LMFDB - Modular curve 20.720.13-20.s.1.1

Invariants

Level:	$20$	$\SL_2$-level:	$10$	Newform level:	$400$
Index:	$720$	$\PSL_2$-index:	$360$
Genus:	$13 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$
Cusps:	$36$ (of which $2$ are rational)	Cusp widths	$10^{36}$	Cusp orbits	$1^{2}\cdot2^{3}\cdot4^{3}\cdot8^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 6$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10A13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	20.720.13.13

Level structure

$\GL_2(\Z/20\Z)$-generators:	$\begin{bmatrix}3&4\\10&1\end{bmatrix}$, $\begin{bmatrix}17&11\\0&9\end{bmatrix}$, $\begin{bmatrix}17&11\\10&19\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup:	$C_2^3.D_4$
Contains $-I$:	no $\quad$ (see 20.360.13.s.1 for the level structure with $-I$)
Cyclic 20-isogeny field degree:	$2$
Cyclic 20-torsion field degree:	$8$
Full 20-torsion field degree:	$64$

Jacobian

Conductor:	$2^{40}\cdot5^{25}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{7}\cdot2^{3}$
Newforms:	50.2.a.a, 50.2.a.b, 50.2.b.a, 80.2.a.b, 400.2.a.c$^{2}$, 400.2.a.d, 400.2.a.f, 400.2.c.b, 400.2.c.c

Models

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

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

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

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
$(0:0:0:1:0:0:0:1:0:0:0:0:0)$, $(0:0:0:0:-1:0:0:0:0:1:0:0:0)$

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 20.90.3.c.1 :

$\displaystyle X$	$=$	$\displaystyle 4x+4y+4z-2w+2t+3u$
$\displaystyle Y$	$=$	$\displaystyle -3x-2y-2z+2w-2t-u-v-2r+s-2a$
$\displaystyle Z$	$=$	$\displaystyle -u$

Equation of the image curve:

$0$

$=$

$ X^{3}Y+X^{2}Y^{2}-XY^{3}-X^{3}Z+7X^{2}YZ+9XY^{2}Z-3Y^{3}Z-3X^{2}Z^{2}+7XYZ^{2}+8Y^{2}Z^{2}+15XZ^{3}+5YZ^{3}-15Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
10.360.4-10.a.1.3	$10$	$2$	$2$	$4$	$0$	$1^{5}\cdot2^{2}$
20.144.1-20.i.1.1	$20$	$5$	$5$	$1$	$0$	$1^{6}\cdot2^{3}$
20.144.1-20.i.2.1	$20$	$5$	$5$	$1$	$0$	$1^{6}\cdot2^{3}$
20.240.5-20.u.1.2	$20$	$3$	$3$	$5$	$0$	$1^{4}\cdot2^{2}$
20.360.4-10.a.1.4	$20$	$2$	$2$	$4$	$0$	$1^{5}\cdot2^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.1440.37-20.bg.1.1	$20$	$2$	$2$	$37$	$3$	$1^{12}\cdot2^{6}$
20.1440.37-20.bk.1.4	$20$	$2$	$2$	$37$	$4$	$1^{12}\cdot2^{6}$
20.1440.37-20.bw.1.1	$20$	$2$	$2$	$37$	$3$	$1^{12}\cdot2^{6}$
20.1440.37-20.ca.1.2	$20$	$2$	$2$	$37$	$3$	$1^{12}\cdot2^{6}$
40.1440.37-40.ka.1.2	$40$	$2$	$2$	$37$	$6$	$1^{12}\cdot2^{6}$
40.1440.37-40.lc.1.2	$40$	$2$	$2$	$37$	$8$	$1^{12}\cdot2^{6}$
40.1440.37-40.pl.1.1	$40$	$2$	$2$	$37$	$7$	$1^{12}\cdot2^{6}$
40.1440.37-40.qn.1.1	$40$	$2$	$2$	$37$	$7$	$1^{12}\cdot2^{6}$
60.1440.37-60.it.1.1	$60$	$2$	$2$	$37$	$7$	$1^{12}\cdot2^{6}$
60.1440.37-60.jb.1.3	$60$	$2$	$2$	$37$	$5$	$1^{12}\cdot2^{6}$
60.1440.37-60.oc.1.1	$60$	$2$	$2$	$37$	$9$	$1^{12}\cdot2^{6}$
60.1440.37-60.ok.1.3	$60$	$2$	$2$	$37$	$6$	$1^{12}\cdot2^{6}$
60.2160.73-60.bjm.1.6	$60$	$3$	$3$	$73$	$11$	$1^{24}\cdot2^{18}$
60.2880.85-60.mz.1.5	$60$	$4$	$4$	$85$	$8$	$1^{36}\cdot2^{18}$

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