Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $600$ | ||
| 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: | 60.48.1.115 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}4&37\\21&14\end{bmatrix}$, $\begin{bmatrix}7&30\\42&1\end{bmatrix}$, $\begin{bmatrix}10&11\\3&10\end{bmatrix}$, $\begin{bmatrix}29&2\\42&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.1.x.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: | $2^{3}\cdot3\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 600.2.a.h |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 608x - 5712 $ |
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 |
|---|
| $(0:1:0)$, $(-17:0:1)$, $(-12:0:1)$, $(28:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^6}\cdot\frac{20x^{2}y^{6}+189375x^{2}y^{4}z^{2}+478750000x^{2}y^{2}z^{4}+400791015625x^{2}z^{6}+830xy^{6}z+5501250xy^{4}z^{3}+13943515625xy^{2}z^{5}+11787343750000xz^{7}+y^{8}+11830y^{6}z^{2}+50854375y^{4}z^{4}+106223984375y^{2}z^{6}+84734218750000z^{8}}{z^{4}y^{2}(40x^{2}+1160xz+y^{2}+8160z^{2})}$ |
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.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.24.0-6.a.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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.b.1.9 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.e.1.10 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.j.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.k.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.y.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.ba.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.bc.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.be.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.3-60.np.1.7 | $60$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 60.240.9-60.do.1.8 | $60$ | $5$ | $5$ | $9$ | $1$ | $1^{8}$ |
| 60.288.9-60.fs.1.7 | $60$ | $6$ | $6$ | $9$ | $1$ | $1^{8}$ |
| 60.480.17-60.nc.1.28 | $60$ | $10$ | $10$ | $17$ | $1$ | $1^{16}$ |
| 120.96.1-120.gl.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.jy.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.zb.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ze.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bky.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ble.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.blk.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.blq.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 180.144.3-180.bg.1.8 | $180$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 180.144.5-180.p.1.9 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.144.5-180.t.1.12 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |