Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $3600$ | ||
| Index: | $360$ | $\PSL_2$-index: | $180$ | ||||
| Genus: | $14 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $30^{2}\cdot60^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 60A14 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.360.14.312 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&26\\28&7\end{bmatrix}$, $\begin{bmatrix}15&59\\28&15\end{bmatrix}$, $\begin{bmatrix}19&44\\4&31\end{bmatrix}$, $\begin{bmatrix}31&19\\4&59\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.180.14.br.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{34}\cdot3^{24}\cdot5^{26}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{14}$ |
| Newforms: | 36.2.a.a, 50.2.a.b$^{2}$, 225.2.a.c$^{3}$, 900.2.a.f, 3600.2.a.b, 3600.2.a.e, 3600.2.a.l, 3600.2.a.m, 3600.2.a.w, 3600.2.a.x, 3600.2.a.y |
Models
Canonical model in $\mathbb{P}^{ 13 }$ defined by 66 equations
| $ 0 $ | $=$ | $ y d - s c $ |
| $=$ | $x e - t b$ | |
| $=$ | $v^{2} - v r + c d$ | |
| $=$ | $x d + y r - u c$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 960 x^{8} y^{2} + 5625 x^{4} y^{4} z^{2} - 50625 x^{2} y^{8} - 150 x^{2} y^{2} z^{6} + z^{10} $ |
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 |
|---|
| $(0:0:0:0:0:0:0:1:0:0:0:0:0:0)$, $(0:0:0:0:0:0:-1:-1:0:1:0:0:0:0)$ |
Maps to other modular curves
Map of degree 6 from the canonical model of this modular curve to the canonical model of the modular curve 60.30.3.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4z+v-r-a-2c$ |
| $\displaystyle Z$ | $=$ | $\displaystyle c$ |
Equation of the image curve:
| $0$ | $=$ | $ 225X^{4}-64X^{3}Z-48X^{2}YZ-12XY^{2}Z-Y^{3}Z+69X^{2}Z^{2}-48XYZ^{2}-6Y^{2}Z^{2}-48XZ^{3}-12YZ^{3}+56Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.180.14.br.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{4}{15}c$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2y$ |
Equation of the image curve:
| $0$ | $=$ | $ -960X^{8}Y^{2}+5625X^{4}Y^{4}Z^{2}-50625X^{2}Y^{8}-150X^{2}Y^{2}Z^{6}+Z^{10} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.72.2-60.x.1.5 | $60$ | $5$ | $5$ | $2$ | $0$ | $1^{12}$ |
| 60.120.4-60.p.1.4 | $60$ | $3$ | $3$ | $4$ | $1$ | $1^{10}$ |
| 60.180.7-60.c.1.3 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{7}$ |
| 60.180.7-60.c.1.14 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{7}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.720.27-60.jq.1.1 | $60$ | $2$ | $2$ | $27$ | $10$ | $1^{13}$ |
| 60.720.27-60.jr.1.4 | $60$ | $2$ | $2$ | $27$ | $13$ | $1^{13}$ |
| 60.720.27-60.kg.1.3 | $60$ | $2$ | $2$ | $27$ | $9$ | $1^{13}$ |
| 60.720.27-60.kh.1.1 | $60$ | $2$ | $2$ | $27$ | $10$ | $1^{13}$ |
| 60.720.27-60.kw.1.2 | $60$ | $2$ | $2$ | $27$ | $11$ | $1^{13}$ |
| 60.720.27-60.kx.1.1 | $60$ | $2$ | $2$ | $27$ | $7$ | $1^{13}$ |
| 60.720.27-60.lm.1.1 | $60$ | $2$ | $2$ | $27$ | $9$ | $1^{13}$ |
| 60.720.27-60.ln.1.2 | $60$ | $2$ | $2$ | $27$ | $11$ | $1^{13}$ |
| 60.1080.40-60.cx.1.4 | $60$ | $3$ | $3$ | $40$ | $13$ | $1^{26}$ |
| 60.1440.53-60.bug.1.5 | $60$ | $4$ | $4$ | $53$ | $20$ | $1^{31}\cdot2^{4}$ |