LMFDB - Modular curve 40.480.13-40.oz.1.13

Invariants

Level:	$40$	$\SL_2$-level:	$40$	Newform level:	$1600$
Index:	$480$	$\PSL_2$-index:	$240$
Genus:	$13 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)	Cusp widths	$5^{8}\cdot10^{4}\cdot40^{4}$	Cusp orbits	$4^{2}\cdot8$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$6$
$\Q$-gonality:	$5 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 8$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	40G13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	40.480.13.1443

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}5&29\\8&5\end{bmatrix}$, $\begin{bmatrix}9&18\\32&1\end{bmatrix}$, $\begin{bmatrix}15&19\\28&25\end{bmatrix}$, $\begin{bmatrix}27&33\\4&19\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.240.13.oz.1 for the level structure with $-I$)
Cyclic 40-isogeny field degree:	$12$
Cyclic 40-torsion field degree:	$96$
Full 40-torsion field degree:	$1536$

Jacobian

Conductor:	$2^{54}\cdot5^{26}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{13}$
Newforms:	50.2.a.b$^{4}$, 100.2.a.a, 1600.2.a.a$^{2}$, 1600.2.a.c, 1600.2.a.f, 1600.2.a.o, 1600.2.a.q$^{2}$, 1600.2.a.v

Models

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

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

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

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=13,17$, and therefore no rational points.

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 20.60.3.c.1 :

$\displaystyle X$	$=$	$\displaystyle 5x+5y-2z-2w-t$
$\displaystyle Y$	$=$	$\displaystyle -3z-3w+t$
$\displaystyle Z$	$=$	$\displaystyle z+w+3t$

Equation of the image curve:

$0$

$=$

$ 2X^{4}-4X^{3}Y+6X^{2}Y^{2}-4XY^{3}+2Y^{4}+4X^{3}Z+17X^{2}YZ-17XY^{2}Z-4Y^{3}Z+5X^{2}Z^{2}+18XYZ^{2}+5Y^{2}Z^{2}+3XZ^{3}-3YZ^{3}-2Z^{4} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
5.20.0.b.1	$5$	$24$	$12$	$0$	$0$	full Jacobian
8.24.0-8.p.1.7	$8$	$20$	$20$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
40.120.4-40.br.1.13	$40$	$4$	$4$	$4$	$2$	$1^{9}$
40.240.5-20.m.1.3	$40$	$2$	$2$	$5$	$0$	$1^{8}$
40.240.5-20.m.1.8	$40$	$2$	$2$	$5$	$0$	$1^{8}$
40.240.7-40.co.1.18	$40$	$2$	$2$	$7$	$2$	$1^{6}$
40.240.7-40.co.1.23	$40$	$2$	$2$	$7$	$2$	$1^{6}$
40.240.7-40.cx.1.19	$40$	$2$	$2$	$7$	$4$	$1^{6}$
40.240.7-40.cx.1.22	$40$	$2$	$2$	$7$	$4$	$1^{6}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
40.960.29-40.hc.1.12	$40$	$2$	$2$	$29$	$9$	$1^{16}$
40.960.29-40.ib.1.7	$40$	$2$	$2$	$29$	$9$	$1^{16}$
40.960.29-40.qs.1.2	$40$	$2$	$2$	$29$	$15$	$1^{16}$
40.960.29-40.qu.1.8	$40$	$2$	$2$	$29$	$15$	$1^{16}$
40.960.29-40.yp.1.4	$40$	$2$	$2$	$29$	$14$	$1^{16}$
40.960.29-40.yr.1.3	$40$	$2$	$2$	$29$	$14$	$1^{16}$
40.960.29-40.yx.1.3	$40$	$2$	$2$	$29$	$9$	$1^{16}$
40.960.29-40.yz.1.4	$40$	$2$	$2$	$29$	$9$	$1^{16}$
40.1440.37-40.mv.1.3	$40$	$3$	$3$	$37$	$14$	$1^{24}$

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