Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.4.58 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&28\\46&23\end{bmatrix}$, $\begin{bmatrix}19&2\\40&49\end{bmatrix}$, $\begin{bmatrix}29&44\\44&13\end{bmatrix}$, $\begin{bmatrix}59&34\\56&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.4.j.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{8}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a$^{2}$, 900.2.a.g, 3600.2.a.e |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 5 y^{2} - 4 z^{2} + 2 z w - w^{2} $ |
| $=$ | $60 x^{3} + y z^{2} - y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 180 y^{4} z^{2} + 300 y^{2} z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^2}\cdot\frac{(13z^{4}-10z^{3}w+9z^{2}w^{2}-4zw^{3}+w^{4})^{3}}{z^{4}(z-w)^{4}(4z^{2}-2zw+w^{2})^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.72.4.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}-180Y^{4}Z^{2}+300Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.48.0-60.g.1.7 | $60$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 60.72.2-60.b.1.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.72.2-60.b.1.6 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.72.2-60.c.1.10 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.72.2-60.c.1.13 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.72.2-12.d.1.6 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.288.7-60.bv.1.1 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 60.288.7-60.by.1.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.288.7-60.cp.1.12 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.288.7-60.cs.1.6 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 60.288.7-60.do.1.2 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.288.7-60.ds.1.4 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.288.7-60.ed.1.6 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.288.7-60.ef.1.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.720.28-60.o.1.11 | $60$ | $5$ | $5$ | $28$ | $10$ | $1^{24}$ |
| 60.864.31-60.o.1.4 | $60$ | $6$ | $6$ | $31$ | $6$ | $1^{27}$ |
| 60.1440.55-60.fg.1.9 | $60$ | $10$ | $10$ | $55$ | $17$ | $1^{51}$ |
| 120.288.7-120.ip.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.jk.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.nr.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.om.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.tm.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.uo.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.xf.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.xt.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 180.432.16-180.j.1.2 | $180$ | $3$ | $3$ | $16$ | $?$ | not computed |