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Properties

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

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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: 24.48.0.319

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}13&0\\4&5\end{bmatrix}$, $\begin{bmatrix}13&2\\8&5\end{bmatrix}$, $\begin{bmatrix}17&10\\4&13\end{bmatrix}$, $\begin{bmatrix}21&20\\4&19\end{bmatrix}$, $\begin{bmatrix}23&16\\16&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.0-24.p.1.1, 24.96.0-24.p.1.2, 24.96.0-24.p.1.3, 24.96.0-24.p.1.4, 24.96.0-24.p.1.5, 24.96.0-24.p.1.6, 24.96.0-24.p.1.7, 24.96.0-24.p.1.8, 24.96.0-24.p.1.9, 24.96.0-24.p.1.10, 24.96.0-24.p.1.11, 24.96.0-24.p.1.12, 24.96.0-24.p.1.13, 24.96.0-24.p.1.14, 24.96.0-24.p.1.15, 24.96.0-24.p.1.16, 120.96.0-24.p.1.1, 120.96.0-24.p.1.2, 120.96.0-24.p.1.3, 120.96.0-24.p.1.4, 120.96.0-24.p.1.5, 120.96.0-24.p.1.6, 120.96.0-24.p.1.7, 120.96.0-24.p.1.8, 120.96.0-24.p.1.9, 120.96.0-24.p.1.10, 120.96.0-24.p.1.11, 120.96.0-24.p.1.12, 120.96.0-24.p.1.13, 120.96.0-24.p.1.14, 120.96.0-24.p.1.15, 120.96.0-24.p.1.16, 168.96.0-24.p.1.1, 168.96.0-24.p.1.2, 168.96.0-24.p.1.3, 168.96.0-24.p.1.4, 168.96.0-24.p.1.5, 168.96.0-24.p.1.6, 168.96.0-24.p.1.7, 168.96.0-24.p.1.8, 168.96.0-24.p.1.9, 168.96.0-24.p.1.10, 168.96.0-24.p.1.11, 168.96.0-24.p.1.12, 168.96.0-24.p.1.13, 168.96.0-24.p.1.14, 168.96.0-24.p.1.15, 168.96.0-24.p.1.16, 264.96.0-24.p.1.1, 264.96.0-24.p.1.2, 264.96.0-24.p.1.3, 264.96.0-24.p.1.4, 264.96.0-24.p.1.5, 264.96.0-24.p.1.6, 264.96.0-24.p.1.7, 264.96.0-24.p.1.8, 264.96.0-24.p.1.9, 264.96.0-24.p.1.10, 264.96.0-24.p.1.11, 264.96.0-24.p.1.12, 264.96.0-24.p.1.13, 264.96.0-24.p.1.14, 264.96.0-24.p.1.15, 264.96.0-24.p.1.16, 312.96.0-24.p.1.1, 312.96.0-24.p.1.2, 312.96.0-24.p.1.3, 312.96.0-24.p.1.4, 312.96.0-24.p.1.5, 312.96.0-24.p.1.6, 312.96.0-24.p.1.7, 312.96.0-24.p.1.8, 312.96.0-24.p.1.9, 312.96.0-24.p.1.10, 312.96.0-24.p.1.11, 312.96.0-24.p.1.12, 312.96.0-24.p.1.13, 312.96.0-24.p.1.14, 312.96.0-24.p.1.15, 312.96.0-24.p.1.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Models

Smooth plane model Smooth plane model

$ 0 $ $=$ $ x^{2} - 6 y^{2} - 24 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

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.24.0.d.1 $8$ $2$ $2$ $0$ $0$
24.24.0.h.1 $24$ $2$ $2$ $0$ $0$
24.24.0.l.1 $24$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
24.96.1.a.1 $24$ $2$ $2$ $1$
24.96.1.d.1 $24$ $2$ $2$ $1$
24.96.1.g.1 $24$ $2$ $2$ $1$
24.96.1.j.1 $24$ $2$ $2$ $1$
24.96.1.ba.1 $24$ $2$ $2$ $1$
24.96.1.bb.1 $24$ $2$ $2$ $1$
24.96.1.bc.1 $24$ $2$ $2$ $1$
24.96.1.bd.1 $24$ $2$ $2$ $1$
24.144.8.ec.2 $24$ $3$ $3$ $8$
24.192.7.co.1 $24$ $4$ $4$ $7$
120.96.1.ly.1 $120$ $2$ $2$ $1$
120.96.1.lz.1 $120$ $2$ $2$ $1$
120.96.1.ma.2 $120$ $2$ $2$ $1$
120.96.1.mb.2 $120$ $2$ $2$ $1$
120.96.1.mo.2 $120$ $2$ $2$ $1$
120.96.1.mp.2 $120$ $2$ $2$ $1$
120.96.1.mq.1 $120$ $2$ $2$ $1$
120.96.1.mr.1 $120$ $2$ $2$ $1$
120.240.16.dc.2 $120$ $5$ $5$ $16$
120.288.15.cbl.1 $120$ $6$ $6$ $15$
168.96.1.ly.1 $168$ $2$ $2$ $1$
168.96.1.lz.1 $168$ $2$ $2$ $1$
168.96.1.ma.1 $168$ $2$ $2$ $1$
168.96.1.mb.1 $168$ $2$ $2$ $1$
168.96.1.mo.1 $168$ $2$ $2$ $1$
168.96.1.mp.1 $168$ $2$ $2$ $1$
168.96.1.mq.1 $168$ $2$ $2$ $1$
168.96.1.mr.1 $168$ $2$ $2$ $1$
168.384.23.io.2 $168$ $8$ $8$ $23$
264.96.1.ly.1 $264$ $2$ $2$ $1$
264.96.1.lz.1 $264$ $2$ $2$ $1$
264.96.1.ma.1 $264$ $2$ $2$ $1$
264.96.1.mb.1 $264$ $2$ $2$ $1$
264.96.1.mo.1 $264$ $2$ $2$ $1$
264.96.1.mp.1 $264$ $2$ $2$ $1$
264.96.1.mq.1 $264$ $2$ $2$ $1$
264.96.1.mr.1 $264$ $2$ $2$ $1$
312.96.1.ly.1 $312$ $2$ $2$ $1$
312.96.1.lz.1 $312$ $2$ $2$ $1$
312.96.1.ma.1 $312$ $2$ $2$ $1$
312.96.1.mb.1 $312$ $2$ $2$ $1$
312.96.1.mo.1 $312$ $2$ $2$ $1$
312.96.1.mp.1 $312$ $2$ $2$ $1$
312.96.1.mq.1 $312$ $2$ $2$ $1$
312.96.1.mr.1 $312$ $2$ $2$ $1$