Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.5.959 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&25\\20&59\end{bmatrix}$, $\begin{bmatrix}21&20\\50&41\end{bmatrix}$, $\begin{bmatrix}23&40\\48&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.5.ie.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $64$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{17}\cdot3^{8}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 360.2.f.c, 400.2.a.c, 720.2.a.h, 1800.2.a.v |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ z w - w^{2} + t^{2} $ |
| $=$ | $3 x^{2} - z t - t^{2}$ | |
| $=$ | $5 y^{2} + z^{2} - 4 z t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 36 x^{4} z^{4} + 144 x^{3} z^{5} + 600 x^{2} y^{4} z^{2} + 360 x^{2} y^{2} z^{4} - 96 x^{2} z^{6} + \cdots + 59 z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(z^{6}-4z^{5}t+16zt^{5}+16t^{6})^{3}}{t^{10}z^{5}(z-4t)(z+t)^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.ie.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ -36X^{4}Z^{4}+144X^{3}Z^{5}+600X^{2}Y^{4}Z^{2}+360X^{2}Y^{2}Z^{4}-96X^{2}Z^{6}-1200XY^{4}Z^{3}-720XY^{2}Z^{5}-96XZ^{7}-625Y^{8}+350Y^{4}Z^{4}+360Y^{2}Z^{6}+59Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.144.1-20.i.1.1 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-20.i.1.7 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-60.y.2.6 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-60.y.2.13 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.3-60.xu.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.xu.2.15 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.864.29-60.bps.2.9 | $60$ | $3$ | $3$ | $29$ | $2$ | $1^{12}\cdot2^{6}$ |
| 60.1152.33-60.if.2.7 | $60$ | $4$ | $4$ | $33$ | $5$ | $1^{14}\cdot2^{7}$ |
| 60.1440.37-60.it.1.1 | $60$ | $5$ | $5$ | $37$ | $7$ | $1^{16}\cdot2^{8}$ |