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Properties

Label 24.48.0.ba.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 $2^{5}$
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: 8O0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.0.441

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&6\\0&19\end{bmatrix}$, $\begin{bmatrix}13&6\\0&1\end{bmatrix}$, $\begin{bmatrix}17&10\\8&5\end{bmatrix}$, $\begin{bmatrix}17&12\\8&11\end{bmatrix}$, $\begin{bmatrix}19&14\\16&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.0-24.ba.1.1, 24.96.0-24.ba.1.2, 24.96.0-24.ba.1.3, 24.96.0-24.ba.1.4, 24.96.0-24.ba.1.5, 24.96.0-24.ba.1.6, 24.96.0-24.ba.1.7, 24.96.0-24.ba.1.8, 24.96.0-24.ba.1.9, 24.96.0-24.ba.1.10, 24.96.0-24.ba.1.11, 24.96.0-24.ba.1.12, 48.96.0-24.ba.1.1, 48.96.0-24.ba.1.2, 48.96.0-24.ba.1.3, 48.96.0-24.ba.1.4, 48.96.0-24.ba.1.5, 48.96.0-24.ba.1.6, 48.96.0-24.ba.1.7, 48.96.0-24.ba.1.8, 120.96.0-24.ba.1.1, 120.96.0-24.ba.1.2, 120.96.0-24.ba.1.3, 120.96.0-24.ba.1.4, 120.96.0-24.ba.1.5, 120.96.0-24.ba.1.6, 120.96.0-24.ba.1.7, 120.96.0-24.ba.1.8, 120.96.0-24.ba.1.9, 120.96.0-24.ba.1.10, 120.96.0-24.ba.1.11, 120.96.0-24.ba.1.12, 168.96.0-24.ba.1.1, 168.96.0-24.ba.1.2, 168.96.0-24.ba.1.3, 168.96.0-24.ba.1.4, 168.96.0-24.ba.1.5, 168.96.0-24.ba.1.6, 168.96.0-24.ba.1.7, 168.96.0-24.ba.1.8, 168.96.0-24.ba.1.9, 168.96.0-24.ba.1.10, 168.96.0-24.ba.1.11, 168.96.0-24.ba.1.12, 240.96.0-24.ba.1.1, 240.96.0-24.ba.1.2, 240.96.0-24.ba.1.3, 240.96.0-24.ba.1.4, 240.96.0-24.ba.1.5, 240.96.0-24.ba.1.6, 240.96.0-24.ba.1.7, 240.96.0-24.ba.1.8, 264.96.0-24.ba.1.1, 264.96.0-24.ba.1.2, 264.96.0-24.ba.1.3, 264.96.0-24.ba.1.4, 264.96.0-24.ba.1.5, 264.96.0-24.ba.1.6, 264.96.0-24.ba.1.7, 264.96.0-24.ba.1.8, 264.96.0-24.ba.1.9, 264.96.0-24.ba.1.10, 264.96.0-24.ba.1.11, 264.96.0-24.ba.1.12, 312.96.0-24.ba.1.1, 312.96.0-24.ba.1.2, 312.96.0-24.ba.1.3, 312.96.0-24.ba.1.4, 312.96.0-24.ba.1.5, 312.96.0-24.ba.1.6, 312.96.0-24.ba.1.7, 312.96.0-24.ba.1.8, 312.96.0-24.ba.1.9, 312.96.0-24.ba.1.10, 312.96.0-24.ba.1.11, 312.96.0-24.ba.1.12
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} - 3 y^{2} - 3 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.i.1 $8$ $2$ $2$ $0$ $0$
24.24.0.h.1 $24$ $2$ $2$ $0$ $0$
24.24.0.by.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.f.1 $24$ $2$ $2$ $1$
24.96.1.bq.1 $24$ $2$ $2$ $1$
24.96.1.cb.1 $24$ $2$ $2$ $1$
24.96.1.cf.1 $24$ $2$ $2$ $1$
24.144.8.ff.2 $24$ $3$ $3$ $8$
24.192.7.dk.1 $24$ $4$ $4$ $7$
48.96.1.a.2 $48$ $2$ $2$ $1$
48.96.1.j.1 $48$ $2$ $2$ $1$
48.96.1.m.2 $48$ $2$ $2$ $1$
48.96.1.p.1 $48$ $2$ $2$ $1$
48.96.3.bn.1 $48$ $2$ $2$ $3$
48.96.3.bs.2 $48$ $2$ $2$ $3$
48.96.3.ca.1 $48$ $2$ $2$ $3$
48.96.3.cm.2 $48$ $2$ $2$ $3$
120.96.1.os.2 $120$ $2$ $2$ $1$
120.96.1.ow.2 $120$ $2$ $2$ $1$
120.96.1.pr.1 $120$ $2$ $2$ $1$
120.96.1.pz.1 $120$ $2$ $2$ $1$
120.240.16.dy.1 $120$ $5$ $5$ $16$
120.288.15.ccs.1 $120$ $6$ $6$ $15$
168.96.1.os.1 $168$ $2$ $2$ $1$
168.96.1.ow.1 $168$ $2$ $2$ $1$
168.96.1.pr.1 $168$ $2$ $2$ $1$
168.96.1.pz.1 $168$ $2$ $2$ $1$
168.384.23.jr.2 $168$ $8$ $8$ $23$
240.96.1.g.1 $240$ $2$ $2$ $1$
240.96.1.p.1 $240$ $2$ $2$ $1$
240.96.1.bb.1 $240$ $2$ $2$ $1$
240.96.1.be.1 $240$ $2$ $2$ $1$
240.96.3.hx.2 $240$ $2$ $2$ $3$
240.96.3.ib.2 $240$ $2$ $2$ $3$
240.96.3.is.1 $240$ $2$ $2$ $3$
240.96.3.jd.2 $240$ $2$ $2$ $3$
264.96.1.os.1 $264$ $2$ $2$ $1$
264.96.1.ow.1 $264$ $2$ $2$ $1$
264.96.1.pr.1 $264$ $2$ $2$ $1$
264.96.1.pz.1 $264$ $2$ $2$ $1$
312.96.1.os.1 $312$ $2$ $2$ $1$
312.96.1.ow.1 $312$ $2$ $2$ $1$
312.96.1.pr.1 $312$ $2$ $2$ $1$
312.96.1.pz.1 $312$ $2$ $2$ $1$