Modular curve 24.384.5-24.bx.1.5

• Label: 24.384.5-24.bx.1.5
• Level: $24$
• Index: $384$
• Genus: $5$
• Analytic rank: $0$
• Cusps: $24$
• $\Q$-cusps: $0$

Invariants

Level:	$24$		$\SL_2$-level:	$12$		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:	$24$ (none of which are rational)		Cusp widths	$4^{12}\cdot12^{12}$		Cusp orbits	$2^{4}\cdot4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	12E5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.384.5.930

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}1&18\\0&7\end{bmatrix}$, $\begin{bmatrix}13&20\\10&15\end{bmatrix}$, $\begin{bmatrix}19&18\\16&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup:	$C_2^4.D_6$
Contains $-I$:	no $\quad$ (see 24.192.5.bx.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree:	$4$
Cyclic 24-torsion field degree:	$16$
Full 24-torsion field degree:	$192$

Jacobian

Conductor:	$2^{27}\cdot3^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	72.2.a.a, 192.2.a.b, 192.2.c.a, 576.2.a.d

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 108 x^{6} y^{2} - 54 x^{5} y z^{2} + 36 x^{4} y^{4} + 162 x^{4} y^{2} z^{2} + 9 x^{4} z^{4} - 72 x^{3} y^{5} + \cdots + z^{8} $

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle 2^4\,\frac{364xw^{23}-4004xw^{21}t^{2}+26040xw^{19}t^{4}-114240xw^{17}t^{6}+367872xw^{15}t^{8}-891072xw^{13}t^{10}+1616512xw^{11}t^{12}-2155520xw^{9}t^{14}+1972224xw^{7}t^{16}-1086464xw^{5}t^{18}+292864xw^{3}t^{20}-24576xwt^{22}-364yw^{23}+4004yw^{21}t^{2}-26040yw^{19}t^{4}+114240yw^{17}t^{6}-367872yw^{15}t^{8}+891072yw^{13}t^{10}-1616512yw^{11}t^{12}+2155520yw^{9}t^{14}-1972224yw^{7}t^{16}+1086464yw^{5}t^{18}-292864yw^{3}t^{20}+24576ywt^{22}+365w^{24}-4380w^{22}t^{2}+29916w^{20}t^{4}-138560w^{18}t^{6}+470880w^{16}t^{8}-1212480w^{14}t^{10}+2367424w^{12}t^{12}-3457536w^{10}t^{14}+3608064w^{8}t^{16}-2446336w^{6}t^{18}+924672w^{4}t^{20}-147456w^{2}t^{22}+4096t^{24}}{t^{4}w^{6}(w^{2}-2t^{2})^{3}(162xw^{7}-486xw^{5}t^{2}+396xw^{3}t^{4}-72xwt^{6}-162yw^{7}+486yw^{5}t^{2}-396yw^{3}t^{4}+72ywt^{6}+162w^{8}-648w^{6}t^{2}+801w^{4}t^{4}-306w^{2}t^{6}+16t^{8})}$

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

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle t$

Equation of the image curve:

$0$

$=$

$ 108X^{6}Y^{2}-54X^{5}YZ^{2}+36X^{4}Y^{4}+162X^{4}Y^{2}Z^{2}+9X^{4}Z^{4}-72X^{3}Y^{5}-36X^{3}Y^{3}Z^{2}-72X^{3}YZ^{4}+36X^{2}Y^{6}+6X^{2}Y^{4}Z^{2}+42X^{2}Y^{2}Z^{4}+12X^{2}Z^{6}-6XY^{5}Z^{2}-6XYZ^{6}+Y^{4}Z^{4}+Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
12.192.1-12.d.1.8	$12$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-12.d.1.2	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.cl.4.6	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.cl.4.15	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.cn.4.5	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.1-24.cn.4.16	$24$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
24.192.3-24.bh.1.10	$24$	$2$	$2$	$3$	$0$	$2$
24.192.3-24.bh.1.11	$24$	$2$	$2$	$3$	$0$	$2$
24.192.3-24.bw.2.4	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.bw.2.14	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.bz.1.4	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.bz.1.13	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.cb.1.1	$24$	$2$	$2$	$3$	$0$	$1^{2}$
24.192.3-24.cb.1.13	$24$	$2$	$2$	$3$	$0$	$1^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
24.1152.25-24.bv.1.8	$24$	$3$	$3$	$25$	$2$	$1^{10}\cdot2^{5}$