Modular curve 40.48.0.t.1

• Label: 40.48.0.t.1
• Level: $40$
• Index: $48$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $10$
• $\Q$-cusps: $0$

Invariants

Level:	$40$		$\SL_2$-level:	$8$
Index:	$48$		$\PSL_2$-index:	$48$
Genus:	$0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps:	$10$ (none of which are rational)		Cusp widths	$2^{4}\cdot4^{2}\cdot8^{4}$		Cusp orbits	$2^{5}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1 \le \gamma \le 2$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	8O0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	40.48.0.312

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}7&32\\14&13\end{bmatrix}$, $\begin{bmatrix}23&24\\32&33\end{bmatrix}$, $\begin{bmatrix}25&4\\36&37\end{bmatrix}$, $\begin{bmatrix}39&8\\4&5\end{bmatrix}$, $\begin{bmatrix}39&32\\6&7\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	40.96.0-40.t.1.1, 40.96.0-40.t.1.2, 40.96.0-40.t.1.3, 40.96.0-40.t.1.4, 40.96.0-40.t.1.5, 40.96.0-40.t.1.6, 40.96.0-40.t.1.7, 40.96.0-40.t.1.8, 40.96.0-40.t.1.9, 40.96.0-40.t.1.10, 40.96.0-40.t.1.11, 40.96.0-40.t.1.12, 40.96.0-40.t.1.13, 40.96.0-40.t.1.14, 40.96.0-40.t.1.15, 40.96.0-40.t.1.16, 120.96.0-40.t.1.1, 120.96.0-40.t.1.2, 120.96.0-40.t.1.3, 120.96.0-40.t.1.4, 120.96.0-40.t.1.5, 120.96.0-40.t.1.6, 120.96.0-40.t.1.7, 120.96.0-40.t.1.8, 120.96.0-40.t.1.9, 120.96.0-40.t.1.10, 120.96.0-40.t.1.11, 120.96.0-40.t.1.12, 120.96.0-40.t.1.13, 120.96.0-40.t.1.14, 120.96.0-40.t.1.15, 120.96.0-40.t.1.16, 280.96.0-40.t.1.1, 280.96.0-40.t.1.2, 280.96.0-40.t.1.3, 280.96.0-40.t.1.4, 280.96.0-40.t.1.5, 280.96.0-40.t.1.6, 280.96.0-40.t.1.7, 280.96.0-40.t.1.8, 280.96.0-40.t.1.9, 280.96.0-40.t.1.10, 280.96.0-40.t.1.11, 280.96.0-40.t.1.12, 280.96.0-40.t.1.13, 280.96.0-40.t.1.14, 280.96.0-40.t.1.15, 280.96.0-40.t.1.16
Cyclic 40-isogeny field degree:	$12$
Cyclic 40-torsion field degree:	$192$
Full 40-torsion field degree:	$15360$

Models

Smooth plane model Smooth plane model

$ 0 $

$=$

$ 32 x^{2} - 40 y^{2} + 5 z^{2} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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
8.24.0.e.1	$8$	$2$	$2$	$0$	$0$
40.24.0.i.1	$40$	$2$	$2$	$0$	$0$
40.24.0.l.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
40.96.1.r.1	$40$	$2$	$2$	$1$
40.96.1.u.1	$40$	$2$	$2$	$1$
40.96.1.w.1	$40$	$2$	$2$	$1$
40.96.1.z.1	$40$	$2$	$2$	$1$
40.96.1.be.1	$40$	$2$	$2$	$1$
40.96.1.bf.1	$40$	$2$	$2$	$1$
40.96.1.bg.1	$40$	$2$	$2$	$1$
40.96.1.bh.1	$40$	$2$	$2$	$1$
40.240.16.z.2	$40$	$5$	$5$	$16$
40.288.15.da.2	$40$	$6$	$6$	$15$
40.480.31.el.2	$40$	$10$	$10$	$31$
120.96.1.im.2	$120$	$2$	$2$	$1$
120.96.1.in.2	$120$	$2$	$2$	$1$
120.96.1.iq.1	$120$	$2$	$2$	$1$
120.96.1.ir.1	$120$	$2$	$2$	$1$
120.96.1.js.1	$120$	$2$	$2$	$1$
120.96.1.jt.1	$120$	$2$	$2$	$1$
120.96.1.jw.2	$120$	$2$	$2$	$1$
120.96.1.jx.2	$120$	$2$	$2$	$1$
120.144.8.np.2	$120$	$3$	$3$	$8$
120.192.7.ht.2	$120$	$4$	$4$	$7$
280.96.1.lq.2	$280$	$2$	$2$	$1$
280.96.1.lr.2	$280$	$2$	$2$	$1$
280.96.1.ls.1	$280$	$2$	$2$	$1$
280.96.1.lt.1	$280$	$2$	$2$	$1$
280.96.1.mg.1	$280$	$2$	$2$	$1$
280.96.1.mh.1	$280$	$2$	$2$	$1$
280.96.1.mi.2	$280$	$2$	$2$	$1$
280.96.1.mj.2	$280$	$2$	$2$	$1$
280.384.23.ix.2	$280$	$8$	$8$	$23$