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Properties

Label 30.90.3.a.1
Level $30$
Index $90$
Genus $3$
Analytic rank $1$
Cusps $9$
$\Q$-cusps $1$

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Invariants

Level: $30$ $\SL_2$-level: $10$ Newform level: $900$
Index: $90$ $\PSL_2$-index:$90$
Genus: $3 = 1 + \frac{ 90 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 9 }{2}$
Cusps: $9$ (of which $1$ is rational) Cusp widths $10^{9}$ Cusp orbits $1\cdot2^{2}\cdot4$
Elliptic points: $4$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2 \le \gamma \le 3$
$\overline{\Q}$-gonality: $2 \le \gamma \le 3$
Rational cusps: $1$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 10C3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 30.90.3.2

Level structure

$\GL_2(\Z/30\Z)$-generators: $\begin{bmatrix}5&1\\4&25\end{bmatrix}$, $\begin{bmatrix}25&6\\24&5\end{bmatrix}$, $\begin{bmatrix}25&28\\24&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 30-isogeny field degree: $8$
Cyclic 30-torsion field degree: $64$
Full 30-torsion field degree: $1536$

Jacobian

Conductor: $2^{4}\cdot3^{4}\cdot5^{6}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}$
Newforms: 50.2.a.b, 450.2.a.c, 900.2.a.b

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $ $=$ $ 28 x^{4} - 5 x^{3} y + 24 x^{3} z + x^{2} y^{2} + 7 x^{2} y z - 2 x^{2} z^{2} + 4 x y^{3} + \cdots + 2 y z^{3} $
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Rational points

This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1)$

Maps to other modular curves

$j$-invariant map of degree 90 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^{10}}\cdot\frac{517149223430117977997168392270617387557x^{3}y^{20}+5893581042120264453763147239537769476020x^{3}y^{19}z+21443108543269168554012136965031136077880x^{3}y^{18}z^{2}+15360867859620582790384369594295754117600x^{3}y^{17}z^{3}-77600731379175501429050828732569199332800x^{3}y^{16}z^{4}-53984977086819148250467725623961677402112x^{3}y^{15}z^{5}+826298797498894278641836402032546027125760x^{3}y^{14}z^{6}-543648538867227122585333418008194862453760x^{3}y^{13}z^{7}-8737490582659989585174822747985414496716800x^{3}y^{12}z^{8}+22202855425415382061175306813945701471846400x^{3}y^{11}z^{9}+13365629236189726499109027685700074433871872x^{3}y^{10}z^{10}-161159789436736831167721057993706018530263040x^{3}y^{9}z^{11}+375975842460968829062354336928126837444771840x^{3}y^{8}z^{12}-499537749352269437226916348175745843816038400x^{3}y^{7}z^{13}+436829918455466543159920420936483509711667200x^{3}y^{6}z^{14}-262278331307670859865133766136889709210632192x^{3}y^{5}z^{15}+108579349026793774953907965461735717700894720x^{3}y^{4}z^{16}-30312596554943889323728302003892788941291520x^{3}y^{3}z^{17}+5410076018889745870460107716209898684416000x^{3}y^{2}z^{18}-552543631021749222028085889313728888832000x^{3}yz^{19}+24348019100653531050875264644198424379392x^{3}z^{20}-320217933718476854759673780659601907693x^{2}y^{21}-3555456759345565368727491059365512031311x^{2}y^{20}z-11883674472050167099086542958594508070190x^{2}y^{19}z^{2}-7001271220514783336750339427097834248980x^{2}y^{18}z^{3}+27927472951061183129464533689267627730240x^{2}y^{17}z^{4}+101002992262554481448376834569652519267648x^{2}y^{16}z^{5}+152100102389452177287168920652743595837696x^{2}y^{15}z^{6}-1204668534368438250388645894229612920792320x^{2}y^{14}z^{7}-954089397102991032400492800159430598115840x^{2}y^{13}z^{8}+11895174331399413335355881891912794763064320x^{2}y^{12}z^{9}-16136385622852327726940112679563685257781248x^{2}y^{11}z^{10}-28387376749921137853642475634220116650704896x^{2}y^{10}z^{11}+132296750915576972015689846528642288689479680x^{2}y^{9}z^{12}-227945474834485875517407121129822219228938240x^{2}y^{8}z^{13}+236532531066944423110773967987708515924049920x^{2}y^{7}z^{14}-163118525694537232632135748804386684009971712x^{2}y^{6}z^{15}+76659400317341812790372614114911517855973376x^{2}y^{5}z^{16}-24336321829063243311560323803153182427709440x^{2}y^{4}z^{17}+5026263611553109297359837804141338553221120x^{2}y^{3}z^{18}-624999328179355768921749446586943049564160x^{2}y^{2}z^{19}+39904605945817061113326554418394800586752x^{2}yz^{20}-856912575115316044747057613104895492096x^{2}z^{21}+164270869771864970990098255388857369868xy^{22}+2195879377339554871075803401366687419769xy^{21}z+9602409128531452487670404140248909675930xy^{20}z^{2}+11415706114129291905217640533630655770645xy^{19}z^{3}-14022736796782847181471347965876602336560xy^{18}z^{4}+4098325117397418477622473421464844338672xy^{17}z^{5}-44043749740406752335135653201887266991344xy^{16}z^{6}-685660589776966359183135510289693293108480xy^{15}z^{7}+1364711525570767178392631753648222600559360xy^{14}z^{8}+4742298703197202018291336334408166462552320xy^{13}z^{9}-18286407011664436022143199687186892253115392xy^{12}z^{10}+10522860666808746720553760283550924401659904xy^{11}z^{11}+55926591770068327336911311571241810943303680xy^{10}z^{12}-160355564042679078039986114266189817331875840xy^{9}z^{13}+221827194529160296391985055144668083044024320xy^{8}z^{14}-193142404673416385623317403673568043492835328xy^{7}z^{15}+112560306359010177897064161503230120260796416xy^{6}z^{16}-44011129113957487756618536260313186545172480xy^{5}z^{17}+11065541787585953035488492285222895820472320xy^{4}z^{18}-1578337933715993969462295689188346492354560xy^{3}z^{19}+73620589715613594027047426147715140026368xy^{2}z^{20}+9073007143478434216716652438429603201024xyz^{21}-974488443585822720540534984168251064320xz^{22}-606407210040891590193133714931712y^{23}-82131948044474750368405517175567827590y^{22}z-863932638889860463329172815638301319686y^{21}z^{2}-2757172367756403672695660649196281085402y^{20}z^{3}-1250180799056498189679468840971128084800y^{19}z^{4}+4271448932866308848776240212202409476352y^{18}z^{5}-10425632594134921608064885673606059406880y^{17}z^{6}+32156111189189525820041203955682155959296y^{16}z^{7}+226529092238002816011518098740820352151552y^{15}z^{8}-720264286433268230788706217778647396211200y^{14}z^{9}-696653176587050736201401445764272424630272y^{13}z^{10}+6069713253303028676454114724347239428300800y^{12}z^{11}-10352329471948145962334043807255938969092096y^{11}z^{12}+3265424664563647079198809238119354470432768y^{10}z^{13}+15386670514964180197960173754336168656896000y^{9}z^{14}-30842722376263307073544059023901224604008448y^{8}z^{15}+30976659174169419853938631438297244243066880y^{7}z^{16}-19748217409752177563609122273121964417613824y^{6}z^{17}+8406996904055159102888563097055202243510272y^{5}z^{18}-2379014692149047975627067531700100241817600y^{4}z^{19}+427817782505053866498073505599303490469888y^{3}z^{20}-43953013810738129119567417782215597096960y^{2}z^{21}+194897682260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Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.45.1.a.1 $10$ $2$ $2$ $1$ $0$ $1^{2}$
30.30.1.b.1 $30$ $3$ $3$ $1$ $1$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
30.180.7.b.1 $30$ $2$ $2$ $7$ $1$ $1^{4}$
30.180.7.d.1 $30$ $2$ $2$ $7$ $2$ $1^{4}$
30.180.7.g.1 $30$ $2$ $2$ $7$ $1$ $1^{4}$
30.180.7.h.1 $30$ $2$ $2$ $7$ $2$ $1^{4}$
30.270.16.i.1 $30$ $3$ $3$ $16$ $6$ $1^{11}\cdot2$
30.360.22.f.1 $30$ $4$ $4$ $22$ $4$ $1^{19}$
60.180.7.e.1 $60$ $2$ $2$ $7$ $2$ $1^{4}$
60.180.7.x.1 $60$ $2$ $2$ $7$ $3$ $1^{4}$
60.180.7.bs.1 $60$ $2$ $2$ $7$ $2$ $1^{4}$
60.180.7.bz.1 $60$ $2$ $2$ $7$ $3$ $1^{4}$
60.180.8.i.1 $60$ $2$ $2$ $8$ $2$ $1^{5}$
60.180.8.l.1 $60$ $2$ $2$ $8$ $3$ $1^{5}$
60.180.8.q.1 $60$ $2$ $2$ $8$ $1$ $1^{5}$
60.180.8.t.1 $60$ $2$ $2$ $8$ $2$ $1^{5}$
60.180.8.z.1 $60$ $2$ $2$ $8$ $2$ $1^{5}$
60.180.8.bc.1 $60$ $2$ $2$ $8$ $1$ $1^{5}$
60.180.8.bi.1 $60$ $2$ $2$ $8$ $2$ $1^{5}$
60.180.8.bk.1 $60$ $2$ $2$ $8$ $5$ $1^{5}$
60.180.10.z.1 $60$ $2$ $2$ $10$ $3$ $1^{7}$
60.180.10.bb.1 $60$ $2$ $2$ $10$ $2$ $1^{7}$
60.180.10.ca.1 $60$ $2$ $2$ $10$ $5$ $1^{7}$
60.180.10.cd.1 $60$ $2$ $2$ $10$ $3$ $1^{7}$
60.180.10.dn.1 $60$ $2$ $2$ $10$ $5$ $1^{7}$
60.180.10.dq.1 $60$ $2$ $2$ $10$ $3$ $1^{7}$
60.180.10.dv.1 $60$ $2$ $2$ $10$ $2$ $1^{7}$
60.180.10.dy.1 $60$ $2$ $2$ $10$ $3$ $1^{7}$
120.180.7.j.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.u.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.cw.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.df.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.fq.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.fz.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.go.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.7.gx.1 $120$ $2$ $2$ $7$ $?$ not computed
120.180.8.y.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.bh.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.bw.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.cf.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.cw.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.df.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.dy.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.8.ee.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.10.dp.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.dv.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.gs.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.hb.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.la.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.lj.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.ly.1 $120$ $2$ $2$ $10$ $?$ not computed
120.180.10.mh.1 $120$ $2$ $2$ $10$ $?$ not computed
210.180.7.bc.1 $210$ $2$ $2$ $7$ $?$ not computed
210.180.7.bd.1 $210$ $2$ $2$ $7$ $?$ not computed
210.180.7.bf.1 $210$ $2$ $2$ $7$ $?$ not computed
210.180.7.bg.1 $210$ $2$ $2$ $7$ $?$ not computed
330.180.7.y.1 $330$ $2$ $2$ $7$ $?$ not computed
330.180.7.z.1 $330$ $2$ $2$ $7$ $?$ not computed
330.180.7.bb.1 $330$ $2$ $2$ $7$ $?$ not computed
330.180.7.bc.1 $330$ $2$ $2$ $7$ $?$ not computed