Modular curve 24.96.1-24.il.1.5

• Label: 24.96.1-24.il.1.5
• Level: $24$
• Index: $96$
• Genus: $1$
• Analytic rank: $1$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$24$		$\SL_2$-level:	$12$		Newform level:	$576$
Index:	$96$		$\PSL_2$-index:	$48$
Genus:	$1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.96.1.1480

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}1&23\\0&17\end{bmatrix}$, $\begin{bmatrix}5&3\\12&19\end{bmatrix}$, $\begin{bmatrix}13&1\\6&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup:	Group 768.69820
Contains $-I$:	no $\quad$ (see 24.48.1.il.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree:	$4$
Cyclic 24-torsion field degree:	$32$
Full 24-torsion field degree:	$768$

Jacobian

Conductor:	$2^{6}\cdot3^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	576.2.a.b

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $	$=$	$ x^{2} + 2 y z $
	$=$	$3 x^{2} + 7 y^{2} - 2 y z - y w + z^{2} + z w + w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 28 x^{4} + 2 x^{2} y z + 16 x^{2} z^{2} + y^{2} z^{2} + y z^{3} + z^{4} $

Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{7^2}\cdot\frac{9448990857216yz^{11}+1574903079936yz^{10}w-10236058642944yz^{9}w^{2}-2842888534272yz^{8}w^{3}+2762569259136yz^{7}w^{4}+904262885568yz^{6}w^{5}-204328601568yz^{5}w^{6}-86704187184yz^{4}w^{7}+3203128800yz^{3}w^{8}+2866505760yz^{2}w^{9}+54686664yzw^{10}-25019280yw^{11}-1349860517888z^{12}-1574903079936z^{11}w-337805056512z^{10}w^{2}+1343061764352z^{9}w^{3}+1324551125376z^{8}w^{4}+18659654208z^{7}w^{5}-298304772192z^{6}w^{6}-52388950608z^{5}w^{7}+16408031328z^{4}w^{8}+4263770448z^{3}w^{9}-82382832z^{2}w^{10}-85917024zw^{11}-6205977w^{12}}{z^{2}(72yz^{9}+732yz^{8}w+3642yz^{7}w^{2}+12075yz^{6}w^{3}+29946yz^{5}w^{4}+59052yz^{4}w^{5}-1647114yz^{3}w^{6}-1027437yz^{2}w^{7}+57474yzw^{8}+55071yw^{9}-72z^{10}-732z^{9}w-3630z^{8}w^{2}-11951z^{7}w^{3}-29316z^{6}w^{4}-56910z^{5}w^{5}+158011z^{4}w^{6}+291345z^{3}w^{7}+324915z^{2}w^{8}+118387zw^{9}+3078w^{10})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.il.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle 2w$
$\displaystyle Z$	$=$	$\displaystyle 2z$

Equation of the image curve:

$0$

$=$

$ 28X^{4}+2X^{2}YZ+16X^{2}Z^{2}+Y^{2}Z^{2}+YZ^{3}+Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.48.0-6.b.1.2	$12$	$2$	$2$	$0$	$0$	full Jacobian
24.48.0-6.b.1.5	$24$	$2$	$2$	$0$	$0$	full Jacobian
24.48.0-24.bx.1.7	$24$	$2$	$2$	$0$	$0$	full Jacobian
24.48.0-24.bx.1.14	$24$	$2$	$2$	$0$	$0$	full Jacobian
24.48.1-24.eq.1.6	$24$	$2$	$2$	$1$	$1$	dimension zero
24.48.1-24.eq.1.15	$24$	$2$	$2$	$1$	$1$	dimension zero

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
24.288.5-24.hf.1.2	$24$	$3$	$3$	$5$	$2$	$1^{4}$
72.288.5-72.bf.1.6	$72$	$3$	$3$	$5$	$?$	not computed
72.288.9-72.co.1.7	$72$	$3$	$3$	$9$	$?$	not computed
72.288.9-72.cp.1.4	$72$	$3$	$3$	$9$	$?$	not computed
120.480.17-120.bpx.1.2	$120$	$5$	$5$	$17$	$?$	not computed