Modular curve 24.48.1.ii.1

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

Invariants

Level:	$24$		$\SL_2$-level:	$12$		Newform level:	$576$
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:	$1$
$\Q$-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:	24.48.1.550

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}1&22\\6&23\end{bmatrix}$, $\begin{bmatrix}5&3\\12&1\end{bmatrix}$, $\begin{bmatrix}7&9\\6&7\end{bmatrix}$, $\begin{bmatrix}13&2\\12&11\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	24.96.1-24.ii.1.1, 24.96.1-24.ii.1.2, 24.96.1-24.ii.1.3, 24.96.1-24.ii.1.4, 24.96.1-24.ii.1.5, 24.96.1-24.ii.1.6, 24.96.1-24.ii.1.7, 24.96.1-24.ii.1.8, 120.96.1-24.ii.1.1, 120.96.1-24.ii.1.2, 120.96.1-24.ii.1.3, 120.96.1-24.ii.1.4, 120.96.1-24.ii.1.5, 120.96.1-24.ii.1.6, 120.96.1-24.ii.1.7, 120.96.1-24.ii.1.8, 168.96.1-24.ii.1.1, 168.96.1-24.ii.1.2, 168.96.1-24.ii.1.3, 168.96.1-24.ii.1.4, 168.96.1-24.ii.1.5, 168.96.1-24.ii.1.6, 168.96.1-24.ii.1.7, 168.96.1-24.ii.1.8, 264.96.1-24.ii.1.1, 264.96.1-24.ii.1.2, 264.96.1-24.ii.1.3, 264.96.1-24.ii.1.4, 264.96.1-24.ii.1.5, 264.96.1-24.ii.1.6, 264.96.1-24.ii.1.7, 264.96.1-24.ii.1.8, 312.96.1-24.ii.1.1, 312.96.1-24.ii.1.2, 312.96.1-24.ii.1.3, 312.96.1-24.ii.1.4, 312.96.1-24.ii.1.5, 312.96.1-24.ii.1.6, 312.96.1-24.ii.1.7, 312.96.1-24.ii.1.8
Cyclic 24-isogeny field degree:	$4$
Cyclic 24-torsion field degree:	$32$
Full 24-torsion field degree:	$1536$

Jacobian

Conductor:	$2^{6}\cdot3^{2}$
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 $
	$=$	$12 x^{2} - 3 y^{2} - 6 y z - 27 z^{2} + w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 36 x^{4} - 20 x^{2} z^{2} - 3 y^{2} z^{2} + 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:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle \frac{2}{3}w$
$\displaystyle Z$	$=$	$\displaystyle 6z$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{1}{3^3}\cdot\frac{271724544yz^{11}+121927680yz^{9}w^{2}+19035648yz^{7}w^{4}+1154304yz^{5}w^{6}+18864yz^{3}w^{8}+72yzw^{10}+268738560z^{12}+110979072z^{10}w^{2}+13996800z^{8}w^{4}+387072z^{6}w^{6}-25344z^{4}w^{8}-504z^{2}w^{10}-w^{12}}{w^{2}z^{6}(5832yz^{3}+54yzw^{2}+5832z^{4}-189z^{2}w^{2}-w^{4})}$

Modular covers

Hi

Sorry, your browser does not support the nearby lattice.

Cover information Click on a modular curve in the diagram to see information about it.

See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.24.0.h.1	$12$	$2$	$2$	$0$	$0$	full Jacobian
24.12.0.q.1	$24$	$4$	$4$	$0$	$0$	full Jacobian
24.24.0.bw.1	$24$	$2$	$2$	$0$	$0$	full Jacobian
24.24.1.eq.1	$24$	$2$	$2$	$1$	$1$	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
24.144.5.fs.1	$24$	$3$	$3$	$5$	$3$	$1^{4}$
72.144.5.bc.1	$72$	$3$	$3$	$5$	$?$	not computed
72.144.9.ch.1	$72$	$3$	$3$	$9$	$?$	not computed
72.144.9.ck.1	$72$	$3$	$3$	$9$	$?$	not computed
120.240.17.bpu.1	$120$	$5$	$5$	$17$	$?$	not computed
120.288.17.bkey.1	$120$	$6$	$6$	$17$	$?$	not computed