Modular curve 48.192.3-48.bv.1.5

• Label: 48.192.3-48.bv.1.5
• Level: $48$
• Index: $192$
• Genus: $3$
• Analytic rank: $0$
• Cusps: $12$
• $\Q$-cusps: $0$

Invariants

Level:	$48$		$\SL_2$-level:	$16$		Newform level:	$576$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)		Cusp widths	$4^{8}\cdot16^{4}$		Cusp orbits	$2^{4}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	16J3
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	48.192.3.2659

Level structure

$\GL_2(\Z/48\Z)$-generators:	$\begin{bmatrix}31&2\\0&25\end{bmatrix}$, $\begin{bmatrix}31&44\\24&25\end{bmatrix}$, $\begin{bmatrix}33&26\\16&21\end{bmatrix}$, $\begin{bmatrix}43&14\\16&25\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 48.96.3.bv.1 for the level structure with $-I$)
Cyclic 48-isogeny field degree:	$8$
Cyclic 48-torsion field degree:	$128$
Full 48-torsion field degree:	$6144$

Jacobian

Conductor:	$2^{17}\cdot3^{2}$
Simple:	no
Squarefree:	yes
Decomposition:	$1\cdot2$
Newforms:	64.2.b.a, 288.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 304 x^{8} - 240 x^{7} y - 368 x^{7} z + 1380 x^{6} y^{2} - 648 x^{6} y z + 388 x^{6} z^{2} + 720 x^{5} y^{3} + \cdots + z^{8} $

Geometric Weierstrass model Geometric Weierstrass model

$ w^{2} $	$=$	$ -9 x^{2} y z - 6 y z^{3} $
$0$	$=$	$3 x^{2} + y^{2} + z^{2}$

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{1}{2^4\cdot3^4\cdot5}\cdot\frac{23514624000000000000xw^{11}-67128652800000000000xw^{10}u-312960049920000000000xw^{9}u^{2}-212502170304000000000xw^{8}u^{3}-567189627350400000000xw^{7}u^{4}-430082083172448000000xw^{6}u^{5}-838207928606385600000xw^{5}u^{6}-354648079468422000000xw^{4}u^{7}-868138828900536348000xw^{3}u^{8}-889650213088470471000xw^{2}u^{9}+1092869861031520025706xwu^{10}-4291477908406963525476xu^{11}+65088849814312500000yt^{11}-160449484583868750000yt^{10}u-131609152370407500000yt^{9}u^{2}+2811021878388369375000yt^{8}u^{3}-8091242669751678675000yt^{7}u^{4}+12565963595239708747500yt^{6}u^{5}-14241473752157072109000yt^{5}u^{6}+12068669398467368255850yt^{4}u^{7}-4936827993266865391290yt^{3}u^{8}+1807716547140839209701yt^{2}u^{9}-10116589421489100870762ytu^{10}+6802503190340841961488yu^{11}-196011187200000000000z^{2}w^{9}u-286886810880000000000z^{2}w^{8}u^{2}+12803292864000000000z^{2}w^{7}u^{3}-495900458409600000000z^{2}w^{6}u^{4}-218524952030112000000z^{2}w^{5}u^{5}-378633670929230400000z^{2}w^{4}u^{6}-264501065026342800000z^{2}w^{3}u^{7}+3481403283834588000z^{2}w^{2}u^{8}-1191478519926512073000z^{2}wu^{9}+2140394559288307796214z^{2}u^{10}-47029248000000000000zw^{11}-260452454400000000000zw^{10}u+7065584640000000000zw^{9}u^{2}+280402000128000000000zw^{8}u^{3}-198530340595200000000zw^{7}u^{4}+73566810631296000000zw^{6}u^{5}+332756205449299200000zw^{5}u^{6}-170916133903070400000zw^{4}u^{7}+977322000113036496000zw^{3}u^{8}-784263531592690380000zw^{2}u^{9}+1289862221594072006088zwu^{10}+68984053021125000000zt^{11}-251207137188900000000zt^{10}u+174477532255177500000zt^{9}u^{2}+2504747471154619500000zt^{8}u^{3}-9501235732389017250000zt^{7}u^{4}+16092322907338339200000zt^{6}u^{5}-14707308072946385835000zt^{5}u^{6}+7570312096771018089000zt^{4}u^{7}-3694470641992707043500zt^{3}u^{8}+5174945629339914351000zt^{2}u^{9}-9789154336855622872476ztu^{10}+6997179011750889467976zu^{11}-47775744000000000000w^{12}-196906982400000000000w^{11}u+44423043840000000000w^{10}u^{2}+7774895808000000000w^{9}u^{3}-297997911619200000000w^{8}u^{4}-18926255846304000000w^{7}u^{5}+125468066108491200000w^{6}u^{6}-323126076116250000000w^{5}u^{7}+836404076662088796000w^{4}u^{8}-786412648887034473000w^{3}u^{9}+704068203754273325238w^{2}u^{10}+5022095189062500000wt^{11}+266921056871831250000wt^{10}u-1316495579864377500000wt^{9}u^{2}+2999627325267123375000wt^{8}u^{3}+29520891751679325000wt^{7}u^{4}-8777298461758216852500wt^{6}u^{5}+8352210821806459971000wt^{5}u^{6}+4075567105366585463850wt^{4}u^{7}-5777111044262808475290wt^{3}u^{8}-4230275939627338339299wt^{2}u^{9}+1039720940299631883420wtu^{10}+6056735532059545488wu^{11}+32236925445250000000t^{12}-45914621024737500000t^{11}u-234654032778900000000t^{10}u^{2}+1715838973111884250000t^{9}u^{3}-3244021513529898150000t^{8}u^{4}+1053203497507961505000t^{7}u^{5}+3775556675671193688000t^{6}u^{6}-3898835608014239969700t^{5}u^{7}-181622408219302012020t^{4}u^{8}+1551284166702816314038t^{3}u^{9}+109065623407962732156t^{2}u^{10}-2279274277423205568876tu^{11}+260620821615298927894u^{12}}{1399680000000xw^{7}u^{4}+2713824000000xw^{6}u^{5}-1058961600000xw^{5}u^{6}+378438480000xw^{4}u^{7}+3759827479200xw^{3}u^{8}-2426873863320xw^{2}u^{9}+4718746507482xwu^{10}+10330894679868xu^{11}-30375000000yt^{11}+206550000000yt^{10}u-348705000000yt^{9}u^{2}-699354000000yt^{8}u^{3}+2320639725000yt^{7}u^{4}-8517612037500yt^{6}u^{5}+20850264495000yt^{5}u^{6}-37742918490750yt^{4}u^{7}+55071326801070yt^{3}u^{8}-72205598792403yt^{2}u^{9}+57165954198726ytu^{10}-29127160438704yu^{11}-622080000000z^{2}w^{6}u^{4}-2558304000000z^{2}w^{5}u^{5}+566481600000z^{2}w^{4}u^{6}-193369680000z^{2}w^{3}u^{7}+1862160904800z^{2}w^{2}u^{8}-2881084389480z^{2}wu^{9}+2056995871398z^{2}u^{10}-2799360000000zw^{7}u^{4}-3328128000000zw^{6}u^{5}+1406419200000zw^{5}u^{6}-2671911360000zw^{4}u^{7}-195236438400zw^{3}u^{8}+564003505440zw^{2}u^{9}-4033732415544zwu^{10}-1615569150000zt^{7}u^{4}+74053860000zt^{6}u^{5}+14241846861000zt^{5}u^{6}-43169346029400zt^{4}u^{7}+63911812801620zt^{3}u^{8}-67647264497640zt^{2}u^{9}+50666229159108ztu^{10}-28382247096888zu^{11}-207360000000w^{8}u^{4}-567648000000w^{7}u^{5}+1149163200000w^{6}u^{6}+1621445040000w^{5}u^{7}+2683411221600w^{4}u^{8}+3140425439640w^{3}u^{9}-5061801585114w^{2}u^{10}-30375000000wt^{11}+206550000000wt^{10}u-348705000000wt^{9}u^{2}-699354000000wt^{8}u^{3}+4623826425000wt^{7}u^{4}-16767851317500wt^{6}u^{5}+22385792367000wt^{5}u^{6}+81836292450wt^{4}u^{7}-22809525401490wt^{3}u^{8}+8355295157157wt^{2}u^{9}+2738331704220wtu^{10}+581183745936wu^{11}-33750000000t^{12}+330750000000t^{11}u-1352700000000t^{10}u^{2}+2996190000000t^{9}u^{3}-4537807100000t^{8}u^{4}+539929985000t^{7}u^{5}+9309521626000t^{6}u^{6}-14663241556900t^{5}u^{7}+3547568465740t^{4}u^{8}+4780509126566t^{3}u^{9}-8255901700628t^{2}u^{10}+116155724468tu^{11}+39710256838u^{12}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.bv.1 :

$\displaystyle X$	$=$	$\displaystyle t$
$\displaystyle Y$	$=$	$\displaystyle 2w$
$\displaystyle Z$	$=$	$\displaystyle 2u$

Equation of the image curve:

$0$

$=$

$ 304X^{8}-240X^{7}Y+1380X^{6}Y^{2}+720X^{5}Y^{3}+360X^{4}Y^{4}-368X^{7}Z-648X^{6}YZ-2844X^{5}Y^{2}Z-936X^{4}Y^{3}Z-468X^{3}Y^{4}Z+388X^{6}Z^{2}+1476X^{5}YZ^{2}+2934X^{4}Y^{2}Z^{2}+756X^{3}Y^{3}Z^{2}+378X^{2}Y^{4}Z^{2}-284X^{5}Z^{3}-1536X^{4}YZ^{3}-1608X^{3}Y^{2}Z^{3}-288X^{2}Y^{3}Z^{3}-144XY^{4}Z^{3}+187X^{4}Z^{4}+864X^{3}YZ^{4}+522X^{2}Y^{2}Z^{4}+72XY^{3}Z^{4}+36Y^{4}Z^{4}-92X^{3}Z^{5}-288X^{2}YZ^{5}-72XY^{2}Z^{5}+34X^{2}Z^{6}+48XYZ^{6}+12Y^{2}Z^{6}-8XZ^{7}+Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
16.96.2-16.d.1.10	$16$	$2$	$2$	$2$	$0$	$1$
24.96.0-24.bc.2.1	$24$	$2$	$2$	$0$	$0$	full Jacobian
48.96.0-24.bc.2.6	$48$	$2$	$2$	$0$	$0$	full Jacobian
48.96.1-48.b.1.2	$48$	$2$	$2$	$1$	$0$	$2$
48.96.1-48.b.1.12	$48$	$2$	$2$	$1$	$0$	$2$
48.96.2-16.d.1.9	$48$	$2$	$2$	$2$	$0$	$1$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
48.384.5-48.bh.3.3	$48$	$2$	$2$	$5$	$1$	$1^{2}$
48.384.5-48.bj.2.5	$48$	$2$	$2$	$5$	$0$	$1^{2}$
48.384.5-48.ci.1.2	$48$	$2$	$2$	$5$	$0$	$1^{2}$
48.384.5-48.cj.2.3	$48$	$2$	$2$	$5$	$1$	$1^{2}$
48.576.19-48.ka.2.2	$48$	$3$	$3$	$19$	$1$	$1^{8}\cdot2^{2}\cdot4$
48.768.21-48.fr.1.18	$48$	$4$	$4$	$21$	$1$	$1^{8}\cdot2^{3}\cdot4$
240.384.5-240.js.1.3	$240$	$2$	$2$	$5$	$?$	not computed
240.384.5-240.jt.1.3	$240$	$2$	$2$	$5$	$?$	not computed
240.384.5-240.la.1.3	$240$	$2$	$2$	$5$	$?$	not computed
240.384.5-240.lb.1.3	$240$	$2$	$2$	$5$	$?$	not computed