Modular curve 40.24.0.bk.1

• Label: 40.24.0.bk.1
• Level: $40$
• Index: $24$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $6$
• $\Q$-cusps: $0$

Invariants

Level:	$40$		$\SL_2$-level:	$8$
Index:	$24$		$\PSL_2$-index:	$24$
Genus:	$0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (none of which are rational)		Cusp widths	$2^{4}\cdot8^{2}$		Cusp orbits	$2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	8G0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	40.24.0.222

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}7&20\\36&13\end{bmatrix}$, $\begin{bmatrix}7&20\\39&13\end{bmatrix}$, $\begin{bmatrix}21&36\\24&15\end{bmatrix}$, $\begin{bmatrix}39&32\\35&37\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	40.48.0-40.bk.1.1, 40.48.0-40.bk.1.2, 40.48.0-40.bk.1.3, 40.48.0-40.bk.1.4, 40.48.0-40.bk.1.5, 40.48.0-40.bk.1.6, 40.48.0-40.bk.1.7, 40.48.0-40.bk.1.8, 120.48.0-40.bk.1.1, 120.48.0-40.bk.1.2, 120.48.0-40.bk.1.3, 120.48.0-40.bk.1.4, 120.48.0-40.bk.1.5, 120.48.0-40.bk.1.6, 120.48.0-40.bk.1.7, 120.48.0-40.bk.1.8, 280.48.0-40.bk.1.1, 280.48.0-40.bk.1.2, 280.48.0-40.bk.1.3, 280.48.0-40.bk.1.4, 280.48.0-40.bk.1.5, 280.48.0-40.bk.1.6, 280.48.0-40.bk.1.7, 280.48.0-40.bk.1.8
Cyclic 40-isogeny field degree:	$12$
Cyclic 40-torsion field degree:	$192$
Full 40-torsion field degree:	$30720$

Models

Smooth plane model Smooth plane model

$ 0 $

$=$

$ 80 x^{2} + 8 y^{2} + 4 y z + 3 z^{2} $

Rational points

This modular curve has no real points, and therefore no rational points.

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
8.12.0.m.1	$8$	$2$	$2$	$0$	$0$
40.12.0.s.1	$40$	$2$	$2$	$0$	$0$
40.12.0.ba.1	$40$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
40.120.8.ck.1	$40$	$5$	$5$	$8$
40.144.7.ed.1	$40$	$6$	$6$	$7$
40.240.15.fi.1	$40$	$10$	$10$	$15$
120.72.4.ji.1	$120$	$3$	$3$	$4$
120.96.3.oa.1	$120$	$4$	$4$	$3$
280.192.11.hg.1	$280$	$8$	$8$	$11$