LMFDB - Modular curve 60.180.13.nr.1

Invariants

Level:	$60$	$\SL_2$-level:	$60$	Newform level:	$3600$
Index:	$180$	$\PSL_2$-index:	$180$
Genus:	$13 = 1 + \frac{ 180 }{12} - \frac{ 6 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps:	$3$ (of which $1$ is rational)	Cusp widths	$60^{3}$	Cusp orbits	$1\cdot2$
Elliptic points:	$6$ of order $2$ and $0$ of order $3$
Analytic rank:	$7$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$1$
Rational CM points:	yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label:	60B13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.180.13.318

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}7&43\\32&37\end{bmatrix}$, $\begin{bmatrix}9&31\\28&3\end{bmatrix}$, $\begin{bmatrix}41&49\\26&11\end{bmatrix}$, $\begin{bmatrix}47&13\\14&29\end{bmatrix}$, $\begin{bmatrix}49&43\\22&11\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 60-isogeny field degree:	$48$
Cyclic 60-torsion field degree:	$768$
Full 60-torsion field degree:	$12288$

Jacobian

Conductor:	$2^{27}\cdot3^{22}\cdot5^{26}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{13}$
Newforms:	50.2.a.b, 225.2.a.c$^{2}$, 225.2.a.d, 400.2.a.a, 450.2.a.g, 900.2.a.d, 900.2.a.g, 1800.2.a.k, 1800.2.a.l, 1800.2.a.r, 3600.2.a.p, 3600.2.a.r

Models

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

$ 0 $	$=$	$ 2 x y - y u - y v + y a - z r + u s $
	$=$	$x w + x r + x b + z u + 2 z v - w u - u b + v r + v b - r a - s^{2} + s c + a b$
	$=$	$x b + y t - 2 y s + u r + v r + v b - r a - s^{2} + s c + a b$
	$=$	$y w + y r + 2 y b + w s + t r$
	$=$	$\cdots$

[2*x*y - y*u - y*v + y*a - z*r + u*s, x*w + x*r + x*b + z*u + 2*z*v - w*u - u*b + v*r + v*b - r*a - s^2 + s*c + a*b, x*b + y*t - 2*y*s + u*r + v*r + v*b - r*a - s^2 + s*c + a*b, y*w + y*r + 2*y*b + w*s + t*r, 2*y*r - w*t + w*s + w*c, y*w + y*r - y*d + w*c + t*b, x*r + y*t - y*s - y*c, x*u + 2*x*v - y*w - y*b + w*c + t*b, 2*y*w + y*r - y*d - w*c - t*b - u*a, z*a + w*v + w*a + v*r + v*b - r*a - s^2 + s*c + a*b, x*w + x*r + x*b + y*s - y*c - z*a, x*u + x*v + x*a - y*z + y*r + y*b + z*s + z*c, x*v - x*a + y*w + y*r - z*t + z*s - z*c + w*s + t*r, 2*x*a - y*w - y*b + w*t + w*s + t*r - t*d, 2*x*a + y*w + y*b + w*t - t*b - t*d + u*v + 2*v^2 + v*a + r*s + r*c - s*b + a^2 + b*c, x*v + x*a + y*z + y*b - y*d - z*t + z*s - z*c + w*t - w*s + t*r - t*d + u^2 - u*v + v^2 + r*s + s*b + a^2 + b*c, y*r + y*b + w*c + t*r - t*b - t*d + u*a + v*a + s*b + 2*s*d - a^2 + b*c, x*a + y*z + y*w + y*r + y*b + y*d + 2*w*t - t*r - t*b + u^2 - u*v + v^2 + r*s + s*b - s*d + a^2, x*v + x*a + y*z + y*r - y*b + y*d - z*s - z*c + w*t - w*s + w*c - t*d - u^2 - 2*u*v, x*v - x*a + y*z + y*d - z*t + w*t - t*r + t*b + t*d - 2*u^2 - s*b - s*d - b*c, x*v + x*a - y*r + y*b + y*d - z*t - z*s + z*c, x*r - y^2 - y*s + y*c - w*u - w*v - u*b, x*w + 2*y^2, x*y - x*t + 2*x*s + y*v + y*a - z*r - t*a + u*s, y*u + y*v - y*a + w^2 + w*r + w*b + t*u + t*v - u*s, y*v + 3*y*a + t*v + u*s, 2*x*y + z*w + z*r - w^2 - w*r - w*b - t*v, x*w + 2*x*b - y*t - y*s + y*c - t^2 + t*s + t*c, x*r - y^2 - z*a - w*a + t*s - t*c + u*r + v*r + v*b - r*a - s^2 + s*c + a*b, x*u + y*z - y*r + y*d + z*t + z*s + z*c - w*s + u*a - v^2 - r*s + s*d - a^2, x*w + x*b + y^2 + y*t + y*s - y*c + z*a - w*u - w*a + t^2 - t*s - t*c + u*d, x*w + x*r + x*b - y*s - w*u - w*v - t*c + u*d + v*d + r*a + s^2 + s*c + 2*a*b, y*z + 2*y*r + y*b + y*d + w*t - t*b + 2*u^2 + u*v - u*a + v^2 - v*a - r*s + r*c - 2*s*b, x*w + x*r + y*s + y*c + z*a + 2*w*u + t^2 - t*s - t*c - 2*u*b + v*r - v*d - r*a - s^2 + s*c, 2*x*w - x*b - y^2 + y*t - y*c + z*u + z*a + t^2 + t*s - r*a - 2*s^2, x*w + x*b - y*s - z*u + 2*z*v + w*v - 2*t^2 - t*s + v*r + v*b + s^2 + s*c + a*b, 2*x*y + y*v - y*a - z*r - w^2 + w*d - 2*t*a + u*c, 2*x*y + x*s - x*c + z*w - z*b - z*d + w^2 - w*d - t*v, x*s - x*c + z*w - z*r + z*b - z*d + w^2 + w*r + w*b - t*a, x*s - x*c - y*v + y*a + z*w + z*r - z*b - z*d + t*u + t*v - u*s, x*t - 2*y*u - 2*z^2, y*u + y*v - y*a + w^2 - w*r + w*b - t*v + u*c + 2*v*s + 2*v*c - r^2 - r*b + r*d, y*u + y*v - y*a - 2*w^2 + w*r + w*b - w*d - t*v + t*a - u*s + u*c + 2*v*c - s*a - a*c - b^2 - b*d, x*y + x*t - 2*x*s - w*r - 2*w*b + u*s + 2*v*s - r^2 - 2*r*b + s*a + a*c - b^2 + b*d, y*v + y*a + 2*w^2 + w*r - w*b + w*d + t*v - t*a - u*s + u*c - 2*v*s + r^2 - r*d - 2*s*a, x*r + x*b - y^2 - y*s + 2*z*u + z*a + 2*w*v + 2*t^2 - t*c + v*r + v*b - v*d + s^2 - a*b - a*d + c^2, x*y + x*s - x*c + 2*y*a - 2*z*w + z*b - z*d + w*r - w*b - 3*t*u + t*v - t*a - r*b - r*d - b^2, x*t + y*u - 2*z^2 - z*r + z*b - w^2 - w*r + w*b + w*d + 2*t*u - 2*t*v - t*a + u*s + u*c + v*c + r*b + r*d - s*a + b^2, x*y + x*t - x*s + x*c + y*u - y*v - y*a - 2*z^2 + 2*z*w - 2*z*b + z*d + 2*w^2 - w*r + w*d + u*c + v*s - s*a + b^2, 2*x*w - x*r - y^2 + y*s + 4*z*u - 3*t^2 - t*s - a*b, 4*x*u - 3*x*v - y*z + y*b + y*d - 3*z*t - z*s - z*c + v^2 - v*a - s*b - s*d, x*w + 3*y*t + 2*y*s + 3*y*c + z*u + w*a - t^2 + 3*t*s + t*c - u*r + v*d + a*b, 8*x*t + 6*x*s + 2*x*c + y*v - y*a + 6*z^2 - z*w + 2*z*d + 2*t*u + t*a - u*c + v*s + v*c - s*a - a*c, 30*x*z - x*w - x*r - x*b + y*s - w*v + t*c + v*d - r*a - s^2 - s*c + a*d, 30*x^2 - x*a + y*z - y*w - y*b + w*t + w*c + t*b + t*d + 2*u^2 - v^2 - r*s - a^2 - b*c - c*d]

Rational points

This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:0:0:0:0:0:0:0:0:1)$

Maps to other modular curves

Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve 30.45.3.a.1 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle -z$
$\displaystyle Z$	$=$	$\displaystyle -y-z+u$

Equation of the image curve:

$0$

$=$

$ 7X^{4}-6X^{3}Y+5X^{2}Y^{2}-5XY^{3}-Y^{4}-3X^{3}Z-3X^{2}YZ-5XY^{2}Z+2Y^{3}Z-3X^{2}Z^{2}+3XYZ^{2}-Y^{2}Z^{2}-XZ^{3} $

Modular covers

Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.36.1.ft.1	$60$	$5$	$5$	$1$	$0$	$1^{12}$
60.60.4.cj.1	$60$	$3$	$3$	$4$	$1$	$1^{9}$
60.90.6.x.1	$60$	$2$	$2$	$6$	$5$	$1^{7}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.360.26.h.1	$60$	$2$	$2$	$26$	$12$	$1^{13}$
60.360.26.p.1	$60$	$2$	$2$	$26$	$11$	$1^{13}$
60.360.26.v.1	$60$	$2$	$2$	$26$	$10$	$1^{13}$
60.360.26.bd.1	$60$	$2$	$2$	$26$	$9$	$1^{13}$
60.360.26.br.1	$60$	$2$	$2$	$26$	$11$	$1^{13}$
60.360.26.bz.1	$60$	$2$	$2$	$26$	$11$	$1^{13}$
60.360.26.ch.1	$60$	$2$	$2$	$26$	$14$	$1^{13}$
60.360.26.cp.1	$60$	$2$	$2$	$26$	$12$	$1^{13}$
60.360.27.ced.1	$60$	$2$	$2$	$27$	$8$	$1^{14}$
60.360.27.cel.1	$60$	$2$	$2$	$27$	$14$	$1^{14}$
60.360.27.cet.1	$60$	$2$	$2$	$27$	$11$	$1^{14}$
60.360.27.cff.1	$60$	$2$	$2$	$27$	$11$	$1^{14}$
60.360.27.cfl.1	$60$	$2$	$2$	$27$	$13$	$1^{14}$
60.360.27.cft.1	$60$	$2$	$2$	$27$	$11$	$1^{14}$
60.360.27.cgd.1	$60$	$2$	$2$	$27$	$10$	$1^{14}$
60.360.27.cgl.1	$60$	$2$	$2$	$27$	$12$	$1^{14}$
60.360.28.j.1	$60$	$2$	$2$	$28$	$10$	$1^{15}$
60.360.28.cc.1	$60$	$2$	$2$	$28$	$11$	$1^{15}$
60.360.28.cy.1	$60$	$2$	$2$	$28$	$11$	$1^{15}$
60.360.28.df.1	$60$	$2$	$2$	$28$	$12$	$1^{15}$
60.360.28.dt.1	$60$	$2$	$2$	$28$	$13$	$1^{15}$
60.360.28.ea.1	$60$	$2$	$2$	$28$	$14$	$1^{15}$
60.360.28.ek.1	$60$	$2$	$2$	$28$	$12$	$1^{15}$
60.360.28.er.1	$60$	$2$	$2$	$28$	$13$	$1^{15}$
60.540.37.sd.1	$60$	$3$	$3$	$37$	$16$	$1^{24}$
60.720.55.bsc.1	$60$	$4$	$4$	$55$	$22$	$1^{36}\cdot2^{3}$

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