Modular curve 24.16.0.a.1

• Label: 24.16.0.a.1
• Level: $24$
• Index: $16$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $2$
• $\Q$-cusps: $2$

Invariants

Level:	$24$		$\SL_2$-level:	$12$
Index:	$16$		$\PSL_2$-index:	$16$
Genus:	$0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$
Cusps:	$2$ (all of which are rational)		Cusp widths	$4\cdot12$		Cusp orbits	$1^{2}$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	12B0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.16.0.11

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}5&6\\0&5\end{bmatrix}$, $\begin{bmatrix}10&1\\9&17\end{bmatrix}$, $\begin{bmatrix}11&12\\18&1\end{bmatrix}$, $\begin{bmatrix}19&1\\9&16\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	24.32.0-24.a.1.1, 24.32.0-24.a.1.2, 24.32.0-24.a.1.3, 24.32.0-24.a.1.4, 24.32.0-24.a.1.5, 24.32.0-24.a.1.6, 24.32.0-24.a.1.7, 24.32.0-24.a.1.8, 120.32.0-24.a.1.1, 120.32.0-24.a.1.2, 120.32.0-24.a.1.3, 120.32.0-24.a.1.4, 120.32.0-24.a.1.5, 120.32.0-24.a.1.6, 120.32.0-24.a.1.7, 120.32.0-24.a.1.8, 168.32.0-24.a.1.1, 168.32.0-24.a.1.2, 168.32.0-24.a.1.3, 168.32.0-24.a.1.4, 168.32.0-24.a.1.5, 168.32.0-24.a.1.6, 168.32.0-24.a.1.7, 168.32.0-24.a.1.8, 264.32.0-24.a.1.1, 264.32.0-24.a.1.2, 264.32.0-24.a.1.3, 264.32.0-24.a.1.4, 264.32.0-24.a.1.5, 264.32.0-24.a.1.6, 264.32.0-24.a.1.7, 264.32.0-24.a.1.8, 312.32.0-24.a.1.1, 312.32.0-24.a.1.2, 312.32.0-24.a.1.3, 312.32.0-24.a.1.4, 312.32.0-24.a.1.5, 312.32.0-24.a.1.6, 312.32.0-24.a.1.7, 312.32.0-24.a.1.8
Cyclic 24-isogeny field degree:	$12$
Cyclic 24-torsion field degree:	$96$
Full 24-torsion field degree:	$4608$

Models

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

Rational points

This modular curve has infinitely many rational points, including 10 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 16 to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^{18}}\cdot\frac{(x+y)^{16}(x^{4}+192y^{4})^{3}(x^{4}+1728y^{4})}{y^{12}x^{4}(x+y)^{16}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
6.8.0.a.1	$6$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
24.48.2.b.2	$24$	$3$	$3$	$2$
24.48.3.e.1	$24$	$3$	$3$	$3$
24.64.1.a.2	$24$	$4$	$4$	$1$
72.48.0.a.1	$72$	$3$	$3$	$0$
72.48.2.a.1	$72$	$3$	$3$	$2$
72.48.2.b.2	$72$	$3$	$3$	$2$
72.48.3.a.2	$72$	$3$	$3$	$3$
72.48.4.a.1	$72$	$3$	$3$	$4$
120.80.4.c.2	$120$	$5$	$5$	$4$
120.96.7.e.2	$120$	$6$	$6$	$7$
120.160.11.i.1	$120$	$10$	$10$	$11$
168.48.2.j.2	$168$	$3$	$3$	$2$
168.48.2.k.1	$168$	$3$	$3$	$2$
168.128.7.g.1	$168$	$8$	$8$	$7$
264.192.15.g.2	$264$	$12$	$12$	$15$
312.48.2.l.2	$312$	$3$	$3$	$2$
312.48.2.m.1	$312$	$3$	$3$	$2$
312.224.15.e.1	$312$	$14$	$14$	$15$