LMFDB - Modular curve 60.288.13.sf.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:	$5$
$\Q$-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:	60.288.13.630

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}1&15\\24&43\end{bmatrix}$, $\begin{bmatrix}23&20\\8&19\end{bmatrix}$, $\begin{bmatrix}38&35\\43&14\end{bmatrix}$, $\begin{bmatrix}53&10\\5&11\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 60-isogeny field degree:	$24$
Cyclic 60-torsion field degree:	$384$
Full 60-torsion field degree:	$7680$

Jacobian

Conductor:	$2^{50}\cdot3^{18}\cdot5^{17}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{7}\cdot2^{3}$
Newforms:	80.2.a.a, 80.2.c.a, 240.2.f.c, 720.2.a.a, 720.2.a.i, 720.2.f.d, 900.2.a.b, 3600.2.a.be, 3600.2.a.f, 3600.2.a.u

Models

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

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

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

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.ud.1 :

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

Equation of the image curve:

$0$	$=$	$ 3X^{2}-YW-2ZW-YT-2ZT $
	$=$	$ Y^{2}-2YZ-2Z^{2}-3XW+3XT $
	$=$	$ 3XY+W^{2}+8WT+T^{2} $

Modular covers

Cover information

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

See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
30.72.1.o.1	$30$	$4$	$4$	$1$	$1$	$1^{6}\cdot2^{3}$
60.144.5.ud.1	$60$	$2$	$2$	$5$	$1$	$1^{4}\cdot2^{2}$
60.144.7.ud.1	$60$	$2$	$2$	$7$	$1$	$1^{4}\cdot2$
60.144.7.uj.1	$60$	$2$	$2$	$7$	$5$	$2^{3}$

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.ea.1	$60$	$2$	$2$	$41$	$8$	$1^{14}\cdot2^{7}$
60.576.41.eh.1	$60$	$2$	$2$	$41$	$9$	$1^{14}\cdot2^{7}$
60.576.41.he.2	$60$	$2$	$2$	$41$	$11$	$1^{14}\cdot2^{7}$
60.576.41.hj.1	$60$	$2$	$2$	$41$	$6$	$1^{14}\cdot2^{7}$
60.576.41.jp.2	$60$	$2$	$2$	$41$	$8$	$1^{14}\cdot2^{7}$
60.576.41.jw.1	$60$	$2$	$2$	$41$	$8$	$1^{14}\cdot2^{7}$
60.576.41.lp.1	$60$	$2$	$2$	$41$	$10$	$1^{14}\cdot2^{7}$
60.576.41.lw.1	$60$	$2$	$2$	$41$	$8$	$1^{14}\cdot2^{7}$
60.864.53.js.2	$60$	$3$	$3$	$53$	$12$	$1^{20}\cdot2^{10}$
60.1440.101.bj.1	$60$	$5$	$5$	$101$	$26$	$1^{42}\cdot2^{23}$

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