Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1200$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 60.48.1.88 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}4&57\\15&22\end{bmatrix}$, $\begin{bmatrix}19&26\\24&29\end{bmatrix}$, $\begin{bmatrix}46&27\\51&46\end{bmatrix}$, $\begin{bmatrix}50&39\\3&28\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.1.y.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $96$ |
| Full 60-torsion field degree: | $46080$ |
Jacobian
| Conductor: | $2^{4}\cdot3\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1200.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 608x + 5712 $ |
Rational points
This modular curve has infinitely many rational points, including 12 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^6}\cdot\frac{20x^{2}y^{6}-189375x^{2}y^{4}z^{2}+478750000x^{2}y^{2}z^{4}-400791015625x^{2}z^{6}-830xy^{6}z+5501250xy^{4}z^{3}-13943515625xy^{2}z^{5}+11787343750000xz^{7}-y^{8}+11830y^{6}z^{2}-50854375y^{4}z^{4}+106223984375y^{2}z^{6}-84734218750000z^{8}}{z^{4}y^{2}(40x^{2}-1160xz-y^{2}+8160z^{2})}$ |
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.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.24.0-6.a.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ | 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.a.1.19 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.h.1.11 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.i.1.4 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.l.1.8 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.z.1.5 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.bb.1.3 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.bd.1.6 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.96.1-60.bf.1.8 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.144.3-60.nq.1.5 | $60$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
| 60.240.9-60.dp.1.10 | $60$ | $5$ | $5$ | $9$ | $1$ | $1^{8}$ |
| 60.288.9-60.ft.1.4 | $60$ | $6$ | $6$ | $9$ | $4$ | $1^{8}$ |
| 60.480.17-60.nd.1.3 | $60$ | $10$ | $10$ | $17$ | $2$ | $1^{16}$ |
| 120.96.1-120.gm.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.kh.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.yy.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.zh.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.blb.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.blh.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bln.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.blt.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 180.144.3-180.bh.1.13 | $180$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 180.144.5-180.q.1.4 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.144.5-180.u.1.13 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |