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Properties

Label 231.168.5-21.a.1.8
Level $231$
Index $168$
Genus $5$
Cusps $6$
$\Q$-cusps $0$

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Invariants

Level: $231$ $\SL_2$-level: $21$ Newform level: $147$
Index: $168$ $\PSL_2$-index:$84$
Genus: $5 = 1 + \frac{ 84 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $7^{3}\cdot21^{3}$ Cusp orbits $3^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 8$
$\overline{\Q}$-gonality: $2 \le \gamma \le 5$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 21A5

Level structure

$\GL_2(\Z/231\Z)$-generators: $\begin{bmatrix}27&178\\175&225\end{bmatrix}$, $\begin{bmatrix}28&129\\67&107\end{bmatrix}$, $\begin{bmatrix}43&78\\88&125\end{bmatrix}$, $\begin{bmatrix}188&73\\84&106\end{bmatrix}$
Contains $-I$: no $\quad$ (see 21.84.5.a.1 for the level structure with $-I$)
Cyclic 231-isogeny field degree: $96$
Cyclic 231-torsion field degree: $11520$
Full 231-torsion field degree: $7603200$

Models

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

$ 0 $ $=$ $ 4 x^{2} + 2 x y + 4 x z - x w - 3 x t - y z - 2 y w + 2 y t + z^{2} - z t - 2 w^{2} + 3 w t $
$=$ $3 x^{2} + 3 x y - 5 x z + x w - 7 x t + 3 y^{2} - 3 y z - 2 y t - 3 z^{2} - 2 z w + z t + w t + 2 t^{2}$
$=$ $x^{2} + x y + y^{2} - 4 y z + 2 y w - 3 y t + 3 z^{2} - 3 z w + 8 z t - w^{2} - 4 w t + 3 t^{2}$
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve CM $j$-invariant $j$-heightCanonical model
27.a3 $-3$$0$ $0.000$$(3/2:1/2:-3/2:-1:1)$

Maps to other modular curves

$j$-invariant map of degree 84 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{-702487137621278967624722625809260z^{3}w^{8}+3335971554958269552334153219881120y^{2}w^{9}-5504626453183132801535358685194635yzw^{9}+4792253648155308560975846921572475z^{2}w^{9}-5937978852201489327138963258765635xw^{10}+1230034505094317069697906395164440yw^{10}-4521517746431703806381909676685150zw^{10}-4100165505349184314010669117524360w^{11}+38096504137238064808196842164259520z^{3}w^{7}t+52927693895941635031567158632225250y^{2}w^{8}t+4396656626285444989870451216890870yzw^{8}t-108144608838612834889244363022113100z^{2}w^{8}t-106310651705117663458233488764885275xw^{9}t-34516466560776557863037192105429810yw^{9}t+46575713957209650205860066644758915zw^{9}t-39454495599535904921562918494054425w^{10}t+2018761010103966673602533120975925560z^{3}w^{6}t^{2}-694228995250279850358084956217184920y^{2}w^{7}t^{2}+2608600782734684503836807932260122736yzw^{7}t^{2}-3557059926192120893566331276449167748z^{2}w^{7}t^{2}+394549018046252190841571321863232014xw^{8}t^{2}-1892395483548597755974918799411528874yw^{8}t^{2}+1214097488705783101274447085717510748zw^{8}t^{2}+895383695458490436430562097283754042w^{9}t^{2}-6133942522535492491803031881661607800z^{3}w^{5}t^{3}-2239160789726370804007028976096784320y^{2}w^{6}t^{3}+7268377772795698117807487916716562880yzw^{6}t^{3}+10405005511563727126371410125127879020z^{2}w^{6}t^{3}+1178498394570300692065326299573268692xw^{7}t^{3}-3220398623850944455407398936834214952yw^{7}t^{3}-7110954113393953539535499848426318752zw^{7}t^{3}+3376733926692069990535615263439790432w^{8}t^{3}-98342970969518918730267948701043007240z^{3}w^{4}t^{4}+61092939651787834704806890167906598380y^{2}w^{5}t^{4}-143130147924765178724565983225221855496yzw^{5}t^{4}+140071311340747745951760200077302444528z^{2}w^{5}t^{4}-66380452639011986785069254288186082704xw^{6}t^{4}+102969018962812385112930071629932885004yw^{6}t^{4}-47638569519625473121690532721108485488zw^{6}t^{4}-56607041651838999301001576078033459600w^{7}t^{4}+1025488494100669253249828405557824593000z^{3}w^{3}t^{5}-312654415480928548503542574071989387620y^{2}w^{4}t^{5}+848840584975879466377194905525078947822yzw^{4}t^{5}-1742753273801639946543321374046734902066z^{2}w^{4}t^{5}+344292044516511701065798209377788034386xw^{5}t^{5}-691187357095714817460368522976641672976yw^{5}t^{5}+812736162163887386144800364169254425168zw^{5}t^{5}+267129606125874979985207953633879223112w^{6}t^{5}-4418748528675179567108889122673588198040z^{3}w^{2}t^{6}+616395296936866962511924283416074053760y^{2}w^{3}t^{6}-2681845997578161227433813132542043220304yzw^{3}t^{6}+8721918643639130659609129254129120679852z^{2}w^{3}t^{6}-559180306095372908778672337820523338782xw^{4}t^{6}+2624977353247648770592714878417305697032yw^{4}t^{6}-5192813821920769378306975117883917533006zw^{4}t^{6}+97039704203856875710339520371069059134w^{5}t^{6}+9168534160532818300969366677558150987680z^{3}wt^{7}-548479827584284459149470565844266897240y^{2}w^{2}t^{7}+3837108256039742844762720529212710285272yzw^{2}t^{7}-22615099007326766286465594963330453613296z^{2}w^{2}t^{7}+1082619272330112307531540250912357764140xw^{3}t^{7}-4548804278668831893644054942386238125580yw^{3}t^{7}+18270325338210133783085120188861816371704zw^{3}t^{7}-3974938459451679391905765508300670012240w^{4}t^{7}-11427057186899154716750095439642101447600z^{3}t^{8}-885969775265822028853206314207058263730y^{2}wt^{8}-3924981898002991750293969026700702285017yzwt^{8}+36557210263873280295575955370423632964001z^{2}wt^{8}+901498631801311952924442549488009833801xw^{2}t^{8}+5732659913689699710322238305393384666404yw^{2}t^{8}-36216486835149845621333201441421783697670zw^{2}t^{8}+12226986099926025560274334221140227951002w^{3}t^{8}+413572153426228324419280688066823150900y^{2}t^{9}-4311684495301309387652675663694716367225yzt^{9}-22674609705227994318860958436570830249925z^{2}t^{9}+3378810882169238071738724628475462037146xwt^{9}+2400793785443867283820486324941011707784ywt^{9}+43609779420097692589310913847235188735969zwt^{9}-21181946483945677562167973681548109169375w^{2}t^{9}+3292402968749036203299285934895102471475xt^{10}-1432853679978501508772670416866437046650yt^{10}-5145566525164554788826939673032073043525zt^{10}+4994316440381700826794095248696975437221wt^{10}+344324362518669055523912830240097117650t^{11}}{31690662207500387449983760400z^{3}w^{8}+168035529466984263599159240430y^{2}w^{9}-941326509996833174580311865462yzw^{9}+867510634925012512246659543346z^{2}w^{9}+48960297387859943841946320892xw^{10}+644778696313474138324728026548yw^{10}-746976799179179241443440885976zw^{10}-164797632294780362579197161554w^{11}-205970392610841597374168513680z^{3}w^{7}t-859240754887539643346791630710y^{2}w^{8}t+5135714001158100740499423539019yzw^{8}t-4943928029577073903543386713487z^{2}w^{8}t+143917380245428392699526455047xw^{9}t-4406634215659224393748232503622yw^{9}t+6411331073797006853849461082526zw^{9}t-339247174351806197058894911626w^{10}t+587998281268934180233828579760z^{3}w^{6}t^{2}+1418443716775453535378239801920y^{2}w^{7}t^{2}-10099712985926863232126614358152yzw^{7}t^{2}+10570215357837744685839181322136z^{2}w^{7}t^{2}-1916964770787393344912962577665xw^{8}t^{2}+12033786833578064329634507394320yw^{8}t^{2}-21412711796462115750156312379639zw^{8}t^{2}+5947969873346380559966681931505w^{9}t^{2}-1116968707326519571904383062640z^{3}w^{5}t^{3}+1681521100300837597240328446560y^{2}w^{6}t^{3}-814405847180858564399394575552yzw^{6}t^{3}-3504628125196024372847615962104z^{2}w^{6}t^{3}+4664854382329852056526471189968xw^{7}t^{3}-10655935024776852707332503684888yw^{7}t^{3}+29508870766671684011144715884208zw^{7}t^{3}-19012543916483175016476787335190w^{8}t^{3}+104934861743413995773364056480z^{3}w^{4}t^{4}-8968301815805047533438783298260y^{2}w^{5}t^{4}+37296944744836611571608281363262yzw^{5}t^{4}-27304314098516083887238513363326z^{2}w^{5}t^{4}-3846720166634473550851531556006xw^{6}t^{4}-21294929410334775888630062445644yw^{6}t^{4}+14316208842444539970894042677200zw^{6}t^{4}+19383850821099881558267503453836w^{7}t^{4}-210967222412149864281942917680z^{3}w^{3}t^{5}+19330941978152059826381026695480y^{2}w^{4}t^{5}-86712012635012389985991226539310yzw^{4}t^{5}+74063500330893871790302005471570z^{2}w^{4}t^{5}-10439035314443687379696506536396xw^{5}t^{5}+84963725999615414848079061988736yw^{5}t^{5}-131068074800770165668890424832834zw^{5}t^{5}+22260295478424117222999220050910w^{6}t^{5}-1918217926867427703784503910960z^{3}w^{2}t^{6}-23052225897848480146574597546640y^{2}w^{3}t^{6}+108930380706014594614757838129784yzw^{3}t^{6}-93775465079034533077510134894672z^{2}w^{3}t^{6}+31819719447325126363197175167486xw^{4}t^{6}-147597982438217210066933167807416yw^{4}t^{6}+267485803416966536446785097038282zw^{4}t^{6}-112641862782265600803607983094198w^{5}t^{6}+2433914265747429984803693263520z^{3}wt^{7}+11246005946850934294782222302880y^{2}w^{2}t^{7}-60722964060113962768199294136972yzw^{2}t^{7}+61022485537551087502474533434196z^{2}w^{2}t^{7}-38774796138946012157263114613420xw^{3}t^{7}+135676675768433818286553009791140yw^{3}t^{7}-288567760159720880214886050971204zw^{3}t^{7}+187748956640744411797192203691764w^{4}t^{7}-2248222624077300192072338980z^{3}t^{8}-12365536277570935357211546406060y^{2}wt^{8}+52992170789573556389220421883573yzwt^{8}-51439921055505739838939064139629z^{2}wt^{8}+32285517327972359529061309176353xw^{2}t^{8}-94567690542626414981579316215848yw^{2}t^{8}+204350502643319888418107098466258zw^{2}t^{8}-162174047930035410530475426458540w^{3}t^{8}+5479287213031107681509756864010y^{2}t^{9}-23606947345287250733721060006074yzt^{9}+19766381056213153920841346996492z^{2}t^{9}-24746546728647259456532207681575xwt^{9}+69550649932155629861723705111870ywt^{9}-152562677802499561541794658324473zwt^{9}+126787694963929672352821083997399w^{2}t^{9}+8384747296456742636587582407422xt^{10}-20701487261861615778643234462662yt^{10}+49478456196958113693129474172828zt^{10}-74899172393715365170960433151782wt^{10}+17041560564666606473554992545412t^{11}}$

Modular covers

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

Factor curve Level Index Degree Genus Rank
$X_{\mathrm{ns}}^+(7)$ $7$ $8$ $4$ $0$ $0$
33.8.0-3.a.1.1 $33$ $21$ $21$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
33.8.0-3.a.1.1 $33$ $21$ $21$ $0$ $0$

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

Covering curve Level Index Degree Genus
231.336.9-21.a.1.4 $231$ $2$ $2$ $9$
231.336.9-231.a.1.2 $231$ $2$ $2$ $9$
231.336.9-21.b.1.3 $231$ $2$ $2$ $9$
231.336.9-231.b.1.8 $231$ $2$ $2$ $9$
231.336.9-21.c.1.4 $231$ $2$ $2$ $9$
231.336.9-231.c.1.6 $231$ $2$ $2$ $9$
231.336.9-21.d.1.3 $231$ $2$ $2$ $9$
231.336.9-231.d.1.5 $231$ $2$ $2$ $9$
231.504.16-21.a.1.1 $231$ $3$ $3$ $16$