Modular curve 24.48.0.bd.1

• Label: 24.48.0.bd.1
• Level: $24$
• Index: $48$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $10$
• $\Q$-cusps: $0$

Invariants

Level:	$24$		$\SL_2$-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	$2^{3}\cdot4$
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:	24.48.0.829

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}1&15\\20&1\end{bmatrix}$, $\begin{bmatrix}9&13\\8&15\end{bmatrix}$, $\begin{bmatrix}19&21\\16&13\end{bmatrix}$, $\begin{bmatrix}23&9\\8&1\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	24.96.0-24.bd.1.1, 24.96.0-24.bd.1.2, 24.96.0-24.bd.1.3, 24.96.0-24.bd.1.4, 24.96.0-24.bd.1.5, 24.96.0-24.bd.1.6, 24.96.0-24.bd.1.7, 24.96.0-24.bd.1.8, 48.96.0-24.bd.1.1, 48.96.0-24.bd.1.2, 48.96.0-24.bd.1.3, 48.96.0-24.bd.1.4, 48.96.0-24.bd.1.5, 48.96.0-24.bd.1.6, 48.96.0-24.bd.1.7, 48.96.0-24.bd.1.8, 120.96.0-24.bd.1.1, 120.96.0-24.bd.1.2, 120.96.0-24.bd.1.3, 120.96.0-24.bd.1.4, 120.96.0-24.bd.1.5, 120.96.0-24.bd.1.6, 120.96.0-24.bd.1.7, 120.96.0-24.bd.1.8, 168.96.0-24.bd.1.1, 168.96.0-24.bd.1.2, 168.96.0-24.bd.1.3, 168.96.0-24.bd.1.4, 168.96.0-24.bd.1.5, 168.96.0-24.bd.1.6, 168.96.0-24.bd.1.7, 168.96.0-24.bd.1.8, 240.96.0-24.bd.1.1, 240.96.0-24.bd.1.2, 240.96.0-24.bd.1.3, 240.96.0-24.bd.1.4, 240.96.0-24.bd.1.5, 240.96.0-24.bd.1.6, 240.96.0-24.bd.1.7, 240.96.0-24.bd.1.8, 264.96.0-24.bd.1.1, 264.96.0-24.bd.1.2, 264.96.0-24.bd.1.3, 264.96.0-24.bd.1.4, 264.96.0-24.bd.1.5, 264.96.0-24.bd.1.6, 264.96.0-24.bd.1.7, 264.96.0-24.bd.1.8, 312.96.0-24.bd.1.1, 312.96.0-24.bd.1.2, 312.96.0-24.bd.1.3, 312.96.0-24.bd.1.4, 312.96.0-24.bd.1.5, 312.96.0-24.bd.1.6, 312.96.0-24.bd.1.7, 312.96.0-24.bd.1.8
Cyclic 24-isogeny field degree:	$4$
Cyclic 24-torsion field degree:	$32$
Full 24-torsion field degree:	$1536$

Models

Smooth plane model Smooth plane model

$ 0 $

$=$

$ 8 x^{2} + 6 y^{2} - 3 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.k.1	$8$	$2$	$2$	$0$	$0$
24.24.0.by.2	$24$	$2$	$2$	$0$	$0$
24.24.0.bz.2	$24$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
24.144.8.fs.2	$24$	$3$	$3$	$8$
24.192.7.dt.1	$24$	$4$	$4$	$7$
48.96.1.s.1	$48$	$2$	$2$	$1$
48.96.1.t.1	$48$	$2$	$2$	$1$
48.96.1.u.1	$48$	$2$	$2$	$1$
48.96.1.x.1	$48$	$2$	$2$	$1$
48.96.1.y.2	$48$	$2$	$2$	$1$
48.96.1.bb.2	$48$	$2$	$2$	$1$
48.96.1.bc.2	$48$	$2$	$2$	$1$
48.96.1.bd.2	$48$	$2$	$2$	$1$
120.240.16.ef.1	$120$	$5$	$5$	$16$
120.288.15.dsh.1	$120$	$6$	$6$	$15$
168.384.23.kc.1	$168$	$8$	$8$	$23$
240.96.1.cj.1	$240$	$2$	$2$	$1$
240.96.1.ck.1	$240$	$2$	$2$	$1$
240.96.1.co.1	$240$	$2$	$2$	$1$
240.96.1.cr.1	$240$	$2$	$2$	$1$
240.96.1.cw.2	$240$	$2$	$2$	$1$
240.96.1.cz.2	$240$	$2$	$2$	$1$
240.96.1.dd.2	$240$	$2$	$2$	$1$
240.96.1.de.2	$240$	$2$	$2$	$1$