $2$ | $49$ | $6$ | $1^{30}$ |
| 60.1440.49-60.bwf.1.11 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{30}$ |
| 60.1440.49-60.bwt.1.18 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{30}$ |
| 60.1440.49-60.bwv.1.24 | $60$ | $2$ | $2$ | $49$ | $12$ | $1^{30}$ |
| 60.1440.49-60.bxb.1.6 | $60$ | $2$ | $2$ | $49$ | $5$ | $1^{30}$ |
| 60.1440.49-60.bxd.1.24 | $60$ | $2$ | $2$ | $49$ | $8$ | $1^{30}$ |
| 60.2160.67-30.a.1.7 | $60$ | $3$ | $3$ | $67$ | $6$ | $1^{44}\cdot2^{2}$ |