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: | 60.120.4.78 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&12\\16&11\end{bmatrix}$, $\begin{bmatrix}31&4\\33&19\end{bmatrix}$, $\begin{bmatrix}55&24\\46&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.60.4.p.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| 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.l, 3600.2.a.m |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 15 x^{2} + 105 y^{2} + z^{2} - w^{2} $ |
| $=$ | $15 x^{2} y - x z w - 15 y^{3} + y z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 60 x^{4} z^{2} + 105 x^{2} y^{2} z^{2} - 4 x^{2} z^{4} + 900 y^{6} - 60 y^{4} z^{2} + 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 |
|---|
| $(0:0:1:1)$, $(0:0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{63849600xyz^{7}w-1381235655xyz^{5}w^{3}+1728532650xyz^{3}w^{5}-243768435xyzw^{7}-16588800y^{2}z^{8}+1815232200y^{2}z^{6}w^{2}-6911923125y^{2}z^{4}w^{4}+3375634830y^{2}z^{2}w^{6}-251119185y^{2}w^{8}-13824z^{10}+8089280z^{8}w^{2}-56985708z^{6}w^{4}+76767920z^{4}w^{6}-29922052z^{2}w^{8}+2097152w^{10}}{5400xyz^{7}w-2910xyz^{5}w^{3}-300xyz^{3}w^{5}+210xyzw^{7}+4800y^{2}z^{8}-12375y^{2}z^{6}w^{2}+3375y^{2}z^{4}w^{4}-105y^{2}z^{2}w^{6}-15y^{2}w^{8}+64z^{10}-135z^{8}w^{2}+49z^{6}w^{4}+35z^{4}w^{6}-13z^{2}w^{8}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.60.4.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 60X^{4}Z^{2}+105X^{2}Y^{2}Z^{2}-4X^{2}Z^{4}+900Y^{6}-60Y^{4}Z^{2}+Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.60.2-20.c.1.1 | $20$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.60.2-20.c.1.4 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.24.0-60.h.1.4 | $60$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.360.10-60.x.1.3 | $60$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 60.360.14-60.br.1.3 | $60$ | $3$ | $3$ | $14$ | $6$ | $1^{10}$ |
| 60.480.13-60.gk.1.4 | $60$ | $4$ | $4$ | $13$ | $4$ | $1^{9}$ |
| 60.480.17-60.bb.1.4 | $60$ | $4$ | $4$ | $17$ | $4$ | $1^{13}$ |
| 120.240.8-120.fe.1.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fe.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ff.1.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ff.1.17 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fi.1.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fi.1.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fj.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fj.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fy.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fy.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fz.1.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fz.1.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gc.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gc.1.13 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gd.1.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gd.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |