Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $10^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.120.4.3 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&36\\12&19\end{bmatrix}$, $\begin{bmatrix}35&4\\2&45\end{bmatrix}$, $\begin{bmatrix}47&14\\2&1\end{bmatrix}$, $\begin{bmatrix}57&8\\8&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.60.4.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $18432$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{4}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 3600.2.a.bc, 3600.2.a.bf |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 6 x^{2} + 3 x y + 3 y^{2} - z w - w^{2} $ |
| $=$ | $3 x y^{2} - x z w - 3 y^{3} - y z^{2} - y z w + y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 108 x^{6} - 45 x^{4} y^{2} - 9 x^{4} y z + 54 x^{4} z^{2} - 6 x^{2} y^{4} - 9 x^{2} y^{3} z + \cdots + y^{2} z^{4} $ |
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 |
|---|
| $(0:0:-1:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{3324xyz^{8}-10104xyz^{7}w-51732xyz^{6}w^{2}+144984xyz^{5}w^{3}+708030xyz^{4}w^{4}+848415xyz^{3}w^{5}+160254xyz^{2}w^{6}-244140xyzw^{7}-97656xyw^{8}-10182y^{2}z^{8}-36042y^{2}z^{7}w+31908y^{2}z^{6}w^{2}+371148y^{2}z^{5}w^{3}+621840y^{2}z^{4}w^{4}+130347y^{2}z^{3}w^{5}-523338y^{2}z^{2}w^{6}-439452y^{2}zw^{7}-97656y^{2}w^{8}-2048z^{10}-10240z^{9}w-22486z^{8}w^{2}-35550z^{7}w^{3}-49244z^{6}w^{4}-30620z^{5}w^{5}+44575z^{4}w^{6}+101685z^{3}w^{7}+67842z^{2}w^{8}+15196zw^{9}-216w^{10}}{12xyz^{8}+24xyz^{6}w^{2}+144xyz^{5}w^{3}+120xyz^{4}w^{4}+60xyz^{3}w^{5}+6xyz^{2}w^{6}+15xyzw^{7}+6xyw^{8}-30y^{2}z^{8}-150y^{2}z^{7}w-222y^{2}z^{6}w^{2}-174y^{2}z^{5}w^{3}-90y^{2}z^{4}w^{4}+18y^{2}z^{3}w^{5}+48y^{2}z^{2}w^{6}+27y^{2}zw^{7}+6y^{2}w^{8}+2z^{8}w^{2}+6z^{7}w^{3}+26z^{6}w^{4}+54z^{5}w^{5}+50z^{4}w^{6}+10z^{3}w^{7}-17z^{2}w^{8}-11zw^{9}-2w^{10}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.60.4.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -108X^{6}-45X^{4}Y^{2}-9X^{4}YZ+54X^{4}Z^{2}-6X^{2}Y^{4}-9X^{2}Y^{3}Z+3X^{2}YZ^{3}-6X^{2}Z^{4}+Y^{3}Z^{3}+Y^{2}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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 12.24.0-12.a.1.2 | $12$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-12.a.1.2 | $12$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 20.60.2-10.a.1.1 | $20$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.60.2-10.a.1.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.240.8-60.a.1.2 | $60$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 60.240.8-60.a.1.3 | $60$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 60.240.8-60.c.1.4 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.8-60.c.1.7 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.8-60.g.1.3 | $60$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 60.240.8-60.g.1.7 | $60$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 60.240.8-60.h.1.1 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.8-60.h.1.2 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.360.10-60.a.1.1 | $60$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 60.360.14-60.c.1.1 | $60$ | $3$ | $3$ | $14$ | $6$ | $1^{10}$ |
| 60.480.13-60.y.1.1 | $60$ | $4$ | $4$ | $13$ | $5$ | $1^{9}$ |
| 60.480.17-60.c.1.2 | $60$ | $4$ | $4$ | $17$ | $4$ | $1^{13}$ |
| 120.240.8-120.b.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.b.1.14 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.f.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.f.1.14 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.o.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.o.1.14 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.r.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.r.1.14 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |