Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $720$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $11 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $12^{2}\cdot60^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 60D11 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.11.43 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}6&5\\31&27\end{bmatrix}$, $\begin{bmatrix}11&0\\51&37\end{bmatrix}$, $\begin{bmatrix}39&10\\1&33\end{bmatrix}$, $\begin{bmatrix}39&35\\34&33\end{bmatrix}$, $\begin{bmatrix}59&55\\22&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.11.ba.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{36}\cdot3^{16}\cdot5^{11}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2^{4}$ |
| Newforms: | 45.2.b.a, 80.2.a.a, 80.2.c.a, 720.2.a.a, 720.2.a.i, 720.2.f.a, 720.2.f.g |
Models
Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ x v - y t $ |
| $=$ | $y z + y w + z b - w a$ | |
| $=$ | $x^{2} - x v - x r - w s$ | |
| $=$ | $x^{2} + x u - x v + z s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{6} y^{9} + 240 x^{5} y^{10} + 36 x^{5} y^{7} z^{3} + 2 x^{5} y^{4} z^{6} + 3000 x^{4} y^{11} + \cdots + 100 z^{15} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:1:0:0:0)$, $(0:0:0:0:0:1:0:0:0:0:0)$ |
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 20.48.3.i.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 2z+3w-3t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z+4w+t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2z+2w-2t$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{3}Y-22X^{2}Y^{2}+6XY^{3}+12X^{2}YZ+14XY^{2}Z-6Y^{3}Z-3X^{2}Z^{2}-2XYZ^{2}+5Y^{2}Z^{2}-10YZ^{3}+2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.11.ba.1 :
| $\displaystyle X$ | $=$ | $\displaystyle b$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}v$ |
Equation of the image curve:
| $0$ | $=$ | $ 8X^{6}Y^{9}+240X^{5}Y^{10}+3000X^{4}Y^{11}+20000X^{3}Y^{12}+75000X^{2}Y^{13}+150000XY^{14}+125000Y^{15}+36X^{5}Y^{7}Z^{3}+1332X^{4}Y^{8}Z^{3}+15740X^{3}Y^{9}Z^{3}+81300X^{2}Y^{10}Z^{3}+186000XY^{11}Z^{3}+145000Y^{12}Z^{3}+2X^{5}Y^{4}Z^{6}+82X^{4}Y^{5}Z^{6}+2470X^{3}Y^{6}Z^{6}+24530X^{2}Y^{7}Z^{6}+89500XY^{8}Z^{6}+98400Y^{9}Z^{6}+X^{4}Y^{2}Z^{9}+90X^{3}Y^{3}Z^{9}+1640X^{2}Y^{4}Z^{9}+14000XY^{5}Z^{9}+28300Y^{6}Z^{9}+30X^{2}YZ^{12}+500XY^{2}Z^{12}+3200Y^{3}Z^{12}+100Z^{15} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 20.96.3-20.i.2.6 | $20$ | $3$ | $3$ | $3$ | $0$ | $1^{2}\cdot2^{3}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.96.3-20.i.2.6 | $20$ | $3$ | $3$ | $3$ | $0$ | $1^{2}\cdot2^{3}$ |
| 60.72.2-15.a.2.8 | $60$ | $4$ | $4$ | $2$ | $0$ | $1^{3}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.576.21-60.cs.1.2 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{8}\cdot2$ |
| 60.576.21-60.cu.1.9 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{8}\cdot2$ |
| 60.576.21-60.ew.1.1 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{8}\cdot2$ |
| 60.576.21-60.ey.1.9 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{8}\cdot2$ |
| 60.576.21-60.gw.1.2 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{8}\cdot2$ |
| 60.576.21-60.gy.2.6 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{8}\cdot2$ |
| 60.576.21-60.hm.2.1 | $60$ | $2$ | $2$ | $21$ | $5$ | $1^{8}\cdot2$ |
| 60.576.21-60.ho.2.1 | $60$ | $2$ | $2$ | $21$ | $6$ | $1^{8}\cdot2$ |
| 60.576.21-60.is.2.9 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.it.2.9 | $60$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.ja.1.1 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.jb.1.11 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.jk.1.6 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.jl.1.14 | $60$ | $2$ | $2$ | $21$ | $7$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.js.2.1 | $60$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.jt.2.5 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.21-60.ka.2.3 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.kb.2.1 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.ki.1.12 | $60$ | $2$ | $2$ | $21$ | $5$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.kj.1.4 | $60$ | $2$ | $2$ | $21$ | $7$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.kq.1.10 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.kr.1.1 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.ky.2.5 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{4}\cdot2^{3}$ |
| 60.576.21-60.kz.2.5 | $60$ | $2$ | $2$ | $21$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.864.31-60.jp.1.2 | $60$ | $3$ | $3$ | $31$ | $7$ | $1^{10}\cdot2^{5}$ |
| 60.1440.55-60.bbt.1.4 | $60$ | $5$ | $5$ | $55$ | $13$ | $1^{20}\cdot2^{12}$ |