Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $450$ | ||
| Index: | $720$ | $\PSL_2$-index: | $360$ | ||||
| Genus: | $23 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $15^{8}\cdot30^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30A23 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.720.23.6 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&17\\18&59\end{bmatrix}$, $\begin{bmatrix}7&20\\0&47\end{bmatrix}$, $\begin{bmatrix}11&3\\12&59\end{bmatrix}$, $\begin{bmatrix}23&21\\24&47\end{bmatrix}$, $\begin{bmatrix}23&39\\6&59\end{bmatrix}$, $\begin{bmatrix}43&2\\24&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.360.23.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{9}\cdot3^{30}\cdot5^{46}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{23}$ |
| Newforms: | 50.2.a.b$^{3}$, 75.2.a.a$^{4}$, 75.2.a.b$^{4}$, 150.2.a.b$^{2}$, 225.2.a.b$^{2}$, 225.2.a.c$^{2}$, 225.2.a.e$^{2}$, 450.2.a.a, 450.2.a.c, 450.2.a.d, 450.2.a.e |
Rational points
This modular curve has no $\Q_p$ points for $p=7,13,37,103$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $72$ | $36$ | $0$ | $0$ | full Jacobian |
| 12.72.0-6.a.1.5 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.0-6.a.1.5 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 60.240.7-30.h.1.5 | $60$ | $3$ | $3$ | $7$ | $0$ | $1^{16}$ |
| 60.240.7-30.h.1.14 | $60$ | $3$ | $3$ | $7$ | $0$ | $1^{16}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.1440.45-30.a.1.12 | $60$ | $2$ | $2$ | $45$ | $6$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-60.a.1.9 | $60$ | $2$ | $2$ | $45$ | $14$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-30.b.1.10 | $60$ | $2$ | $2$ | $45$ | $6$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-30.c.1.10 | $60$ | $2$ | $2$ | $45$ | $7$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-30.d.1.11 | $60$ | $2$ | $2$ | $45$ | $9$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-60.d.1.9 | $60$ | $2$ | $2$ | $45$ | $12$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-60.g.1.9 | $60$ | $2$ | $2$ | $45$ | $15$ | $1^{18}\cdot2^{2}$ |
| 60.1440.45-60.j.1.9 | $60$ | $2$ | $2$ | $45$ | $11$ | $1^{18}\cdot2^{2}$ |
| 60.1440.49-30.a.1.16 | $60$ | $2$ | $2$ | $49$ | $7$ | $1^{26}$ |
| 60.1440.49-60.cr.1.18 | $60$ | $2$ | $2$ | $49$ | $13$ | $1^{26}$ |
| 60.1440.49-30.cu.1.12 | $60$ | $2$ | $2$ | $49$ | $5$ | $1^{26}$ |
| 60.1440.49-30.dd.1.6 | $60$ | $2$ | $2$ | $49$ | $9$ | $1^{24}\cdot2$ |
| 60.1440.49-30.de.1.5 | $60$ | $2$ | $2$ | $49$ | $9$ | $1^{24}\cdot2$ |
| 60.1440.49-30.dq.1.5 | $60$ | $2$ | $2$ | $49$ | $8$ | $1^{24}\cdot2$ |
| 60.1440.49-30.dr.1.6 | $60$ | $2$ | $2$ | $49$ | $8$ | $1^{24}\cdot2$ |
| 60.1440.49-30.dy.1.10 | $60$ | $2$ | $2$ | $49$ | $6$ | $1^{26}$ |
| 60.1440.49-30.dz.1.10 | $60$ | $2$ | $2$ | $49$ | $10$ | $1^{26}$ |
| 60.1440.49-60.ga.1.14 | $60$ | $2$ | $2$ | $49$ | $13$ | $1^{26}$ |
| 60.1440.49-60.gc.1.32 | $60$ | $2$ | $2$ | $49$ | $7$ | $1^{26}$ |
| 60.1440.49-60.bau.1.11 | $60$ | $2$ | $2$ | $49$ | $12$ | $1^{26}$ |
| 60.1440.49-60.bax.1.10 | $60$ | $2$ | $2$ | $49$ | $12$ | $1^{26}$ |
| 60.1440.49-60.bay.1.14 | $60$ | $2$ | $2$ | $49$ | $5$ | $1^{26}$ |
| 60.1440.49-60.bdz.1.9 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bec.1.13 | $60$ | $2$ | $2$ | $49$ | $11$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bed.1.15 | $60$ | $2$ | $2$ | $49$ | $9$ | $1^{24}\cdot2$ |
| 60.1440.49-60.beg.1.11 | $60$ | $2$ | $2$ | $49$ | $15$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bej.1.9 | $60$ | $2$ | $2$ | $49$ | $15$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bek.1.13 | $60$ | $2$ | $2$ | $49$ | $9$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bim.1.13 | $60$ | $2$ | $2$ | $49$ | $8$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bin.1.9 | $60$ | $2$ | $2$ | $49$ | $14$ | $1^{24}\cdot2$ |
| 60.1440.49-60.biq.1.9 | $60$ | $2$ | $2$ | $49$ | $14$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bit.1.15 | $60$ | $2$ | $2$ | $49$ | $8$ | $1^{24}\cdot2$ |
| 60.1440.49-60.biu.1.13 | $60$ | $2$ | $2$ | $49$ | $16$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bix.1.13 | $60$ | $2$ | $2$ | $49$ | $16$ | $1^{24}\cdot2$ |
| 60.1440.49-60.bke.1.14 | $60$ | $2$ | $2$ | $49$ | $6$ | $1^{26}$ |
| 60.1440.49-60.bkf.1.10 | $60$ | $2$ | $2$ | $49$ | $10$ | $1^{26}$ |
| 60.1440.49-60.bki.1.9 | $60$ | $2$ | $2$ | $49$ | $10$ | $1^{26}$ |
| 60.1440.49-60.bkl.1.16 | $60$ | $2$ | $2$ | $49$ | $10$ | $1^{26}$ |
| 60.1440.49-60.bkm.1.14 | $60$ | $2$ | $2$ | $49$ | $19$ | $1^{26}$ |
| 60.1440.49-60.bkp.1.11 | $60$ | $2$ | $2$ | $49$ | $19$ | $1^{26}$ |
| 60.1440.53-60.ckq.1.1 | $60$ | $2$ | $2$ | $53$ | $10$ | $1^{30}$ |
| 60.1440.53-60.ckr.1.5 | $60$ | $2$ | $2$ | $53$ | $13$ | $1^{30}$ |
| 60.1440.53-60.dbk.1.5 | $60$ | $2$ | $2$ | $53$ | $15$ | $1^{30}$ |
| 60.1440.53-60.dbl.1.7 | $60$ | $2$ | $2$ | $53$ | $12$ | $1^{30}$ |
| 60.1440.53-60.dlg.1.2 | $60$ | $2$ | $2$ | $53$ | $12$ | $1^{30}$ |
| 60.1440.53-60.dlh.1.6 | $60$ | $2$ | $2$ | $53$ | $14$ | $1^{30}$ |
| 60.1440.53-60.dma.1.6 | $60$ | $2$ | $2$ | $53$ | $14$ | $1^{30}$ |
| 60.1440.53-60.dmb.1.8 | $60$ | $2$ | $2$ | $53$ | $14$ | $1^{30}$ |
| 60.1440.53-60.dxw.1.8 | $60$ | $2$ | $2$ | $53$ | $15$ | $1^{30}$ |
| 60.1440.53-60.dxx.1.6 | $60$ | $2$ | $2$ | $53$ | $17$ | $1^{30}$ |
| 60.1440.53-60.dyq.1.6 | $60$ | $2$ | $2$ | $53$ | $15$ | $1^{30}$ |
| 60.1440.53-60.dyr.1.2 | $60$ | $2$ | $2$ | $53$ | $21$ | $1^{30}$ |
| 60.1440.53-60.dzs.1.7 | $60$ | $2$ | $2$ | $53$ | $9$ | $1^{30}$ |
| 60.1440.53-60.dzt.1.5 | $60$ | $2$ | $2$ | $53$ | $12$ | $1^{30}$ |
| 60.1440.53-60.ebg.1.5 | $60$ | $2$ | $2$ | $53$ | $12$ | $1^{30}$ |
| 60.1440.53-60.ebi.1.1 | $60$ | $2$ | $2$ | $53$ | $15$ | $1^{30}$ |
| 60.2160.67-30.b.1.8 | $60$ | $3$ | $3$ | $67$ | $7$ | $1^{40}\cdot2^{2}$ |