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: | 60.288.7.2983 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&40\\48&11\end{bmatrix}$, $\begin{bmatrix}7&0\\30&23\end{bmatrix}$, $\begin{bmatrix}19&35\\48&59\end{bmatrix}$, $\begin{bmatrix}31&55\\42&13\end{bmatrix}$, $\begin{bmatrix}41&5\\24&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.7.md.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $2$ |
| Cyclic 60-torsion field degree: | $16$ |
| 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 u + x v - y w + y u + y v $ |
| $=$ | $2 x w - x u - x v - z w$ | |
| $=$ | $x w - x u + x v - y w + y t - y u + y v$ | |
| $=$ | $x w - 2 x t + 2 x u - z w + z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{8} z^{2} + x^{7} y^{3} + 9 x^{7} y z^{2} - 18 x^{6} y^{2} z^{2} - 54 x^{6} z^{4} + \cdots + 135 y^{4} z^{6} $ |
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 |
|---|
| $(0:0:0:0:1:0:0)$, $(0:0:0:0:-1:-1:1)$, $(0:0:0:-2:-1:-1:1)$, $(0:0:0:1/2:1/4:0:1)$ |
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.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w+u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w-t+2u+v$ |
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.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{8}Z^{2}+X^{7}Y^{3}+9X^{7}YZ^{2}-18X^{6}Y^{2}Z^{2}-54X^{6}Z^{4}-21X^{5}Y^{3}Z^{2}+63X^{5}YZ^{4}+3X^{4}Y^{4}Z^{2}+324X^{4}Y^{2}Z^{4}+135X^{4}Z^{6}+63X^{3}Y^{3}Z^{4}-945X^{3}YZ^{6}-54X^{2}Y^{4}Z^{4}-810X^{2}Y^{2}Z^{6}+405XY^{3}Z^{6}+2025XYZ^{8}+135Y^{4}Z^{6} $ |
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.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 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.1.5 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.gd.2.2 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.ju.1.1 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.jv.2.4 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.nd.1.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.nf.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.nq.1.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2$ |
| 60.576.13-60.nr.1.4 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{4}\cdot2$ |
| 60.576.17-60.bk.1.1 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.bm.1.9 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.ee.1.9 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.ef.1.9 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gu.1.2 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gw.1.3 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hh.1.10 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hi.1.11 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.jt.1.2 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ju.1.4 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ka.1.10 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kc.1.12 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kj.1.4 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kk.1.8 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kq.1.12 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ks.1.24 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.864.25-60.ia.2.9 | $60$ | $3$ | $3$ | $25$ | $0$ | $1^{8}\cdot2^{5}$ |
| 60.1440.43-60.pm.1.30 | $60$ | $5$ | $5$ | $43$ | $0$ | $1^{16}\cdot2^{10}$ |