LMFDB - Modular curve 60.216.13.cx.1

Invariants

Level:	$60$	$\SL_2$-level:	$60$	Newform level:	$180$
Index:	$216$	$\PSL_2$-index:	$216$
Genus:	$13 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $2$ are rational)	Cusp widths	$3^{4}\cdot12^{2}\cdot15^{4}\cdot60^{2}$	Cusp orbits	$1^{2}\cdot2^{3}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$6$
$\overline{\Q}$-gonality:	$6$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	60N13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.216.13.300

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}9&5\\50&13\end{bmatrix}$, $\begin{bmatrix}19&35\\30&23\end{bmatrix}$, $\begin{bmatrix}31&5\\56&59\end{bmatrix}$, $\begin{bmatrix}31&35\\56&47\end{bmatrix}$, $\begin{bmatrix}41&45\\46&13\end{bmatrix}$, $\begin{bmatrix}51&25\\20&21\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.432.13-60.cx.1.1, 60.432.13-60.cx.1.2, 60.432.13-60.cx.1.3, 60.432.13-60.cx.1.4, 60.432.13-60.cx.1.5, 60.432.13-60.cx.1.6, 60.432.13-60.cx.1.7, 60.432.13-60.cx.1.8, 60.432.13-60.cx.1.9, 60.432.13-60.cx.1.10, 60.432.13-60.cx.1.11, 60.432.13-60.cx.1.12, 60.432.13-60.cx.1.13, 60.432.13-60.cx.1.14, 60.432.13-60.cx.1.15, 60.432.13-60.cx.1.16, 60.432.13-60.cx.1.17, 60.432.13-60.cx.1.18, 60.432.13-60.cx.1.19, 60.432.13-60.cx.1.20, 60.432.13-60.cx.1.21, 60.432.13-60.cx.1.22, 60.432.13-60.cx.1.23, 60.432.13-60.cx.1.24, 60.432.13-60.cx.1.25, 60.432.13-60.cx.1.26, 60.432.13-60.cx.1.27, 60.432.13-60.cx.1.28, 60.432.13-60.cx.1.29, 60.432.13-60.cx.1.30, 60.432.13-60.cx.1.31, 60.432.13-60.cx.1.32, 120.432.13-60.cx.1.1, 120.432.13-60.cx.1.2, 120.432.13-60.cx.1.3, 120.432.13-60.cx.1.4, 120.432.13-60.cx.1.5, 120.432.13-60.cx.1.6, 120.432.13-60.cx.1.7, 120.432.13-60.cx.1.8, 120.432.13-60.cx.1.9, 120.432.13-60.cx.1.10, 120.432.13-60.cx.1.11, 120.432.13-60.cx.1.12, 120.432.13-60.cx.1.13, 120.432.13-60.cx.1.14, 120.432.13-60.cx.1.15, 120.432.13-60.cx.1.16, 120.432.13-60.cx.1.17, 120.432.13-60.cx.1.18, 120.432.13-60.cx.1.19, 120.432.13-60.cx.1.20, 120.432.13-60.cx.1.21, 120.432.13-60.cx.1.22, 120.432.13-60.cx.1.23, 120.432.13-60.cx.1.24, 120.432.13-60.cx.1.25, 120.432.13-60.cx.1.26, 120.432.13-60.cx.1.27, 120.432.13-60.cx.1.28, 120.432.13-60.cx.1.29, 120.432.13-60.cx.1.30, 120.432.13-60.cx.1.31, 120.432.13-60.cx.1.32
Cyclic 60-isogeny field degree:	$8$
Cyclic 60-torsion field degree:	$128$
Full 60-torsion field degree:	$10240$

Jacobian

Conductor:	$2^{10}\cdot3^{26}\cdot5^{11}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{7}\cdot2^{3}$
Newforms:	36.2.a.a$^{2}$, 45.2.b.a$^{3}$, 90.2.a.a$^{2}$, 90.2.a.b$^{2}$, 180.2.a.a

Models

Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations

$ 0 $	$=$	$ x r + t s $
	$=$	$x v - w s$
	$=$	$w r + t v$
	$=$	$y w - y a - u b$
	$=$	$\cdots$

[x*r + t*s, x*v - w*s, w*r + t*v, y*w - y*a - u*b, x*u + y*c - u*b - u*d, x*y - z*s - u*s, y*w - z*v - u*v, x*w - x*a - z*b, y*w - y*t - z*b, x*u - y*z, y*t + z*r + u*r, z*a - u*a + r*s - a*c, z*w + z*t + w*u - w*c - u*a, x*v + x*r - x*b + w*s - s*a, x*w + x*t + x*a + u*b - a*b - a*d - b*c, x*r - x*b + y*b - t*s + r*b + r*d, x*v - x*r + y*b - v*b - v*d + s*a, y*t + z*v - u*v + u*b - v*c, x*w + x*t + t*b + t*d - u*b, x*z - y*u - t*r - r*a + a*b, z*w - z*a - w*u + t*u, w*r - t*v - t*r + t*b - r*a, x*z - y*u - t*v + t*r - t*b + r*a, z*b - w*b - w*d + a*b + a*d + b*c, x*z + x*c - z*b - z*d, y*v + y*r - w*s + s*a, y*a - 2*z*r + u*r - r*c, x*z - y*z - t*v + t*r - t*b - u*b - u*d - 2*r*a + s^2, x*z - x*u - 2*y*z + y*c - u*b - u*d + s^2, y^2 + 2*z*s - u*s + 3*v^2 + s*c, y^2 + z*s - u*s + 3*r^2 - s*c, x*y + y*b + y*d + z*s + u*s + 3*v*b, 2*y^2 + y*b + y*d - v*r + v*b - 2*r*b + b^2, x*y - y^2 - 2*v*r + 2*v*b + 2*r*b - b^2, 2*x*y + 2*y^2 + z*s + u*s - v*r + v*b + r*b + b^2, z*w + 2*z*t - w*u + t*c + u*a, z*w - 2*z*a + w*c - t*u - v*s - a*c, z*t + w*u + w*c + t*u + t*c + u*a - v*s + r*s - a*c, z*t - z*a + w*u + w*c + t*u + t*c - u*a + v*s - s*b + a*c, x*z - x*c + y*u - t*v - t*r - t*b + 2*r*a - a*b, y*w + y*t + y*a - z*v - z*r + u*v - u*r + v*c - b*c, x*u + y*z + w*v - w*r + w*b + u*b + u*d - 2*v*a, 2*w*v + w*r - w*b - v*a, x*z + y*u + z*b + z*d + w*v - w*r + w*b + v*a, y*z - w*v + w*r - w*b + t*r + u*b + u*d - v*a + r*a - s^2 + 2*a*b - b*c - c*d, x^2 - x*y + x*b + x*d + y^2 + 2*v*r + v*b + r*b - 3*b^2 - 2*b*d - d^2, y*s + 2*z^2 - 3*z*u - z*c + u^2 - c^2, x*s - 2*z^2 - z*u - z*c + 3*t*a + u^2 - u*c, x*s - y*s - 2*z^2 - 2*z*u + z*c + w*a - t^2 + u*c - 2*a^2, 3*w*t + 2*w*a + t^2 - a^2, y*s - 2*z^2 - z*u - z*c + 3*w*t - w*a + t^2 + u^2 - u*c + 2*a^2, 3*y*v - 2*y*r + v*b + v*d + r*b + r*d - b^2 - b*d, y*s + z^2 - 3*z*u - 4*u^2, x*s - y*s + z^2 + z*u + 2*z*c + 3*w^2 - 2*w*a - t^2 + 2*u*c - 2*a^2, x*s + y*s - z*u + 2*z*c + w*a - t^2 + 3*t*a - u^2 + 2*u*c - s*b - s*d + a^2]

Singular plane model Singular plane model

$ 0 $

$=$

$ 284765625 x^{16} z^{2} - 607500 x^{14} y^{3} z + 151875000 x^{14} z^{4} - 81 x^{12} y^{6} + \cdots + 24 z^{18} $

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
$(0:0:-1/3:0:0:1/3:0:0:0:0:0:1:0)$, $(0:0:1/2:0:0:-1/2:0:0:0:0:0:1:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle 25c$
$\displaystyle Z$	$=$	$\displaystyle 5y$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 30.108.6.a.1 :

$\displaystyle X$	$=$	$\displaystyle -t$
$\displaystyle Y$	$=$	$\displaystyle -w$
$\displaystyle Z$	$=$	$\displaystyle w+t-a$
$\displaystyle W$	$=$	$\displaystyle -v+r$
$\displaystyle T$	$=$	$\displaystyle -v$
$\displaystyle U$	$=$	$\displaystyle -v+b$

Equation of the image curve:

$0$	$=$	$ YW-XT-YT $
	$=$	$ YW-ZT+YU-ZU $
	$=$	$ YW+XT-YT+ZT-YU $
	$=$	$ 2XW+ZW-XU-YU $
	$=$	$ 3XY+Y^{2}-2XZ-Z^{2} $
	$=$	$ 3WT-2T^{2}+2WU-2TU-U^{2} $
	$=$	$ X^{2}Z+XYZ+2XZ^{2}+YZ^{2}+Z^{3}-W^{2}T+WTU+2WU^{2}-TU^{2}-U^{3} $
	$=$	$ XYZ+Y^{2}Z-XZ^{2}-2YZ^{2}-2W^{2}T+2WT^{2}-WTU+TU^{2} $
	$=$	$ 3X^{3}+4X^{2}Y+XY^{2}+4X^{2}Z+XYZ+Y^{2}Z-3YZ^{2}-Z^{3}-WT^{2}+2T^{3}-W^{2}U+WTU-2T^{2}U-2WU^{2}+2TU^{2}+U^{3} $

Modular covers

Cover information

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
5.12.0.a.2	$5$	$18$	$18$	$0$	$0$	full Jacobian
12.18.1.f.1	$12$	$12$	$12$	$1$	$0$	$1^{6}\cdot2^{3}$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
30.108.6.a.1	$30$	$2$	$2$	$6$	$0$	$1^{5}\cdot2$
60.72.1.ba.1	$60$	$3$	$3$	$1$	$0$	$1^{6}\cdot2^{3}$
60.108.4.c.1	$60$	$2$	$2$	$4$	$0$	$1^{5}\cdot2^{2}$
60.108.7.f.1	$60$	$2$	$2$	$7$	$0$	$2^{3}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.432.25.fw.1	$60$	$2$	$2$	$25$	$0$	$1^{6}\cdot2^{3}$
60.432.25.fx.1	$60$	$2$	$2$	$25$	$2$	$1^{6}\cdot2^{3}$
60.432.25.hh.1	$60$	$2$	$2$	$25$	$0$	$1^{6}\cdot2^{3}$
60.432.25.hi.1	$60$	$2$	$2$	$25$	$4$	$1^{6}\cdot2^{3}$
60.432.25.is.1	$60$	$2$	$2$	$25$	$1$	$1^{6}\cdot2^{3}$
60.432.25.it.1	$60$	$2$	$2$	$25$	$4$	$1^{6}\cdot2^{3}$
60.432.25.jh.1	$60$	$2$	$2$	$25$	$0$	$1^{6}\cdot2^{3}$
60.432.25.ji.1	$60$	$2$	$2$	$25$	$1$	$1^{6}\cdot2^{3}$
60.432.29.f.1	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.nl.1	$60$	$2$	$2$	$29$	$4$	$1^{8}\cdot2^{4}$
60.432.29.bjk.1	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.bjl.1	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.bqa.1	$60$	$2$	$2$	$29$	$2$	$1^{8}\cdot2^{4}$
60.432.29.bqi.1	$60$	$2$	$2$	$29$	$4$	$1^{8}\cdot2^{4}$
60.432.29.brs.1	$60$	$2$	$2$	$29$	$1$	$1^{8}\cdot2^{4}$
60.432.29.brt.2	$60$	$2$	$2$	$29$	$3$	$1^{8}\cdot2^{4}$
60.432.29.bxs.1	$60$	$2$	$2$	$29$	$6$	$1^{8}\cdot2^{4}$
60.432.29.bxt.1	$60$	$2$	$2$	$29$	$2$	$1^{8}\cdot2^{4}$
60.432.29.byy.1	$60$	$2$	$2$	$29$	$3$	$1^{8}\cdot2^{4}$
60.432.29.bza.1	$60$	$2$	$2$	$29$	$6$	$1^{8}\cdot2^{4}$
60.432.29.cau.1	$60$	$2$	$2$	$29$	$2$	$1^{8}\cdot2^{4}$
60.432.29.cav.2	$60$	$2$	$2$	$29$	$0$	$1^{8}\cdot2^{4}$
60.432.29.cbo.1	$60$	$2$	$2$	$29$	$0$	$1^{8}\cdot2^{4}$
60.432.29.cbu.1	$60$	$2$	$2$	$29$	$4$	$1^{8}\cdot2^{4}$
60.1080.73.oz.1	$60$	$5$	$5$	$73$	$7$	$1^{24}\cdot2^{18}$

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