Modular curve 24.288.9.e.2

• Label: 24.288.9.e.2
• Level: $24$
• Index: $288$
• Genus: $9$
• Analytic rank: $2$
• Cusps: $32$
• $\Q$-cusps: $2$

Invariants

Level:	$24$		$\SL_2$-level:	$12$		Newform level:	$576$
Index:	$288$		$\PSL_2$-index:	$288$
Genus:	$9 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$
Cusps:	$32$ (of which $2$ are rational)		Cusp widths	$6^{16}\cdot12^{16}$		Cusp orbits	$1^{2}\cdot2^{5}\cdot4^{3}\cdot8$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$2$
$\Q$-gonality:	$4 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 6$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	12B9
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.288.9.217

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}5&0\\6&13\end{bmatrix}$, $\begin{bmatrix}11&0\\12&11\end{bmatrix}$, $\begin{bmatrix}19&0\\18&5\end{bmatrix}$, $\begin{bmatrix}19&6\\12&1\end{bmatrix}$, $\begin{bmatrix}23&12\\6&5\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup:	$C_2^5:D_4$
Contains $-I$:	yes
Quadratic refinements:	24.576.9-24.e.2.1, 24.576.9-24.e.2.2, 24.576.9-24.e.2.3, 24.576.9-24.e.2.4, 24.576.9-24.e.2.5, 24.576.9-24.e.2.6, 24.576.9-24.e.2.7, 24.576.9-24.e.2.8, 24.576.9-24.e.2.9, 24.576.9-24.e.2.10, 24.576.9-24.e.2.11, 24.576.9-24.e.2.12, 24.576.9-24.e.2.13, 24.576.9-24.e.2.14, 24.576.9-24.e.2.15, 24.576.9-24.e.2.16
Cyclic 24-isogeny field degree:	$4$
Cyclic 24-torsion field degree:	$16$
Full 24-torsion field degree:	$256$

Jacobian

Conductor:	$2^{42}\cdot3^{17}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{5}\cdot2^{2}$
Newforms:	36.2.a.a, 36.2.b.a, 192.2.a.b, 576.2.a.b$^{2}$, 576.2.a.f, 576.2.c.b

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $	$=$	$ z^{2} - t u + t v + u r - u s $
	$=$	$z v - t v + t r - t s$
	$=$	$z^{2} - t u + t v - u r + r^{2} - r s$
	$=$	$z^{2} - z r - t u + u v - v^{2} - v s$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 3 x^{12} - 9 x^{10} y^{2} + 18 x^{10} z^{2} + 9 x^{8} y^{4} + 114 x^{8} y^{2} z^{2} + 36 x^{8} z^{4} + \cdots + 8 y^{6} z^{6} $

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(0:0:0:0:-1:0:0:1:1)$, $(0:0:0:0:0:0:-1:1:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{2}z$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 12.144.3.a.1 :

$\displaystyle X$	$=$	$\displaystyle 3x-y+w$
$\displaystyle Y$	$=$	$\displaystyle 3x+y-w$
$\displaystyle Z$	$=$	$\displaystyle -2y-w$

Equation of the image curve:

$0$

$=$

$ X^{3}Y+XY^{3}-XZ^{3}+YZ^{3} $

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.144.3.a.1	$12$	$2$	$2$	$3$	$0$	$1^{4}\cdot2$
24.96.1.cp.2	$24$	$3$	$3$	$1$	$1$	$1^{4}\cdot2^{2}$
24.96.1.cp.4	$24$	$3$	$3$	$1$	$1$	$1^{4}\cdot2^{2}$
24.144.3.a.1	$24$	$2$	$2$	$3$	$0$	$1^{4}\cdot2$
24.144.5.m.1	$24$	$2$	$2$	$5$	$2$	$2^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
24.576.25.bg.1	$24$	$2$	$2$	$25$	$4$	$1^{8}\cdot2^{4}$
24.576.25.bh.1	$24$	$2$	$2$	$25$	$4$	$1^{8}\cdot2^{4}$
24.576.25.bj.1	$24$	$2$	$2$	$25$	$3$	$1^{8}\cdot2^{4}$
24.576.25.bk.1	$24$	$2$	$2$	$25$	$3$	$1^{8}\cdot2^{4}$
24.576.25.bq.1	$24$	$2$	$2$	$25$	$3$	$1^{8}\cdot2^{4}$
24.576.25.br.1	$24$	$2$	$2$	$25$	$3$	$1^{8}\cdot2^{4}$
24.576.25.bu.1	$24$	$2$	$2$	$25$	$3$	$1^{8}\cdot2^{4}$
24.576.25.bw.2	$24$	$2$	$2$	$25$	$3$	$1^{8}\cdot2^{4}$