Properties

Label 24.48.0-24.d.1.2
Level $24$
Index $48$
Genus $0$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $4$
Index: $48$ $\PSL_2$-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 $4^{6}$ Cusp orbits $2^{3}$
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: 4G0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.0.136

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&10\\20&23\end{bmatrix}$, $\begin{bmatrix}11&18\\10&13\end{bmatrix}$, $\begin{bmatrix}19&10\\14&7\end{bmatrix}$
Contains $-I$: no $\quad$ (see 24.24.0.d.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Models

Smooth plane model Smooth plane model

$ 0 $ $=$ $ 24 x^{2} - y^{2} + y z - z^{2} $
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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

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.24.0-8.a.1.2 $8$ $2$ $2$ $0$ $0$
12.24.0-12.b.1.1 $12$ $2$ $2$ $0$ $0$
24.24.0-8.a.1.3 $24$ $2$ $2$ $0$ $0$
24.24.0-12.b.1.3 $24$ $2$ $2$ $0$ $0$
24.24.0-24.b.1.2 $24$ $2$ $2$ $0$ $0$
24.24.0-24.b.1.6 $24$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
24.144.4-24.g.1.4 $24$ $3$ $3$ $4$
24.192.3-24.be.1.9 $24$ $4$ $4$ $3$
120.240.8-120.d.1.8 $120$ $5$ $5$ $8$
120.288.7-120.cs.1.2 $120$ $6$ $6$ $7$
120.480.15-120.d.1.2 $120$ $10$ $10$ $15$
168.384.11-168.g.1.22 $168$ $8$ $8$ $11$