↑

Properties

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

Related objects

Downloads

Learn more

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 $4^{8}\cdot8^{2}$ 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: 8N0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.0.422

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&0\\20&11\end{bmatrix}$, $\begin{bmatrix}9&16\\4&7\end{bmatrix}$, $\begin{bmatrix}11&8\\12&5\end{bmatrix}$, $\begin{bmatrix}15&20\\10&19\end{bmatrix}$, $\begin{bmatrix}23&16\\2&21\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.0-24.e.1.1, 24.96.0-24.e.1.2, 24.96.0-24.e.1.3, 24.96.0-24.e.1.4, 24.96.0-24.e.1.5, 24.96.0-24.e.1.6, 24.96.0-24.e.1.7, 24.96.0-24.e.1.8, 24.96.0-24.e.1.9, 24.96.0-24.e.1.10, 24.96.0-24.e.1.11, 24.96.0-24.e.1.12, 24.96.0-24.e.1.13, 24.96.0-24.e.1.14, 24.96.0-24.e.1.15, 24.96.0-24.e.1.16, 120.96.0-24.e.1.1, 120.96.0-24.e.1.2, 120.96.0-24.e.1.3, 120.96.0-24.e.1.4, 120.96.0-24.e.1.5, 120.96.0-24.e.1.6, 120.96.0-24.e.1.7, 120.96.0-24.e.1.8, 120.96.0-24.e.1.9, 120.96.0-24.e.1.10, 120.96.0-24.e.1.11, 120.96.0-24.e.1.12, 120.96.0-24.e.1.13, 120.96.0-24.e.1.14, 120.96.0-24.e.1.15, 120.96.0-24.e.1.16, 168.96.0-24.e.1.1, 168.96.0-24.e.1.2, 168.96.0-24.e.1.3, 168.96.0-24.e.1.4, 168.96.0-24.e.1.5, 168.96.0-24.e.1.6, 168.96.0-24.e.1.7, 168.96.0-24.e.1.8, 168.96.0-24.e.1.9, 168.96.0-24.e.1.10, 168.96.0-24.e.1.11, 168.96.0-24.e.1.12, 168.96.0-24.e.1.13, 168.96.0-24.e.1.14, 168.96.0-24.e.1.15, 168.96.0-24.e.1.16, 264.96.0-24.e.1.1, 264.96.0-24.e.1.2, 264.96.0-24.e.1.3, 264.96.0-24.e.1.4, 264.96.0-24.e.1.5, 264.96.0-24.e.1.6, 264.96.0-24.e.1.7, 264.96.0-24.e.1.8, 264.96.0-24.e.1.9, 264.96.0-24.e.1.10, 264.96.0-24.e.1.11, 264.96.0-24.e.1.12, 264.96.0-24.e.1.13, 264.96.0-24.e.1.14, 264.96.0-24.e.1.15, 264.96.0-24.e.1.16, 312.96.0-24.e.1.1, 312.96.0-24.e.1.2, 312.96.0-24.e.1.3, 312.96.0-24.e.1.4, 312.96.0-24.e.1.5, 312.96.0-24.e.1.6, 312.96.0-24.e.1.7, 312.96.0-24.e.1.8, 312.96.0-24.e.1.9, 312.96.0-24.e.1.10, 312.96.0-24.e.1.11, 312.96.0-24.e.1.12, 312.96.0-24.e.1.13, 312.96.0-24.e.1.14, 312.96.0-24.e.1.15, 312.96.0-24.e.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} + 12 z^{2} $
 Copy content Toggle raw display

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

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.24.0.c.1 $8$ $2$ $2$ $0$ $0$
24.24.0.h.2 $24$ $2$ $2$ $0$ $0$
24.24.0.i.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.k.2 $24$ $2$ $2$ $1$
24.96.1.v.2 $24$ $2$ $2$ $1$
24.96.1.bb.2 $24$ $2$ $2$ $1$
24.96.1.bh.2 $24$ $2$ $2$ $1$
24.96.1.bj.2 $24$ $2$ $2$ $1$
24.96.1.bp.2 $24$ $2$ $2$ $1$
24.96.1.bq.2 $24$ $2$ $2$ $1$
24.96.1.br.2 $24$ $2$ $2$ $1$
24.144.8.s.1 $24$ $3$ $3$ $8$
24.192.7.m.2 $24$ $4$ $4$ $7$
120.96.1.dn.1 $120$ $2$ $2$ $1$
120.96.1.dp.1 $120$ $2$ $2$ $1$
120.96.1.dx.2 $120$ $2$ $2$ $1$
120.96.1.ed.2 $120$ $2$ $2$ $1$
120.96.1.en.2 $120$ $2$ $2$ $1$
120.96.1.et.2 $120$ $2$ $2$ $1$
120.96.1.fb.1 $120$ $2$ $2$ $1$
120.96.1.fd.1 $120$ $2$ $2$ $1$
120.240.16.j.2 $120$ $5$ $5$ $16$
120.288.15.o.1 $120$ $6$ $6$ $15$
168.96.1.dn.2 $168$ $2$ $2$ $1$
168.96.1.dp.2 $168$ $2$ $2$ $1$
168.96.1.dx.2 $168$ $2$ $2$ $1$
168.96.1.ed.2 $168$ $2$ $2$ $1$
168.96.1.en.2 $168$ $2$ $2$ $1$
168.96.1.et.2 $168$ $2$ $2$ $1$
168.96.1.fb.2 $168$ $2$ $2$ $1$
168.96.1.fd.2 $168$ $2$ $2$ $1$
168.384.23.n.2 $168$ $8$ $8$ $23$
264.96.1.dn.2 $264$ $2$ $2$ $1$
264.96.1.dp.2 $264$ $2$ $2$ $1$
264.96.1.dx.2 $264$ $2$ $2$ $1$
264.96.1.ed.2 $264$ $2$ $2$ $1$
264.96.1.en.2 $264$ $2$ $2$ $1$
264.96.1.et.2 $264$ $2$ $2$ $1$
264.96.1.fb.2 $264$ $2$ $2$ $1$
264.96.1.fd.2 $264$ $2$ $2$ $1$
312.96.1.dn.2 $312$ $2$ $2$ $1$
312.96.1.dp.2 $312$ $2$ $2$ $1$
312.96.1.dx.2 $312$ $2$ $2$ $1$
312.96.1.ed.2 $312$ $2$ $2$ $1$
312.96.1.en.2 $312$ $2$ $2$ $1$
312.96.1.et.2 $312$ $2$ $2$ $1$
312.96.1.fb.2 $312$ $2$ $2$ $1$
312.96.1.fd.2 $312$ $2$ $2$ $1$