Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $1800$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $9 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $10\cdot20\cdot30\cdot60$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 60B9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.9.308 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}23&21\\28&19\end{bmatrix}$, $\begin{bmatrix}29&0\\20&43\end{bmatrix}$, $\begin{bmatrix}31&48\\58&1\end{bmatrix}$, $\begin{bmatrix}43&12\\28&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.120.9.dm.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{17}\cdot3^{12}\cdot5^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 1800.2.a.a, 1800.2.a.m, 1800.2.a.r$^{2}$, 1800.2.a.t |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v - y t + z t $ |
| $=$ | $t^{2} + t u + t s + v r$ | |
| $=$ | $x r + x s + y t - z r$ | |
| $=$ | $x t + x u - x r - y t + y r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} z^{10} - 300 x^{3} y^{4} z^{7} + 260 x^{3} y^{2} z^{9} - 32 x^{3} z^{11} - 28125 x^{2} y^{8} z^{4} + \cdots + 64 z^{14} $ |
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 |
|---|
| $(0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:0:-1:0:2:1)$, $(0:0:0:0:0:-1:-2:0:1)$, $(0:0:0:0:0:-1:0:-1:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 30.60.4.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x-y-z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-4XY+XZ+3YZ-XW+2W^{2} $ |
| $=$ | $ X^{3}-X^{2}Y+X^{2}Z-2XYZ-Y^{2}Z+YZ^{2}-2X^{2}W-XYW-XZW+XW^{2}+ZW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.120.9.dm.1 :
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 18984375Y^{14}+1265625XY^{12}Z-18984375Y^{12}Z^{2}-1518750XY^{10}Z^{3}-28125X^{2}Y^{8}Z^{4}+7931250Y^{10}Z^{4}+523125XY^{8}Z^{5}+1500X^{2}Y^{6}Z^{6}-1653750Y^{8}Z^{6}-300X^{3}Y^{4}Z^{7}-51750XY^{6}Z^{7}+4825X^{2}Y^{4}Z^{8}+148875Y^{6}Z^{8}+260X^{3}Y^{2}Z^{9}-9850XY^{4}Z^{9}+4X^{4}Z^{10}-1380X^{2}Y^{2}Z^{10}+3025Y^{4}Z^{10}-32X^{3}Z^{11}+2400XY^{2}Z^{11}+96X^{2}Z^{12}-1360Y^{2}Z^{12}-128XZ^{13}+64Z^{14} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.48.1-60.v.1.9 | $60$ | $5$ | $5$ | $1$ | $1$ | $1^{8}$ |
| 60.120.4-30.b.1.2 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 60.120.4-30.b.1.5 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.480.17-60.h.1.14 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 60.480.17-60.l.1.6 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 60.480.17-60.z.1.3 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 60.480.17-60.ba.1.6 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 60.480.17-60.ie.1.2 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 60.480.17-60.if.1.2 | $60$ | $2$ | $2$ | $17$ | $9$ | $1^{8}$ |
| 60.480.17-60.ii.1.7 | $60$ | $2$ | $2$ | $17$ | $9$ | $1^{8}$ |
| 60.480.17-60.ij.1.4 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 60.720.25-60.uz.1.8 | $60$ | $3$ | $3$ | $25$ | $6$ | $1^{16}$ |
| 60.720.27-60.rp.1.6 | $60$ | $3$ | $3$ | $27$ | $8$ | $1^{18}$ |
| 60.960.33-60.pi.1.16 | $60$ | $4$ | $4$ | $33$ | $8$ | $1^{24}$ |
| 120.480.17-120.js.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ql.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bqt.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bqw.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dtv.1.5 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dty.1.5 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.duh.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.duk.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |