Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $720$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $4^{6}\cdot20^{6}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-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: | 20K7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.7.20 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&0\\28&13\end{bmatrix}$, $\begin{bmatrix}27&8\\28&5\end{bmatrix}$, $\begin{bmatrix}35&14\\52&23\end{bmatrix}$, $\begin{bmatrix}53&8\\0&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.144.7.r.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $4$ |
| Cyclic 60-torsion field degree: | $64$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{8}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 20.2.a.a$^{2}$, 40.2.a.a, 180.2.a.a, 360.2.a.a, 720.2.a.e, 720.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ w t - u v $ |
| $=$ | $x w - z u$ | |
| $=$ | $ - x v + z t$ | |
| $=$ | $x t - y v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} - x^{6} y^{2} - 3 x^{6} z^{2} - 6 x^{4} y^{2} z^{2} - 3 x^{2} y^{4} z^{2} - 9 x^{2} y^{2} z^{4} + 9 y^{4} z^{4} $ |
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 |
|---|
| $(0:0:0:0:-1:0:1)$, $(0:0:0:-1:0:1:0)$, $(0:0:0:0:1:0:1)$, $(0:0:0:1:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{12963906z^{2}v^{10}-15625w^{12}+187500w^{10}v^{2}+93750w^{8}v^{4}+1937500w^{6}v^{6}+23390625w^{4}v^{8}+31249w^{2}u^{10}-406243w^{2}u^{9}v+2311769w^{2}u^{8}v^{2}-8247827w^{2}u^{7}v^{3}+20332719w^{2}u^{6}v^{4}-39607693w^{2}u^{5}v^{5}+62947093w^{2}u^{4}v^{6}-80675519w^{2}u^{3}v^{7}+68932916w^{2}u^{2}v^{8}-24243108w^{2}uv^{9}+710166w^{2}v^{10}-t^{12}+12t^{10}v^{2}-810t^{8}v^{4}+12124t^{6}v^{6}-298551t^{4}v^{8}+4592904t^{2}v^{10}-15625u^{12}+156251u^{11}v-375003u^{10}v^{2}+719u^{9}v^{3}+766348u^{8}v^{4}-1579807u^{7}v^{5}+1756231u^{6}v^{6}+776391u^{5}v^{7}-6588188u^{4}v^{8}+34603848u^{3}v^{9}-14380680u^{2}v^{10}+11982150uv^{11}-4321303v^{12}}{v^{2}(2277z^{2}v^{8}-w^{2}u^{8}+3w^{2}u^{7}v+19w^{2}u^{6}v^{2}+15w^{2}u^{5}v^{3}-25w^{2}u^{4}v^{4}-5w^{2}u^{3}v^{5}+154w^{2}u^{2}v^{6}-318w^{2}uv^{7}+307w^{2}v^{8}-t^{8}v^{2}+16t^{6}v^{4}-136t^{4}v^{6}+880t^{2}v^{8}+u^{9}v+u^{8}v^{2}-15u^{7}v^{3}-55u^{6}v^{4}-95u^{5}v^{5}-94u^{4}v^{6}+754u^{3}v^{7}-1839u^{2}v^{8}+1849uv^{9}-759v^{10})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.7.r.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{8}-X^{6}Y^{2}-3X^{6}Z^{2}-6X^{4}Y^{2}Z^{2}-3X^{2}Y^{4}Z^{2}-9X^{2}Y^{2}Z^{4}+9Y^{4}Z^{4} $ |
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 |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.c.1.2 | $12$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.c.1.2 | $12$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 20.144.3-20.b.1.10 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.144.3-20.b.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.144.3-60.a.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.144.3-60.a.1.16 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.144.3-60.b.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{4}$ |
| 60.144.3-60.b.1.10 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.576.13-60.r.1.10 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.r.2.10 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.t.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.t.2.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.cd.1.4 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.cd.2.4 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.cf.1.1 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.576.13-60.cf.2.1 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 60.864.31-60.e.1.20 | $60$ | $3$ | $3$ | $31$ | $3$ | $1^{24}$ |
| 60.1152.37-60.e.1.15 | $60$ | $4$ | $4$ | $37$ | $4$ | $1^{30}$ |
| 60.1440.43-60.l.1.4 | $60$ | $5$ | $5$ | $43$ | $10$ | $1^{36}$ |