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: | 60.288.5.958 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}31&40\\42&11\end{bmatrix}$, $\begin{bmatrix}53&50\\56&31\end{bmatrix}$, $\begin{bmatrix}59&55\\12&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.5.im.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: | $2^{18}\cdot3^{8}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 180.2.a.a, 400.2.a.c, 720.2.f.e, 3600.2.a.h |
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 $ | $=$ | $ 9 x^{4} z^{4} - 36 x^{3} z^{5} - 75 x^{2} y^{4} z^{2} + 39 x^{2} z^{6} + 150 x y^{4} z^{3} + \cdots + 19 z^{8} $ |
Rational points
This modular curve has no real points, 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(z^{6}+4z^{5}t-16zt^{5}+16t^{6})^{3}}{t^{10}z^{5}(z-t)^{2}(z+4t)}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.im.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:
| $0$ | $=$ | $ 9X^{4}Z^{4}-36X^{3}Z^{5}-75X^{2}Y^{4}Z^{2}+39X^{2}Z^{6}+150XY^{4}Z^{3}-6XZ^{7}+625Y^{8}+750Y^{6}Z^{2}+400Y^{4}Z^{4}+150Y^{2}Z^{6}+19Z^{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.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-60.ba.2.6 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-60.ba.2.13 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.3-60.xw.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.xw.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.bqi.2.9 | $60$ | $3$ | $3$ | $29$ | $4$ | $1^{12}\cdot2^{6}$ |
| 60.1152.33-60.in.2.7 | $60$ | $4$ | $4$ | $33$ | $1$ | $1^{14}\cdot2^{7}$ |
| 60.1440.37-60.jb.1.3 | $60$ | $5$ | $5$ | $37$ | $5$ | $1^{16}\cdot2^{8}$ |