Modular curve 24.48.1.iz.1

• Label: 24.48.1.iz.1
• Level: $24$
• Index: $48$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$24$		$\SL_2$-level:	$12$		Newform level:	$192$
Index:	$48$		$\PSL_2$-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:	$0$
$\Q$-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:	24.48.1.543

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}5&9\\12&5\end{bmatrix}$, $\begin{bmatrix}5&17\\12&23\end{bmatrix}$, $\begin{bmatrix}17&18\\12&11\end{bmatrix}$, $\begin{bmatrix}23&2\\18&13\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	24.96.1-24.iz.1.1, 24.96.1-24.iz.1.2, 24.96.1-24.iz.1.3, 24.96.1-24.iz.1.4, 24.96.1-24.iz.1.5, 24.96.1-24.iz.1.6, 24.96.1-24.iz.1.7, 24.96.1-24.iz.1.8, 120.96.1-24.iz.1.1, 120.96.1-24.iz.1.2, 120.96.1-24.iz.1.3, 120.96.1-24.iz.1.4, 120.96.1-24.iz.1.5, 120.96.1-24.iz.1.6, 120.96.1-24.iz.1.7, 120.96.1-24.iz.1.8, 168.96.1-24.iz.1.1, 168.96.1-24.iz.1.2, 168.96.1-24.iz.1.3, 168.96.1-24.iz.1.4, 168.96.1-24.iz.1.5, 168.96.1-24.iz.1.6, 168.96.1-24.iz.1.7, 168.96.1-24.iz.1.8, 264.96.1-24.iz.1.1, 264.96.1-24.iz.1.2, 264.96.1-24.iz.1.3, 264.96.1-24.iz.1.4, 264.96.1-24.iz.1.5, 264.96.1-24.iz.1.6, 264.96.1-24.iz.1.7, 264.96.1-24.iz.1.8, 312.96.1-24.iz.1.1, 312.96.1-24.iz.1.2, 312.96.1-24.iz.1.3, 312.96.1-24.iz.1.4, 312.96.1-24.iz.1.5, 312.96.1-24.iz.1.6, 312.96.1-24.iz.1.7, 312.96.1-24.iz.1.8
Cyclic 24-isogeny field degree:	$4$
Cyclic 24-torsion field degree:	$32$
Full 24-torsion field degree:	$1536$

Jacobian

Conductor:	$2^{6}\cdot3$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	192.2.a.b

Models

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

$ 0 $	$=$	$ 3 x y - z^{2} $
	$=$	$6 x^{2} + 6 x y + 54 y^{2} + 18 z^{2} - w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 3 x^{4} - 2 x^{2} y^{2} + 10 x^{2} z^{2} + 3 z^{4} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{6}w$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}z$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^3}\cdot\frac{(8z^{2}-w^{2})(8945664y^{2}z^{8}+208896y^{2}z^{6}w^{2}-87552y^{2}z^{4}w^{4}+87360y^{2}z^{2}w^{6}-4368y^{2}w^{8}+327680z^{10}+49152z^{8}w^{2}-41472z^{6}w^{4}+31424z^{4}w^{6}-3240z^{2}w^{8}+81w^{10})}{w^{2}z^{4}(96y^{2}z^{4}+12y^{2}z^{2}w^{2}-3y^{2}w^{4}+32z^{6}+2z^{4}w^{2})}$

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
12.24.0.j.1	$12$	$2$	$2$	$0$	$0$	full Jacobian
24.12.0.bf.1	$24$	$4$	$4$	$0$	$0$	full Jacobian
24.24.0.ca.1	$24$	$2$	$2$	$0$	$0$	full Jacobian
24.24.1.es.1	$24$	$2$	$2$	$1$	$0$	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
24.144.5.hz.1	$24$	$3$	$3$	$5$	$1$	$1^{4}$
72.144.5.bt.1	$72$	$3$	$3$	$5$	$?$	not computed
72.144.9.eu.1	$72$	$3$	$3$	$9$	$?$	not computed
72.144.9.ev.1	$72$	$3$	$3$	$9$	$?$	not computed
120.240.17.flv.1	$120$	$5$	$5$	$17$	$?$	not computed
120.288.17.cguj.1	$120$	$6$	$6$	$17$	$?$	not computed