Modular curve 120.192.1-24.bj.1.4

• Label: 120.192.1-24.bj.1.4
• Level: $120$
• Index: $192$
• Genus: $1$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$120$		$\SL_2$-level:	$8$		Newform level:	$576$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$4^{8}\cdot8^{8}$		Cusp orbits	$4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 96$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

8K1

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}9&104\\34&45\end{bmatrix}$, $\begin{bmatrix}21&8\\34&59\end{bmatrix}$, $\begin{bmatrix}25&92\\74&87\end{bmatrix}$, $\begin{bmatrix}91&56\\44&65\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 24.96.1.bj.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$48$
Cyclic 120-torsion field degree:	$1536$
Full 120-torsion field degree:	$184320$

Jacobian

Conductor:	$?$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	576.2.a.c

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 2 x^{4} - 4 x^{3} z - 4 x^{2} y^{2} + 2 x^{2} z^{2} + 4 x y^{2} z + 8 y^{4} + 9 y^{2} z^{2} + 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{50331648xz^{21}w^{2}-528482304xz^{20}w^{3}+2774532096xz^{19}w^{4}-9622781952xz^{18}w^{5}+24475336704xz^{17}w^{6}-48009314304xz^{16}w^{7}+74406494208xz^{15}w^{8}-91672510464xz^{14}w^{9}+88478367744xz^{13}w^{10}-63135621120xz^{12}w^{11}+26228920320xz^{11}w^{12}+6403491840xz^{10}w^{13}-24137401344xz^{9}w^{14}+26457933312xz^{8}w^{15}-19797432960xz^{7}w^{16}+11303098560xz^{6}w^{17}-5010238944xz^{5}w^{18}+1675954896xz^{4}w^{19}-376483752xz^{3}w^{20}+30863628xz^{2}w^{21}+19026330xzw^{22}-7042071xw^{23}-8388608z^{24}+100663296z^{23}w-629145600z^{22}w^{2}+2675965952z^{21}w^{3}-8477736960z^{20}w^{4}+20831010816z^{19}w^{5}-40363884544z^{18}w^{6}+61502128128z^{17}w^{7}-71282884608z^{16}w^{8}+55454203904z^{15}w^{9}-10094690304z^{14}w^{10}-51028475904z^{13}w^{11}+102416744448z^{12}w^{12}-123259404288z^{11}w^{13}+110407351296z^{10}w^{14}-77352152064z^{9}w^{15}+42296006784z^{8}w^{16}-17007942144z^{7}w^{17}+3806245344z^{6}w^{18}+796031328z^{5}w^{19}-1340638608z^{4}w^{20}+783063680z^{3}w^{21}-303405030z^{2}w^{22}+79333686zw^{23}-17267750w^{24}}{w^{8}(32768xz^{15}-245760xz^{14}w+843776xz^{13}w^{2}-1757184xz^{12}w^{3}+2242560xz^{11}w^{4}-1205248xz^{10}w^{5}-1627648xz^{9}w^{6}+4942080xz^{8}w^{7}-6679936xz^{7}w^{8}+5937472xz^{6}w^{9}-3652512xz^{5}w^{10}+1488624xz^{4}w^{11}-326472xz^{3}w^{12}-7524xz^{2}w^{13}+20938xzw^{14}-2967xw^{15}+139264z^{14}w^{2}-974848z^{13}w^{3}+3440640z^{12}w^{4}-7970816z^{11}w^{5}+13240832z^{10}w^{6}-16372224z^{9}w^{7}+15110272z^{8}w^{8}-10057216z^{7}w^{9}+4240800z^{6}w^{10}-427488z^{5}w^{11}-810096z^{4}w^{12}+625920z^{3}w^{13}-221926z^{2}w^{14}+36886zw^{15}-1830w^{16})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.bj.1 :

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

Equation of the image curve:

$0$

$=$

$ 2X^{4}-4X^{2}Y^{2}+8Y^{4}-4X^{3}Z+4XY^{2}Z+2X^{2}Z^{2}+9Y^{2}Z^{2}+3Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
40.96.0-8.d.1.1	$40$	$2$	$2$	$0$	$0$	full Jacobian
120.96.0-8.d.1.4	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.e.2.4	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.e.2.9	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.t.1.4	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.t.1.8	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.u.2.8	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.u.2.16	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.1-24.bb.1.6	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bb.1.8	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bg.2.2	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bg.2.8	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bh.2.4	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bh.2.6	$120$	$2$	$2$	$1$	$?$	dimension zero