LMFDB - Modular curve 40.384.13-40.q.2.12

Invariants

Level:	$40$	$\SL_2$-level:	$40$	Newform level:	$320$
Index:	$384$	$\PSL_2$-index:	$192$
Genus:	$13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)	Cusp widths	$8^{4}\cdot40^{4}$	Cusp orbits	$2^{2}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4 \le \gamma \le 12$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 10$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	40D13
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	40.384.13.19

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}36&17\\5&28\end{bmatrix}$, $\begin{bmatrix}37&29\\3&18\end{bmatrix}$, $\begin{bmatrix}39&4\\20&3\end{bmatrix}$, $\begin{bmatrix}39&12\\28&3\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.192.13.q.2 for the level structure with $-I$)
Cyclic 40-isogeny field degree:	$12$
Cyclic 40-torsion field degree:	$192$
Full 40-torsion field degree:	$1920$

Jacobian

Conductor:	$2^{72}\cdot5^{13}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{5}\cdot2^{2}\cdot4$
Newforms:	80.2.a.a, 80.2.c.a, 320.2.a.b, 320.2.a.c, 320.2.a.e, 320.2.a.f, 320.2.c.c, 320.2.c.d

Models

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

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

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

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 4 from the canonical model of this modular curve to the canonical model of the modular curve 20.48.3.i.2 :

$\displaystyle X$	$=$	$\displaystyle -2w-3t-3v$
$\displaystyle Y$	$=$	$\displaystyle -w-4t+v$
$\displaystyle Z$	$=$	$\displaystyle 2w-2t-2v$

Equation of the image curve:

$0$

$=$

$ 6X^{3}Y-22X^{2}Y^{2}+6XY^{3}+12X^{2}YZ+14XY^{2}Z-6Y^{3}Z-3X^{2}Z^{2}-2XYZ^{2}+5Y^{2}Z^{2}-10YZ^{3}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.96.3-20.i.2.6	$20$	$4$	$4$	$3$	$0$	$1^{4}\cdot2\cdot4$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
40.768.25-40.a.2.8	$40$	$2$	$2$	$25$	$2$	$1^{6}\cdot2^{3}$
40.768.25-40.c.2.6	$40$	$2$	$2$	$25$	$2$	$1^{6}\cdot2^{3}$
40.768.25-40.i.2.7	$40$	$2$	$2$	$25$	$2$	$1^{6}\cdot2^{3}$
40.768.25-40.k.2.8	$40$	$2$	$2$	$25$	$2$	$1^{6}\cdot2^{3}$
40.768.25-40.q.1.3	$40$	$2$	$2$	$25$	$1$	$1^{6}\cdot2^{3}$
40.768.25-40.s.1.1	$40$	$2$	$2$	$25$	$4$	$1^{6}\cdot2^{3}$
40.768.25-40.y.2.7	$40$	$2$	$2$	$25$	$3$	$1^{6}\cdot2^{3}$
40.768.25-40.ba.2.8	$40$	$2$	$2$	$25$	$5$	$1^{6}\cdot2^{3}$
40.1152.37-40.oe.1.3	$40$	$3$	$3$	$37$	$3$	$1^{10}\cdot2^{5}\cdot4$
40.1920.69-40.mw.1.1	$40$	$5$	$5$	$69$	$14$	$1^{24}\cdot2^{14}\cdot4$

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