Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $720$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $2^{3}\cdot6^{3}\cdot10^{3}\cdot30^{3}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30S7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.7.2985 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&15\\54&49\end{bmatrix}$, $\begin{bmatrix}7&20\\54&59\end{bmatrix}$, $\begin{bmatrix}11&5\\54&47\end{bmatrix}$, $\begin{bmatrix}13&25\\6&29\end{bmatrix}$, $\begin{bmatrix}31&35\\12&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.7.md.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $2$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{17}\cdot3^{11}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 15.2.a.a$^{2}$, 30.2.a.a, 720.2.f.c, 720.2.f.f |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w + x v - y t - y v $ |
| $=$ | $x w + x t - y w - y u$ | |
| $=$ | $w^{2} - w t + w u - t^{2} - t v + u v$ | |
| $=$ | $x t + 2 x u + x v + z w - z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 90 x^{7} y + 45 x^{6} y^{2} + 12 x^{6} z^{2} - 18 x^{5} y z^{2} + 15 x^{4} y^{2} z^{2} + \cdots + y^{4} z^{4} $ |
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.
| Canonical model |
|---|
| $(0:0:0:0:0:1:0)$, $(0:0:0:-1:-1:0:1)$, $(0:0:0:-1:1:0:1)$, $(0:0:0:0:-1/2:-1/4:1)$ |
Maps to other modular curves
Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 60.48.3.bc.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 5x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3t+2u+2v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t+u+v$ |
Equation of the image curve:
| $0$ | $=$ | $ 45X^{4}+6X^{2}Y^{2}-2Y^{3}Z+6X^{2}Z^{2}+3Y^{2}Z^{2}+2YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.7.md.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -90X^{7}Y+45X^{6}Y^{2}+12X^{6}Z^{2}-18X^{5}YZ^{2}+15X^{4}Y^{2}Z^{2}-18X^{3}Y^{3}Z^{2}-2X^{3}YZ^{4}+15X^{2}Y^{4}Z^{2}+5X^{2}Y^{2}Z^{4}-4XY^{3}Z^{4}-3Y^{6}Z^{2}+Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.3-30.a.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 60.144.3-30.a.1.9 | $60$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.576.13-60.ch.2.5 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.gd.1.3 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.ju.2.1 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.jv.1.4 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.nd.2.16 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.nf.2.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.nq.2.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2$ |
| 60.576.13-60.nr.2.4 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{4}\cdot2$ |
| 60.576.17-60.bk.2.1 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.bm.2.9 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.ee.2.9 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.ef.2.9 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gu.2.2 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gw.2.2 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hh.2.10 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hi.2.10 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.jt.2.3 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ju.2.7 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ka.2.11 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kc.2.15 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kj.2.4 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kk.2.8 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kq.2.12 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ks.2.24 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.864.25-60.ia.1.13 | $60$ | $3$ | $3$ | $25$ | $0$ | $1^{8}\cdot2^{5}$ |
| 60.1440.43-60.pm.1.30 | $60$ | $5$ | $5$ | $43$ | $0$ | $1^{16}\cdot2^{10}$ |