Modular curve 132.288.9-66.a.1.16

• Label: 132.288.9-66.a.1.16
• Level: $132$
• Index: $288$
• Genus: $9$
• Cusps: $8$
• $\Q$-cusps: $8$

Invariants

Level:	$132$		$\SL_2$-level:	$132$		Newform level:	$66$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (all of which are rational)		Cusp widths	$1\cdot2\cdot3\cdot6\cdot11\cdot22\cdot33\cdot66$		Cusp orbits	$1^{8}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 9$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 9$
Rational cusps:	$8$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

66A9

Level structure

$\GL_2(\Z/132\Z)$-generators:	$\begin{bmatrix}52&53\\57&92\end{bmatrix}$, $\begin{bmatrix}77&16\\56&81\end{bmatrix}$, $\begin{bmatrix}78&17\\115&2\end{bmatrix}$, $\begin{bmatrix}83&70\\48&61\end{bmatrix}$, $\begin{bmatrix}118&33\\105&46\end{bmatrix}$, $\begin{bmatrix}129&28\\86&49\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 66.144.9.a.1 for the level structure with $-I$)
Cyclic 132-isogeny field degree:	$2$
Cyclic 132-torsion field degree:	$80$
Full 132-torsion field degree:	$211200$

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 17 x^{9} - 25 x^{8} y + 8 x^{8} z + 52 x^{7} y^{2} - 26 x^{7} y z - 9 x^{7} z^{2} - 48 x^{6} y^{3} + \cdots + y^{5} z^{4} $

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(0:0:0:0:-1:0:0:0:1)$, $(1:1:1:2:-5:2:-1:0:1)$, $(0:0:0:0:0:-1:0:1:0)$, $(0:0:1:0:0:0:0:1:0)$, $(1:1:1:2:1:2:-1:0:1)$, $(0:0:-1/2:0:0:0:0:1:0)$, $(0:0:0:0:0:0:0:0:1)$, $(0:0:0:0:0:1:0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2}\cdot\frac{1681645448234355963624xvr^{8}+1781544060642224023756xvr^{7}s+2407932029570458339570xvr^{6}s^{2}+1208638204638912458316xvr^{5}s^{3}-2035392822053656433482xvr^{4}s^{4}+684172572007596269864xvr^{3}s^{5}+3083863575726091702858xvr^{2}s^{6}-591313014126077749580xvrs^{7}-313101667566257685776xvs^{8}+167602872231821256144xr^{9}-626922415162943330668xr^{8}s-5107632987155784773310xr^{7}s^{2}-2239637953225020088631xr^{6}s^{3}+4739597103841625573257xr^{5}s^{4}+1576505773095728712933xr^{4}s^{5}-2509715907560666792531xr^{3}s^{6}+658692102734135789649xr^{2}s^{7}+340838392561235165277xrs^{8}-65619171697881542972xs^{9}-45072636453995497680yr^{9}+509357979807270495032yr^{8}s+1105250750494685143016yr^{7}s^{2}-1462352676849513351860yr^{6}s^{3}-658277809847628203028yr^{5}s^{4}+669096950532361946388yr^{4}s^{5}+309552802915521511212yr^{3}s^{6}+297389568188518039364yr^{2}s^{7}-510961519824812738196yrs^{8}-33741065630355285184ys^{9}-265305937342133434920zvr^{8}+1585362039212334814876zvr^{7}s+1587765608499743052562zvr^{6}s^{2}-2827283776519596660606zvr^{5}s^{3}-631928384086595074918zvr^{4}s^{4}+190155992642398022426zvr^{3}s^{5}-883436818119670215806zvr^{2}s^{6}+1086195114739844800402zvrs^{7}-167155159003142132888zvs^{8}+365316090000601896zr^{9}-170605741719437335944zr^{8}s-372616869864134193944zr^{7}s^{2}+1543072713571048929202zr^{6}s^{3}-4752519029292636064344zr^{5}s^{4}-3699228143685700237930zr^{4}s^{5}+688748976234963362492zr^{3}s^{6}+483275473653966291826zr^{2}s^{7}+1336231840484457889492zrs^{8}-401127748895640556112zs^{9}+2171581028977890305328wvr^{8}-142208325204439298960wvr^{7}s-4556125284517500840080wvr^{6}s^{2}+6049295209892614957014wvr^{5}s^{3}+5245432038639220321172wvr^{4}s^{4}-1725545289690783047662wvr^{3}s^{5}+106880142690865198792wvr^{2}s^{6}-1457967840341347444034wvrs^{7}-5694342618534013640wvs^{8}+108679651237127007192wr^{9}-393732619667336742720wr^{8}s-4892377610999617585896wr^{7}s^{2}-2521359667643698309095wr^{6}s^{3}+6357062407975044922071wr^{5}s^{4}+2097019149024952730865wr^{4}s^{5}-2453068897484349410457wr^{3}s^{6}+169350773720150814849wr^{2}s^{7}-168461460522193451919wrs^{8}+112976256141732724596ws^{9}+915403260094930085016tvr^{8}+1428962096456974554364tvr^{7}s-453180509525105561134tvr^{6}s^{2}-487942959777100470400tvr^{5}s^{3}+551501452342285168990tvr^{4}s^{4}+124189507939001371524tvr^{3}s^{5}+182224151275281328378tvr^{2}s^{6}+77079914416111760888tvrs^{7}-153410198547883680tvs^{8}+49236051690323741208tr^{9}-303611796403631575992tr^{8}s-2744632558789248153368tr^{7}s^{2}-1788856261740429238537tr^{6}s^{3}+842704425313777561994tr^{5}s^{4}-142776404389264761539tr^{4}s^{5}-1151933326395985635852tr^{3}s^{6}+562027742880972104851tr^{2}s^{7}+500149465954431728972trs^{8}-123738785946730656ts^{9}+1030265607983716104u^{2}r^{8}+137099185289277284088u^{2}r^{7}s+924877808160374274588u^{2}r^{6}s^{2}-402115066522844562264u^{2}r^{5}s^{3}-1098333250845594241620u^{2}r^{4}s^{4}+478746117169770276000u^{2}r^{3}s^{5}+1164891871439156598372u^{2}r^{2}s^{6}-337848176694017796312u^{2}rs^{7}-367110126078703073760u^{2}s^{8}-549207027484412710560uvr^{8}+1552963545587442320360uvr^{7}s+1491822289665558595736uvr^{6}s^{2}-3621462042630580024671uvr^{5}s^{3}-540274433587752608450uvr^{4}s^{4}+487873041883185123355uvr^{3}s^{5}-951465359439746999212uvr^{2}s^{6}+818786089480864401029uvrs^{7}-398034294820960970956uvs^{8}-1212923011993625088ur^{9}+47193838070624221656ur^{8}s-238281225945215101224ur^{7}s^{2}-3377104270254002303440ur^{6}s^{3}-1508019417524859806357ur^{5}s^{4}+2291069818410784952362ur^{4}s^{5}-254746265937420436855ur^{3}s^{6}-1683775926281755500516ur^{2}s^{7}+353746725297479706695urs^{8}+162116170398927491548us^{9}+2389369671812405912280v^{3}r^{7}+1126142989697513575628v^{3}r^{6}s-3889932739388391577934v^{3}r^{5}s^{2}-1480043635355435956448v^{3}r^{4}s^{3}+1607308442334025202846v^{3}r^{3}s^{4}+991412176003846468260v^{3}r^{2}s^{5}+76892236500149910314v^{3}rs^{6}-36066797233885582472v^{3}s^{7}+1509675398738101072872v^{2}r^{8}+3266305543790841387024v^{2}r^{7}s-6347216408728026115168v^{2}r^{6}s^{2}-7014253817883349058396v^{2}r^{5}s^{3}+3265141353163895494096v^{2}r^{4}s^{4}+996037129238991151580v^{2}r^{3}s^{5}-1215785681877245990376v^{2}r^{2}s^{6}+979135037773963723700v^{2}rs^{7}+257227325251882155808v^{2}s^{8}-27620250007507850568vr^{9}+1695176555997886797092vr^{8}s-3025180438121761464194vr^{7}s^{2}-9199852767381461629215vr^{6}s^{3}-1554353174529710157861vr^{5}s^{4}+3086574905101531926289vr^{4}s^{5}-1639231362678617824397vr^{3}s^{6}-325764608942172423047vr^{2}s^{7}+1099390498248868173275vrs^{8}+162116170657120163116vs^{9}+182658048883954488r^{10}-40748284381904423184r^{9}s-1156465636873041326468r^{8}s^{2}-3999492301439227726800r^{7}s^{3}-2247518636564500984901r^{6}s^{4}+657674402294711212794r^{5}s^{5}-849786789264959979863r^{4}s^{6}-798714633260683773188r^{3}s^{7}+736994676046626546531r^{2}s^{8}+499006651409732699276rs^{9}+24569010336s^{10}}{483031886696198xvr^{8}-374311105241886xvr^{7}s-9987607104137434xvr^{6}s^{2}+56914892652815210xvr^{5}s^{3}-80384273200111538xvr^{4}s^{4}-96577154674336672xvr^{3}s^{5}+274762726155997262xvr^{2}s^{6}-95352814202470870xvrs^{7}-47628407583832924xvs^{8}+422460020146900xr^{9}-466873499130991xr^{8}s+4002353644752376xr^{7}s^{2}-21626339351350614xr^{6}s^{3}+19232239258375834xr^{5}s^{4}+94590504705771098xr^{4}s^{5}-199470893232380589xr^{3}s^{6}+88036868440113219xr^{2}s^{7}+10922193710376084xrs^{8}-12097632320527201xs^{9}-319181585762096yr^{9}+1254563107508664yr^{8}s+2361672545720784yr^{7}s^{2}-32752408736510880yr^{6}s^{3}+84404669307904944yr^{5}s^{4}-60016772548090808yr^{4}s^{5}-54989967213412384yr^{3}s^{6}+82042845735902728yr^{2}s^{7}-19958299103316072yrs^{8}-5681152604515424ys^{9}-343312026076054zvr^{8}+180686554319232zvr^{7}s+19941879689788892zvr^{6}s^{2}-102524165109206428zvr^{5}s^{3}+183437371659642316zvr^{4}s^{4}-16157661535508914zvr^{3}s^{5}-246597926207733970zvr^{2}s^{6}+137732549355950672zvrs^{7}+27587708353260302zvs^{8}-534825300927922zr^{8}s+3580042118347974zr^{7}s^{2}+7762319496338546zr^{6}s^{3}-94045568222204746zr^{5}s^{4}+238952912659565098zr^{4}s^{5}-116430445583391676zr^{3}s^{6}-241534007973864946zr^{2}s^{7}+185735411503696862zrs^{8}+25709999386820888zs^{9}+808260837484340wvr^{8}+1512386910815622wvr^{7}s-37702233518584330wvr^{6}s^{2}+135932744834616482wvr^{5}s^{3}-159582725535753314wvr^{4}s^{4}-101593028298517282wvr^{3}s^{5}+317592525841190576wvr^{2}s^{6}-123463985876345746wvrs^{7}-35703938570723902wvs^{8}+184962274370862wr^{9}+2453397832739655wr^{8}s-10904873577951462wr^{7}s^{2}-4674735643701768wr^{6}s^{3}+84550030984117308wr^{5}s^{4}-92863625570749782wr^{4}s^{5}-87981976650620907wr^{3}s^{6}+156530931402021249wr^{2}s^{7}-50153125350724488wrs^{8}-15550361899191297ws^{9}-108039846575146tvr^{8}+2150251767335438tvr^{7}s-6816746359482350tvr^{6}s^{2}+7468227146031750tvr^{5}s^{3}+14336327944446706tvr^{4}s^{4}-35300730482354412tvr^{3}s^{5}+10083719751996854tvr^{2}s^{6}+7070588407969150tvrs^{7}-95756637425934tr^{9}+1748567656128695tr^{8}s-3041440238402401tr^{7}s^{2}-8651413306939271tr^{6}s^{3}+18761008903667715tr^{5}s^{4}+55405238534907355tr^{4}s^{5}-122980578286631640tr^{3}s^{6}+14876232828129215tr^{2}s^{7}+29551728414188713trs^{8}+185188701913668u^{2}r^{7}s-2859577198227516u^{2}r^{6}s^{2}+12632863090256004u^{2}r^{5}s^{3}-3880366181883612u^{2}r^{4}s^{4}-63855612507169188u^{2}r^{3}s^{5}+90391274616665928u^{2}r^{2}s^{6}-1985689631562540u^{2}rs^{7}-20084566201877388u^{2}s^{8}-587276596740518uvr^{8}+2553795771255249uvr^{7}s+9605144648458753uvr^{6}s^{2}-89596164465308633uvr^{5}s^{3}+210660827716726157uvr^{4}s^{4}-79535975607905183uvr^{3}s^{5}-246236552510860844uvr^{2}s^{6}+179616585823846369uvrs^{7}+29015966374182523uvs^{8}-95756637425934ur^{8}s+1816405126217683ur^{7}s^{2}-3134297313706901ur^{6}s^{3}-17618820139018627ur^{5}s^{4}+60890998259068407ur^{4}s^{5}+3235820575755671ur^{3}s^{6}-143180273601807792ur^{2}s^{7}+68007358095623275urs^{8}+28194588633414653us^{9}+395963962657366v^{3}r^{7}+418970741443150v^{3}r^{6}s+1600696959795218v^{3}r^{5}s^{2}-3079994819186658v^{3}r^{4}s^{3}-7718881792400950v^{3}r^{3}s^{4}+14783367074602644v^{3}r^{2}s^{5}-2192628144274058v^{3}rs^{6}-2955646798613530v^{3}s^{7}-715016133732v^{2}r^{8}+5618329495839388v^{2}r^{7}s-10752031659767636v^{2}r^{6}s^{2}-35649518799788524v^{2}r^{5}s^{3}+153007182064482492v^{2}r^{4}s^{4}-73470555234056320v^{2}r^{3}s^{5}-198063711541493676v^{2}r^{2}s^{6}+109428304744247500v^{2}rs^{7}+38940895088430320v^{2}s^{8}-43447593563564vr^{9}+2395835584489669vr^{8}s+2041733402939730vr^{7}s^{2}-43014012828867116vr^{6}s^{3}+65452205946819520vr^{5}s^{4}+141225663915388884vr^{4}s^{5}-269476672276754715vr^{3}s^{6}-77770928671073821vr^{2}s^{7}+123504420873928054vrs^{8}+28194588633414653vs^{9}+38236246422494r^{9}s+3845101706162131r^{8}s^{2}-18279662593592085r^{7}s^{3}+15037500961133365r^{6}s^{4}+54884122475999047r^{5}s^{5}-45487456946130533r^{4}s^{6}-90442438599529924r^{3}s^{7}+33750978784276131r^{2}s^{8}+29551728414188713rs^{9}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve $X_0(66)$ :

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

Equation of the image curve:

$0$

$=$

$ 17X^{9}-25X^{8}Y+8X^{8}Z+52X^{7}Y^{2}-26X^{7}YZ-9X^{7}Z^{2}-48X^{6}Y^{3}+22X^{6}Y^{2}Z-8X^{6}YZ^{2}-6X^{6}Z^{3}+42X^{5}Y^{4}-38X^{5}Y^{3}Z+22X^{5}Y^{2}Z^{2}+22X^{5}YZ^{3}+X^{5}Z^{4}-23X^{4}Y^{5}+18X^{4}Y^{4}Z-21X^{4}Y^{3}Z^{2}-4X^{4}Y^{2}Z^{3}+X^{4}YZ^{4}-2X^{4}Z^{5}+7X^{3}Y^{6}-24X^{3}Y^{5}Z+12X^{3}Y^{4}Z^{2}+2X^{3}Y^{3}Z^{3}-8X^{3}Y^{2}Z^{4}-4X^{3}YZ^{5}-X^{3}Z^{6}-2X^{2}Y^{7}+6X^{2}Y^{6}Z-4X^{2}Y^{5}Z^{2}+8X^{2}Y^{4}Z^{3}+5X^{2}Y^{3}Z^{4}-2XY^{7}Z+5XY^{6}Z^{2}+2XY^{5}Z^{3}+2Y^{6}Z^{3}+Y^{5}Z^{4} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
12.24.0-6.a.1.6	$12$	$12$	$12$	$0$	$0$
$X_0(11)$	$11$	$24$	$12$	$1$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
12.24.0-6.a.1.6	$12$	$12$	$12$	$0$	$0$