Modular curve 120.192.1-24.bp.2.2

• Label: 120.192.1-24.bp.2.2
• 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	$2^{4}\cdot4^{2}$
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:	$\begin{bmatrix}5&36\\28&109\end{bmatrix}$, $\begin{bmatrix}27&92\\98&109\end{bmatrix}$, $\begin{bmatrix}59&60\\76&97\end{bmatrix}$, $\begin{bmatrix}97&112\\60&91\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 24.96.1.bp.2 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$48$
Cyclic 120-torsion field degree:	$768$
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} $
	$=$	$ - 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 + 8 x^{2} y^{2} - 2 x^{2} z^{2} - 8 x y^{2} z + 4 x z^{3} + 2 y^{4} + 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 to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^2\,\frac{8632565760xz^{23}-99274506240xz^{22}w+562810073088xz^{21}w^{2}-2087437277184xz^{20}w^{3}+5672786214912xz^{19}w^{4}-11995722369024xz^{18}w^{5}+20469457631232xz^{17}w^{6}-28859429385216xz^{16}w^{7}+34154022506496xz^{15}w^{8}-34290823231488xz^{14}w^{9}+29407797496320xz^{13}w^{10}-21625817351424xz^{12}w^{11}+13654902610944xz^{11}w^{12}-7395702574848xz^{10}w^{13}+3424505617536xz^{9}w^{14}-1347686093376xz^{8}w^{15}+446774026368xz^{7}w^{16}-123186318912xz^{6}w^{17}+27745004976xz^{5}w^{18}-4974025560xz^{4}w^{19}+682791168xz^{3}w^{20}-67408200xz^{2}w^{21}+4262016xzw^{22}-129672xw^{23}-6319476736z^{24}+75833720832z^{23}w-452080570368z^{22}w^{2}+1775231045632z^{21}w^{3}-5139500402688z^{20}w^{4}+11645504028672z^{19}w^{5}-21412666417152z^{18}w^{6}+32709378521088z^{17}w^{7}-42174880300800z^{16}w^{8}+46398238676992z^{15}w^{9}-43865148102144z^{14}w^{10}+35792495162880z^{13}w^{11}-25257408732672z^{12}w^{12}+15412470618624z^{11}w^{13}-8115692095104z^{10}w^{14}+3672081205888z^{9}w^{15}-1418270770416z^{8}w^{16}+463167005376z^{7}w^{17}-126209254928z^{6}w^{18}+28170884112z^{5}w^{19}-5017089336z^{4}w^{20}+685574880z^{3}w^{21}-67494648z^{2}w^{22}+4262016zw^{23}-129673w^{24}}{w^{8}(43456xz^{15}-325920xz^{14}w+1193480xz^{13}w^{2}-2814500xz^{12}w^{3}+4758480xz^{11}w^{4}-6087004xz^{10}w^{5}+6066742xz^{9}w^{6}-4783779xz^{8}w^{7}+3001096xz^{7}w^{8}-1494154xz^{6}w^{9}+584082xz^{5}w^{10}-175601xz^{4}w^{11}+39196xz^{3}w^{12}-6114xz^{2}w^{13}+594xzw^{14}-27xw^{15}-31812z^{16}+254496z^{15}w-1011800z^{14}w^{2}+2628920z^{13}w^{3}-4962208z^{12}w^{4}+7176856z^{11}w^{5}-8191942z^{10}w^{6}+7500350z^{9}w^{7}-5549047z^{8}w^{8}+3318736z^{7}w^{9}-1594760z^{6}w^{10}+607586z^{5}w^{11}-179417z^{4}w^{12}+39580z^{3}w^{13}-6132z^{2}w^{14}+594zw^{15}-27w^{16})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.bp.2 :

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

Equation of the image curve:

$0$

$=$

$ 2X^{4}+8X^{2}Y^{2}+2Y^{4}-4X^{3}Z-8XY^{2}Z-2X^{2}Z^{2}+Y^{2}Z^{2}+4XZ^{3}-Z^{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.f.1.6	$40$	$2$	$2$	$0$	$0$	full Jacobian
120.96.0-24.e.1.3	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.e.1.10	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-8.f.1.3	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.v.2.7	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.v.2.15	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.w.1.1	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.0-24.w.1.7	$120$	$2$	$2$	$0$	$?$	full Jacobian
120.96.1-24.bb.1.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bb.1.7	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bi.2.3	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bi.2.8	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bj.1.6	$120$	$2$	$2$	$1$	$?$	dimension zero
120.96.1-24.bj.1.7	$120$	$2$	$2$	$1$	$?$	dimension zero