Modular curve 24.384.5-24.fa.2.8

• Label: 24.384.5-24.fa.2.8
• Level: $24$
• Index: $384$
• Genus: $5$
• Analytic rank: $1$
• Cusps: $24$
• $\Q$-cusps: $4$

Invariants

Level:	$24$		$\SL_2$-level:	$24$		Newform level:	$576$
Index:	$384$		$\PSL_2$-index:	$192$
Genus:	$5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps:	$24$ (of which $4$ are rational)		Cusp widths	$2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$		Cusp orbits	$1^{4}\cdot2^{2}\cdot4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$4$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	24Z5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.384.5.2833

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}5&7\\0&7\end{bmatrix}$, $\begin{bmatrix}11&11\\0&19\end{bmatrix}$, $\begin{bmatrix}13&4\\0&13\end{bmatrix}$, $\begin{bmatrix}13&11\\0&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup:	$D_8:D_6$
Contains $-I$:	no $\quad$ (see 24.192.5.fa.2 for the level structure with $-I$)
Cyclic 24-isogeny field degree:	$1$
Cyclic 24-torsion field degree:	$8$
Full 24-torsion field degree:	$192$

Jacobian

Conductor:	$2^{27}\cdot3^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	24.2.a.a, 192.2.c.a, 576.2.a.b, 576.2.a.d

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - 18 x^{4} z^{4} - 36 x^{3} y^{2} z^{3} - 36 x^{3} z^{5} - 42 x^{2} y^{4} z^{2} - 48 x^{2} y^{2} z^{4} + \cdots + 9 z^{8} $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(0:1/2:-1/2:1:0)$, $(0:1/2:1/2:0:1)$, $(0:-1/2:-1/2:1:0)$, $(0:-1/2:1/2:0:1)$

Maps to other modular curves

$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^3\,\frac{2097152z^{24}-12582912z^{22}t^{2}+12582912z^{21}t^{3}+12582912z^{20}t^{4}-31457280z^{19}t^{5}+52428800z^{18}t^{6}-94371840z^{17}t^{7}+75497472z^{16}t^{8}+62390272z^{15}t^{9}-262668288z^{14}t^{10}+501743616z^{13}t^{11}-757596160z^{12}t^{12}+870580224z^{11}t^{13}-645660672z^{10}t^{14}-93585408z^{9}t^{15}+1672740864z^{8}t^{16}-4737761280z^{7}t^{17}+10583375872z^{6}t^{18}-22048505856z^{5}t^{19}+45481820160z^{4}t^{20}-95108284416z^{3}t^{21}+203012358144z^{2}t^{22}-441993117696zt^{23}-3w^{23}t-39w^{22}t^{2}-241w^{21}t^{3}-1929w^{20}t^{4}-12429w^{19}t^{5}-40577w^{18}t^{6}-131463w^{17}t^{7}-647415w^{16}t^{8}-1434734w^{15}t^{9}+2509002w^{14}t^{10}+25942470w^{13}t^{11}+97882998w^{12}t^{12}+277703094w^{11}t^{13}+699377454w^{10}t^{14}+1687284290w^{9}t^{15}+4055422914w^{8}t^{16}+9753411105w^{7}t^{17}+22971182093w^{6}t^{18}+51370131195w^{5}t^{19}+105459721635w^{4}t^{20}+189503685927w^{3}t^{21}+265781299875w^{2}t^{22}+146450126709wt^{23}+179844046757t^{24}}{t^{3}(262144z^{18}t^{3}-786432z^{17}t^{4}+589824z^{16}t^{5}+917504z^{15}t^{6}-3145728z^{14}t^{7}+5898240z^{13}t^{8}-10092544z^{12}t^{9}+17399808z^{11}t^{10}-30867456z^{10}t^{11}+56688640z^{9}t^{12}-107544576z^{8}t^{13}+209731584z^{7}t^{14}-418512896z^{6}t^{15}+851238912z^{5}t^{16}-1759322112z^{4}t^{17}+3685636096z^{3}t^{18}-7810633728z^{2}t^{19}+16717271040zt^{20}+w^{21}+3w^{20}t-3w^{19}t^{2}-81w^{18}t^{3}-408w^{17}t^{4}-1224w^{16}t^{5}-2464w^{15}t^{6}-5472w^{14}t^{7}-35358w^{13}t^{8}-268442w^{12}t^{9}-1557102w^{11}t^{10}-7183578w^{10}t^{11}-27833472w^{9}t^{12}-93747456w^{8}t^{13}-280082616w^{7}t^{14}-749881448w^{6}t^{15}-1801657803w^{5}t^{16}-3844711329w^{4}t^{17}-7046132023w^{3}t^{18}-9959749389w^{2}t^{19}-5498247168wt^{20}-6778091520t^{21})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.fa.2 :

$\displaystyle X$	$=$	$\displaystyle x+w$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle z$

Equation of the image curve:

$0$

$=$

$ -18X^{4}Z^{4}-36X^{3}Y^{2}Z^{3}-36X^{3}Z^{5}-42X^{2}Y^{4}Z^{2}-48X^{2}Y^{2}Z^{4}+18X^{2}Z^{6}-24XY^{6}Z-36XY^{4}Z^{3}+24XY^{2}Z^{5}+36XZ^{7}-5Y^{8}-8Y^{6}Z^{2}-8Y^{4}Z^{4}+12Y^{2}Z^{6}+9Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
24.192.1-24.da.1.4	$24$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
24.192.1-24.da.1.15	$24$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
24.192.1-24.dg.1.18	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.dg.1.19	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.do.1.8	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.do.1.12	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.3-24.fb.1.10	$24$	$2$	$2$	$3$	$1$	$2$
24.192.3-24.fb.1.23	$24$	$2$	$2$	$3$	$1$	$2$
24.192.3-24.fr.2.5	$24$	$2$	$2$	$3$	$1$	$1^{2}$
24.192.3-24.fr.2.8	$24$	$2$	$2$	$3$	$1$	$1^{2}$
24.192.3-24.gn.4.7	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.gn.4.20	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.gv.4.10	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.gv.4.16	$24$	$2$	$2$	$3$	$0$	$1^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
24.768.13-24.fw.3.4	$24$	$2$	$2$	$13$	$1$	$2^{4}$
24.768.13-24.fx.2.8	$24$	$2$	$2$	$13$	$1$	$2^{4}$
24.768.13-24.fy.3.4	$24$	$2$	$2$	$13$	$1$	$2^{4}$
24.768.13-24.fz.4.2	$24$	$2$	$2$	$13$	$1$	$2^{4}$
24.1152.25-24.eq.1.12	$24$	$3$	$3$	$25$	$3$	$1^{10}\cdot2^{5}$
48.768.13-48.kp.3.8	$48$	$2$	$2$	$13$	$1$	$2^{4}$
48.768.13-48.kp.4.8	$48$	$2$	$2$	$13$	$1$	$2^{4}$
48.768.13-48.kq.3.8	$48$	$2$	$2$	$13$	$1$	$2^{4}$
48.768.13-48.kq.4.8	$48$	$2$	$2$	$13$	$1$	$2^{4}$
48.768.17-48.wd.3.4	$48$	$2$	$2$	$17$	$2$	$1^{6}\cdot2\cdot4$
48.768.17-48.xa.3.4	$48$	$2$	$2$	$17$	$2$	$1^{6}\cdot2\cdot4$
48.768.17-48.bkx.4.8	$48$	$2$	$2$	$17$	$3$	$1^{6}\cdot2\cdot4$
48.768.17-48.blu.4.8	$48$	$2$	$2$	$17$	$2$	$1^{6}\cdot2\cdot4$