Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $1800$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.5.12 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&36\\14&25\end{bmatrix}$, $\begin{bmatrix}19&38\\28&39\end{bmatrix}$, $\begin{bmatrix}31&40\\40&11\end{bmatrix}$, $\begin{bmatrix}43&12\\16&37\end{bmatrix}$, $\begin{bmatrix}59&8\\4&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.5.bl.1 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: | $2^{13}\cdot3^{8}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 20.2.a.a, 360.2.f.c, 900.2.a.b, 1800.2.a.v |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y w - y t + w^{2} $ |
| $=$ | $x^{2} - x y + 2 x z - y z + z^{2} - t^{2}$ | |
| $=$ | $13 x^{2} + 2 x y - 4 x z + 2 y z - y t - 2 z^{2} + 3 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{4} y^{4} + 450 x^{3} y^{5} + 225 x^{2} y^{6} - 120 x^{2} y^{5} z - 450 x^{2} y^{4} z^{2} + \cdots + 361 z^{8} $ |
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 |
|---|
| $(0:1:1:0:0)$, $(0:-1:0:1:0)$, $(0:-1:-1:1:0)$, $(0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{6}+4y^{5}t-16yt^{5}+16t^{6})^{3}}{t^{10}y^{5}(y-t)^{2}(y+4t)}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.bl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 225X^{4}Y^{4}+450X^{3}Y^{5}+225X^{2}Y^{6}-120X^{2}Y^{5}Z-450X^{2}Y^{4}Z^{2}-660X^{2}Y^{3}Z^{3}-330X^{2}Y^{2}Z^{4}-120XY^{6}Z-450XY^{5}Z^{2}-660XY^{4}Z^{3}-330XY^{3}Z^{4}-60Y^{7}Z+16Y^{6}Z^{2}-19Y^{4}Z^{4}+988Y^{3}Z^{5}+2014Y^{2}Z^{6}+1444YZ^{7}+361Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-10.a.1.7 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.144.1-60.cj.2.4 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.144.1-60.cj.2.13 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.144.3-60.xu.2.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 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 |
|---|---|---|---|---|---|---|
| 60.576.9-60.t.1.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 60.576.9-60.t.3.3 | $60$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 60.576.9-60.v.2.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 60.576.9-60.v.4.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 60.576.13-60.cq.2.6 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.cr.1.15 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.cw.1.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.cx.1.13 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.er.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
| 60.576.13-60.ev.2.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
| 60.576.13-60.fd.2.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
| 60.576.13-60.fh.1.14 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
| 60.864.29-60.v.2.4 | $60$ | $3$ | $3$ | $29$ | $3$ | $1^{12}\cdot2^{6}$ |
| 60.1152.33-60.bh.2.8 | $60$ | $4$ | $4$ | $33$ | $5$ | $1^{14}\cdot2^{7}$ |
| 60.1440.37-60.m.1.12 | $60$ | $5$ | $5$ | $37$ | $6$ | $1^{16}\cdot2^{8}$ |