Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.1.136 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}4&43\\45&16\end{bmatrix}$, $\begin{bmatrix}10&43\\21&16\end{bmatrix}$, $\begin{bmatrix}16&45\\21&26\end{bmatrix}$, $\begin{bmatrix}32&9\\27&46\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.1.w.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $46080$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.v |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 5475x - 148750 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-50:0:1)$, $(-35:0:1)$, $(85:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6\cdot5^6}\cdot\frac{60x^{2}y^{6}+15339375x^{2}y^{4}z^{2}+1047026250000x^{2}y^{2}z^{4}+23666308681640625x^{2}z^{6}+7350xy^{6}z+1306125000xy^{4}z^{3}+89389353515625xy^{2}z^{5}+2040759965917968750xz^{7}+y^{8}+312000y^{6}z^{2}+35751375000y^{4}z^{4}+2000370304687500y^{2}z^{6}+42966811672119140625z^{8}}{z^{4}y^{2}(120x^{2}+10200xz+y^{2}+210000z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.24.0-6.a.1.11 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.96.1-60.c.1.11 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.g.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.m.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.p.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.q.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.t.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.u.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.x.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.3-60.lc.1.5 | $60$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 60.240.9-60.dn.1.12 | $60$ | $5$ | $5$ | $9$ | $4$ | $1^{8}$ |
| 60.288.9-60.fr.1.15 | $60$ | $6$ | $6$ | $9$ | $0$ | $1^{8}$ |
| 60.480.17-60.nb.1.29 | $60$ | $10$ | $10$ | $17$ | $5$ | $1^{16}$ |
| 120.96.1-120.gk.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ke.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.zk.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.zt.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bae.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ban.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.baq.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.baz.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 180.144.3-180.bf.1.6 | $180$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 180.144.5-180.o.1.11 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.144.5-180.s.1.4 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |