Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | 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 $6$ are rational) | Cusp widths | $4^{6}\cdot20^{6}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.7.4 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&10\\22&41\end{bmatrix}$, $\begin{bmatrix}1&10\\40&51\end{bmatrix}$, $\begin{bmatrix}13&40\\18&41\end{bmatrix}$, $\begin{bmatrix}33&40\\22&31\end{bmatrix}$, $\begin{bmatrix}43&0\\54&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.7.ei.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{26}\cdot3^{12}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2\cdot4$ |
| Newforms: | 20.2.a.a, 720.2.x.a, 720.2.x.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y z - u v $ |
| $=$ | $y z - w t$ | |
| $=$ | $y z - y u + y v + v^{2}$ | |
| $=$ | $y z + z u + z v - u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 9 x^{4} y^{5} - 27 x^{4} y^{4} z - 27 x^{4} y^{3} z^{2} - 9 x^{4} y^{2} z^{3} + y^{7} z^{2} + \cdots + z^{9} $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1:0:0:0:0)$, $(0:1:0:0:0:0:0)$, $(0:0:-1:0:1:0:0)$, $(0:1:0:1:0:0:0)$, $(0:-1:0:0:0:0:1)$, $(0:0:1:0:0:1:0)$ |
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.f.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-Y^{3}Z+11Y^{2}Z^{2}+YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.7.ei.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -9X^{4}Y^{5}-27X^{4}Y^{4}Z-27X^{4}Y^{3}Z^{2}-9X^{4}Y^{2}Z^{3}+Y^{7}Z^{2}+Y^{6}Z^{3}-12Y^{5}Z^{4}+10Y^{4}Z^{5}+10Y^{3}Z^{6}-8Y^{2}Z^{7}-3YZ^{8}+Z^{9} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 12.12.0-2.a.1.2 | $12$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ | $2\cdot4$ |
| 60.96.3-60.f.2.5 | $60$ | $3$ | $3$ | $3$ | $0$ | $4$ |
| 60.144.1-10.a.1.9 | $60$ | $2$ | $2$ | $1$ | $0$ | $2\cdot4$ |
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.ey.1.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.ez.2.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fa.1.14 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fb.2.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fe.2.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.ff.1.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fg.2.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fh.1.14 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.864.31-60.s.1.31 | $60$ | $3$ | $3$ | $31$ | $0$ | $1^{6}\cdot2^{5}\cdot8$ |
| 60.1152.37-60.s.1.31 | $60$ | $4$ | $4$ | $37$ | $0$ | $1^{6}\cdot2^{4}\cdot4^{2}\cdot8$ |
| 60.1440.43-60.ct.1.16 | $60$ | $5$ | $5$ | $43$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{3}\cdot8$ |