Modular curve 80.480.13-40.nb.1.1

• Label: 80.480.13-40.nb.1.1
• Level: $80$
• Index: $480$
• Genus: $13$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		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	$4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 24$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 13$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40G13

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}0&43\\33&50\end{bmatrix}$, $\begin{bmatrix}26&79\\35&54\end{bmatrix}$, $\begin{bmatrix}43&6\\54&27\end{bmatrix}$, $\begin{bmatrix}46&53\\39&4\end{bmatrix}$, $\begin{bmatrix}54&69\\61&38\end{bmatrix}$, $\begin{bmatrix}65&46\\74&61\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.240.13.nb.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree:	$12$
Cyclic 80-torsion field degree:	$192$
Full 80-torsion field degree:	$24576$

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 614656 x^{8} y^{16} - 3449600 x^{8} y^{14} z^{2} + 2738880 x^{8} y^{12} z^{4} - 924800 x^{8} y^{10} z^{6} + \cdots + z^{24} $

Rational points

This modular curve has no $\Q_p$ points for $p=3,13,19$, 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 :

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

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} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.240.13.nb.1 :

$\displaystyle X$	$=$	$\displaystyle d$
$\displaystyle Y$	$=$	$\displaystyle 5x$
$\displaystyle Z$	$=$	$\displaystyle c$

Equation of the image curve:

$0$

$=$

$ 614656X^{8}Y^{16}-2809856X^{6}Y^{18}+4114432X^{4}Y^{20}-2064384X^{2}Y^{22}+331776Y^{24}-3449600X^{8}Y^{14}Z^{2}+11045888X^{6}Y^{16}Z^{2}-13246976X^{4}Y^{18}Z^{2}+6991872X^{2}Y^{20}Z^{2}-1548288Y^{22}Z^{2}+2738880X^{8}Y^{12}Z^{4}-10515456X^{6}Y^{14}Z^{4}+15967232X^{4}Y^{16}Z^{4}-11282432X^{2}Y^{18}Z^{4}+3262464Y^{20}Z^{4}-924800X^{8}Y^{10}Z^{6}+5186816X^{6}Y^{12}Z^{6}-11744256X^{4}Y^{14}Z^{6}+11501568X^{2}Y^{16}Z^{6}-4098048Y^{18}Z^{6}+159600X^{8}Y^{8}Z^{8}-1510400X^{6}Y^{10}Z^{8}+6029824X^{4}Y^{12}Z^{8}-8134656X^{2}Y^{14}Z^{8}+3411712Y^{16}Z^{8}-14000X^{8}Y^{6}Z^{10}+269120X^{6}Y^{8}Z^{10}-2304064X^{4}Y^{10}Z^{10}+4095488X^{2}Y^{12}Z^{10}-1979392Y^{14}Z^{10}+500X^{8}Y^{4}Z^{12}-27200X^{6}Y^{6}Z^{12}+644032X^{4}Y^{8}Z^{12}-1457408X^{2}Y^{10}Z^{12}+818944Y^{12}Z^{12}+1200X^{6}Y^{4}Z^{14}-122880X^{4}Y^{6}Z^{14}+355840X^{2}Y^{8}Z^{14}-242944Y^{10}Z^{14}+13720X^{4}Y^{4}Z^{16}-56192X^{2}Y^{6}Z^{16}+51184Y^{8}Z^{16}-650X^{4}Y^{2}Z^{18}+5104X^{2}Y^{4}Z^{18}-7456Y^{6}Z^{18}-200X^{2}Y^{2}Z^{20}+712Y^{4}Z^{20}-40Y^{2}Z^{22}+Z^{24} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.240.7-40.cj.1.1	$80$	$2$	$2$	$7$	$?$
80.240.7-40.cj.1.15	$80$	$2$	$2$	$7$	$?$