Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $180$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $3 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $2$ are rational) | Cusp widths | $2^{8}\cdot4^{2}\cdot10^{8}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20R3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.3.3 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&0\\12&31\end{bmatrix}$, $\begin{bmatrix}13&50\\36&31\end{bmatrix}$, $\begin{bmatrix}17&50\\22&11\end{bmatrix}$, $\begin{bmatrix}23&20\\50&1\end{bmatrix}$, $\begin{bmatrix}29&0\\54&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.3.b.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{4}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 20.2.a.a, 180.2.k.c |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 12.12.0-2.a.1.1 | $12$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 60.144.1-10.a.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.576.9-60.i.2.4 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.j.2.4 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.j.3.8 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.k.1.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.k.4.4 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.l.1.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.l.4.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.q.1.4 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.q.3.4 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.r.1.4 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.r.3.8 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.u.2.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.u.3.4 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.v.2.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.9-60.v.4.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.eq.2.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.er.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.es.1.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.et.2.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.ew.2.16 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.ex.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.ey.1.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.576.13-60.ez.2.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 60.864.27-60.e.2.4 | $60$ | $3$ | $3$ | $27$ | $0$ | $1^{6}\cdot2^{5}\cdot8$ |
| 60.1152.29-60.i.2.4 | $60$ | $4$ | $4$ | $29$ | $0$ | $1^{6}\cdot2^{4}\cdot12$ |
| 60.1440.31-60.a.1.12 | $60$ | $5$ | $5$ | $31$ | $2$ | $1^{6}\cdot2^{7}\cdot8$ |