Modular curve 24.12.0-4.b.1.3

• Label: 24.12.0-4.b.1.3
• Level: $24$
• Index: $12$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $3$
• Q\Q-cusps: $1$

Invariants

Level:	$24$		$\SL_2$-level:	$8$
Index:	$12$		$\PSL_2$-index:	$6$
Genus:	$0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps:	$3$ (of which $1$ is rational)		Cusp widths	$1^{2}\cdot4$		Cusp orbits	$1\cdot2$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$1$
Rational CM points:	yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label:	4B0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.12.0.28

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}5&1\\20&21\end{bmatrix}$, $\begin{bmatrix}11&6\\2&17\end{bmatrix}$, $\begin{bmatrix}11&14\\6&1\end{bmatrix}$, $\begin{bmatrix}13&13\\8&13\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 4.6.0.b.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree:	$16$
Cyclic 24-torsion field degree:	$128$
Full 24-torsion field degree:	$6144$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 11629 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 6 to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{x^6 (x^2+48 y^2)^3}{y^4 x^6 (x^2+64 y^2)}$

Modular covers

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

Covering curve	Level	Index	Degree	Genus
24.24.0-4.a.1.3	$24$	$2$	$2$	$0$
24.24.0-4.c.1.2	$24$	$2$	$2$	$0$
24.24.0-8.c.1.2	$24$	$2$	$2$	$0$
24.24.0-12.e.1.2	$24$	$2$	$2$	$0$
24.24.0-12.f.1.2	$24$	$2$	$2$	$0$
24.24.0-8.h.1.1	$24$	$2$	$2$	$0$
24.24.0-24.m.1.1	$24$	$2$	$2$	$0$
24.24.0-24.p.1.1	$24$	$2$	$2$	$0$
24.36.1-12.b.1.5	$24$	$3$	$3$	$1$
24.48.0-12.f.1.7	$24$	$4$	$4$	$0$
72.324.10-36.c.1.4	$72$	$27$	$27$	$10$
120.24.0-20.e.1.3	$120$	$2$	$2$	$0$
120.24.0-60.e.1.6	$120$	$2$	$2$	$0$
120.24.0-20.f.1.3	$120$	$2$	$2$	$0$
120.24.0-60.f.1.8	$120$	$2$	$2$	$0$
120.24.0-40.m.1.3	$120$	$2$	$2$	$0$
120.24.0-120.m.1.2	$120$	$2$	$2$	$0$
120.24.0-40.p.1.3	$120$	$2$	$2$	$0$
120.24.0-120.p.1.2	$120$	$2$	$2$	$0$
120.60.2-20.b.1.1	$120$	$5$	$5$	$2$
120.72.1-20.b.1.11	$120$	$6$	$6$	$1$
120.120.3-20.b.1.11	$120$	$10$	$10$	$3$
168.24.0-28.e.1.2	$168$	$2$	$2$	$0$
168.24.0-84.e.1.6	$168$	$2$	$2$	$0$
168.24.0-28.f.1.2	$168$	$2$	$2$	$0$
168.24.0-84.f.1.8	$168$	$2$	$2$	$0$
168.24.0-56.m.1.2	$168$	$2$	$2$	$0$
168.24.0-168.m.1.5	$168$	$2$	$2$	$0$
168.24.0-56.p.1.2	$168$	$2$	$2$	$0$
168.24.0-168.p.1.7	$168$	$2$	$2$	$0$
168.96.2-28.b.1.11	$168$	$8$	$8$	$2$
168.252.7-28.b.1.15	$168$	$21$	$21$	$7$
168.336.9-28.b.1.15	$168$	$28$	$28$	$9$
264.24.0-44.e.1.3	$264$	$2$	$2$	$0$
264.24.0-132.e.1.6	$264$	$2$	$2$	$0$
264.24.0-44.f.1.1	$264$	$2$	$2$	$0$
264.24.0-132.f.1.8	$264$	$2$	$2$	$0$
264.24.0-88.m.1.3	$264$	$2$	$2$	$0$
264.24.0-264.m.1.5	$264$	$2$	$2$	$0$
264.24.0-88.p.1.3	$264$	$2$	$2$	$0$
264.24.0-264.p.1.5	$264$	$2$	$2$	$0$
264.144.4-44.b.1.13	$264$	$12$	$12$	$4$
312.24.0-52.e.1.2	$312$	$2$	$2$	$0$
312.24.0-156.e.1.7	$312$	$2$	$2$	$0$
312.24.0-52.f.1.2	$312$	$2$	$2$	$0$
312.24.0-156.f.1.8	$312$	$2$	$2$	$0$
312.24.0-104.m.1.2	$312$	$2$	$2$	$0$
312.24.0-312.m.1.5	$312$	$2$	$2$	$0$
312.24.0-104.p.1.2	$312$	$2$	$2$	$0$
312.24.0-312.p.1.3	$312$	$2$	$2$	$0$
312.168.5-52.b.1.9	$312$	$14$	$14$	$5$