LMFDB - Modular curve 60.288.13.rz.1

Invariants

Level:	$60$	$\SL_2$-level:	$60$	Newform level:	$3600$
Index:	$288$	$\PSL_2$-index:	$288$
Genus:	$13 = 1 + \frac{ 288 }{12} - \frac{ 32 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)	Cusp widths	$12^{4}\cdot60^{4}$	Cusp orbits	$2^{2}\cdot4$
Elliptic points:	$32$ of order $2$ and $0$ of order $3$
Analytic rank:	$2$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	60AK13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.288.13.789

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}23&0\\21&41\end{bmatrix}$, $\begin{bmatrix}27&10\\10&57\end{bmatrix}$, $\begin{bmatrix}49&55\\49&38\end{bmatrix}$, $\begin{bmatrix}56&45\\3&19\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 60-isogeny field degree:	$24$
Cyclic 60-torsion field degree:	$192$
Full 60-torsion field degree:	$7680$

Jacobian

Conductor:	$2^{38}\cdot3^{12}\cdot5^{17}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{7}\cdot2^{3}$
Newforms:	40.2.c.a, 45.2.b.a, 80.2.a.a, 100.2.a.a, 120.2.f.a, 400.2.a.c, 720.2.a.a, 720.2.a.i, 1200.2.a.e, 1200.2.a.k

Models

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

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

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

Rational points

This modular curve has no real points, and therefore no rational points.

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 60.144.5.ue.2 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle -x+y$
$\displaystyle Z$	$=$	$\displaystyle -y$
$\displaystyle W$	$=$	$\displaystyle r$
$\displaystyle T$	$=$	$\displaystyle s$

Equation of the image curve:

$0$	$=$	$ X^{2}-YW+ZW-YT+ZT $
	$=$	$ 2YZ-XW+XT $
	$=$	$ XY+XZ-2W^{2}-6WT-2T^{2} $

Modular covers

Cover information

Click on a modular curve in the diagram to see information about it.

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.72.1.fh.2	$60$	$4$	$4$	$1$	$0$	$1^{6}\cdot2^{3}$
60.144.5.ue.2	$60$	$2$	$2$	$5$	$1$	$1^{4}\cdot2^{2}$
60.144.7.uc.1	$60$	$2$	$2$	$7$	$1$	$1^{4}\cdot2$
60.144.7.uh.1	$60$	$2$	$2$	$7$	$2$	$2^{3}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.576.41.ek.1	$60$	$2$	$2$	$41$	$6$	$1^{14}\cdot2^{7}$
60.576.41.en.1	$60$	$2$	$2$	$41$	$5$	$1^{14}\cdot2^{7}$
60.576.41.gm.2	$60$	$2$	$2$	$41$	$3$	$1^{14}\cdot2^{7}$
60.576.41.gr.1	$60$	$2$	$2$	$41$	$8$	$1^{14}\cdot2^{7}$
60.576.41.kf.1	$60$	$2$	$2$	$41$	$5$	$1^{14}\cdot2^{7}$
60.576.41.kk.1	$60$	$2$	$2$	$41$	$9$	$1^{14}\cdot2^{7}$
60.576.41.ko.2	$60$	$2$	$2$	$41$	$3$	$1^{14}\cdot2^{7}$
60.576.41.ks.1	$60$	$2$	$2$	$41$	$5$	$1^{14}\cdot2^{7}$
60.864.53.hw.1	$60$	$3$	$3$	$53$	$8$	$1^{20}\cdot2^{10}$
60.1440.101.bo.1	$60$	$5$	$5$	$101$	$19$	$1^{42}\cdot2^{23}$

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