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Properties

Label 30.72.1.g.2
Level $30$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $30$ $\SL_2$-level: $10$ Newform level: $180$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $2^{6}\cdot10^{6}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 30.72.1.11

Level structure

$\GL_2(\Z/30\Z)$-generators: $\begin{bmatrix}3&25\\22&1\end{bmatrix}$, $\begin{bmatrix}9&20\\16&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 30-isogeny field degree: $4$
Cyclic 30-torsion field degree: $32$
Full 30-torsion field degree: $1920$

Jacobian

Conductor: $2^{2}\cdot3^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 180.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 8 x^{2} - 3 x y - 6 x z - 4 x w + y w + 2 z w + w^{2} $
$=$ $3 x^{2} + x y + 2 x z + 2 x w - 5 y^{2} - 5 y z - 2 y w - 5 z^{2} - 4 z w - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 269 x^{4} - 360 x^{3} y - 252 x^{3} z + 135 x^{2} y^{2} + 300 x^{2} y z + 131 x^{2} z^{2} - 90 x y^{2} z + \cdots + 4 z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 3^3\,\frac{25247917921545694563089975055886487809134929688xz^{17}+270168536014303676795497473463277752419909933912xz^{16}w+1331912912622284772281159934859563352692533079651xz^{15}w^{2}+4061373409187202198511404688309322228314863777495xz^{14}w^{3}+8669536886193245311505968904383548602980277150043xz^{13}w^{4}+13886241625062677293157447320222952823568657691871xz^{12}w^{5}+17428190135345258840784833728180364515967841564003xz^{11}w^{6}+17608865326878295402730767531783567301926830300135xz^{10}w^{7}+14538010162837509129164825092422774153674574825075xz^{9}w^{8}+9859147010523796235271214276792514393776334889087xz^{8}w^{9}+5476194592103381132238977953830661451824324836171xz^{7}w^{10}+2466124203119453882105482651442273247525473497955xz^{6}w^{11}+884641076688124406609404045061204646311763192843xz^{5}w^{12}+245973383494441524713504222077999752304631916675xz^{4}w^{13}+50825890170409189718424396972942830742732574463xz^{3}w^{14}+7282285244996933873854110311211104026811421663xz^{2}w^{15}+635570371847802932636568552460985146328014431xzw^{16}+24618428020802236315627028727290854743569647xw^{17}-22005286733245610294046420153349179862628745309y^{2}z^{16}-236407636699627160836301966287377616142416358658y^{2}z^{15}w-1119010124307388076703787084564301550142001454654y^{2}z^{14}w^{2}-3175134945474485082633036169380340480059195380322y^{2}z^{13}w^{3}-6165712802690244681697813326786846983744023725801y^{2}z^{12}w^{4}-8831210378937921475543449745638540860786838297102y^{2}z^{11}w^{5}-9774173101393113810759367864095327432023145347964y^{2}z^{10}w^{6}-8599703289777405557234190046011859003612458850446y^{2}z^{9}w^{7}-6102600553653132357956340456338581347580089829803y^{2}z^{8}w^{8}-3501802305944941563801963757401932011179163436394y^{2}z^{7}w^{9}-1611091082985048375988320147404993720602689237306y^{2}z^{6}w^{10}-582496445705486148500942431853349960307353269682y^{2}z^{5}w^{11}-159867953229383536168758897209991240134054582754y^{2}z^{4}w^{12}-31418504449759944169990164469279353051426962522y^{2}z^{3}w^{13}-3967320951027010835774527928054720724964109766y^{2}z^{2}w^{14}-248238548809000321686355190372601307097484642y^{2}zw^{15}-1324025629590263104578733574399466221241224y^{2}w^{16}-22005286733245610294046420153349179862628745309yz^{17}-243742732277375697600984106338494009429959273761yz^{16}w-1196018160082188084339123991164244355642951166264yz^{15}w^{2}-3530624273690817117882042209827298391642835098120yz^{14}w^{3}-7142094638485253801736399494679562730460871634277yz^{13}w^{4}-10647318498166265881675083690493849609647035386551yz^{12}w^{5}-12230049381774540749614902554853026974835673187206yz^{11}w^{6}-11115370383024094764935710118397258557785930785162yz^{10}w^{7}-8096751824423999308860126293425015307087384397555yz^{9}w^{8}-4731273463454894428911012078673602439262114246313yz^{8}w^{9}-2192872520307707460135914292071651786682912919188yz^{7}w^{10}-784474927329618123611540605153234907391880312700yz^{6}w^{11}-204644451120480182909887402772789761346197013106yz^{5}w^{12}-33726412410625937595800263036533605799570813110yz^{4}w^{13}-1528149442495404279300493199410664136457956292yz^{3}w^{14}+726573676825687880529428802494034917414243892yz^{2}w^{15}+173501361571481433404544390656763717683852941yzw^{16}+13323841495398301757076067373424582748533553yw^{17}-22014602378237522697215237784871789975310343942z^{18}-259549694365590940305709143864041559329970090558z^{17}w-1367564982335641203700174190576440830185647457268z^{16}w^{2}-4371453873495107239302720862858014571472219296353z^{15}w^{3}-9655924372997116560008636996549073036640451138589z^{14}w^{4}-15859456026561482163334977941725205252770673329845z^{13}w^{5}-20283099038397362008641359905483993331575693825629z^{12}w^{6}-20804592298438153841474881385695155243991493092329z^{11}w^{7}-17418460772558581894136231947821415776874986972861z^{10}w^{8}-12001233921070820262036705844336424977033155697701z^{9}w^{9}-6806317543450132492907824251712665325338184677021z^{8}w^{10}-3155858276650762889076185096422809270466883379721z^{7}w^{11}-1179849740864704657936267597942348791388568326938z^{6}w^{12}-347806745013667928521117172975448282824882415971z^{5}w^{13}-78083021953231046313767907333857035892971594304z^{4}w^{14}-12617601404720514050546077107635415022066913809z^{3}w^{15}-1325208760746043907988036425050457315703459512z^{2}w^{16}-71859954281669450105131299927025857569359021zw^{17}-724594555224752196088245436557205319897008w^{18}}{13869630734914309387963701657600000000000xz^{17}-153290454110478752304221326540800000000000xz^{16}w+25113526694731496976139170235457367150929688xz^{15}w^{2}+159787754071353914750182175679042339233068920xz^{14}w^{3}+452907924046202820827955608075497050726107931xz^{13}w^{4}+777144279513824409497807389627490015564610291xz^{12}w^{5}+944964934741829829389158627498546561485574506xz^{11}w^{6}+924147823097178867758357335696334025414953274xz^{10}w^{7}+750608035702255319403596110001811741831376887xz^{9}w^{8}+425280779588926969671044195881083251221581045xz^{8}w^{9}+71246345988914438996533519657780917223124484xz^{7}w^{10}-106253881907189729755877300661360818781833784xz^{6}w^{11}-89726283642602592328301253661387735559439558xz^{5}w^{12}-25554018018951275703220609149698431306285896xz^{4}w^{13}+1102416217134338105775360489330225514117755xz^{3}w^{14}+2240563007381920753572220696896904741392843xz^{2}w^{15}+417651605799435652035648146994156211388511xzw^{16}+16766716257620006630901013781722915516279xw^{17}-12091230871400603622397929062400000000000y^{2}z^{16}+186115181659450270339010081587200000000000y^{2}z^{15}w-33352572935310115308996602937113959625490618y^{2}z^{14}w^{2}-189760693662629083714902556179514966725056904y^{2}z^{13}w^{3}-456598290862439997249699363610866051288519597y^{2}z^{12}w^{4}-605298790985313306601760568605767724215887756y^{2}z^{11}w^{5}-488762009405877716848823257180751174389276254y^{2}z^{10}w^{6}-280867372580103742279699635535376727734417592y^{2}z^{9}w^{7}-142787543013755231090246162842642618666978440y^{2}z^{8}w^{8}-20265520082615186711189472261491399840095632y^{2}z^{7}w^{9}+84137635138468113393537645405603164849794650y^{2}z^{6}w^{10}+93629142610253923048106233593060491550828516y^{2}z^{5}w^{11}+44522837147362967758133298596467847551256667y^{2}z^{4}w^{12}+11304223723170704787401013965183954180313918y^{2}z^{3}w^{13}+2332492018060985068529666343670961797115199y^{2}z^{2}w^{14}+624068822539630033886077494866671054498908y^{2}zw^{15}+93493758509391588454417221029756362553307y^{2}w^{16}-12091230871400603622397929062400000000000yz^{17}+182084771368983402464877438566400000000000yz^{16}w-23974033399465863635906217208193446943891985yz^{15}w^{2}-147409260547652513125595578394081565650302365yz^{14}w^{3}-398840782324695608184982474075568589072899529yz^{13}w^{4}-636374842920013511511825608449671997658335209yz^{12}w^{5}-686130291467329486528285156682037514771573167yz^{11}w^{6}-573835201189198397382908316044329159923499095yz^{10}w^{7}-408407485168447513022582067338654890131430992yz^{9}w^{8}-206448299940920427222814201575224791474604982yz^{8}w^{9}-17274272181012793497086205130318703163614730yz^{7}w^{10}+63162565207824664097084510744715503202368394yz^{6}w^{11}+49159590762142849357278527493288330321951141yz^{5}w^{12}+20991267597971625333965164887038410593786561yz^{4}w^{13}+8183261043520066529063037374555818724687217yz^{3}w^{14}+3099014171918558424083204389039261191556611yz^{2}w^{15}+752356039006776574165909123646629379337993yzw^{16}+75815608415499630955065492867323699476121yw^{17}-12091230871400603622397929062400000000000z^{18}+173431150833545098128090228326400000000000z^{17}w-33177807656180079862398480770432359625490618z^{16}w^{2}-216913160920980352335551602576455916754731422z^{15}w^{3}-633974318263173668844603059609274557286786378z^{14}w^{4}-1117387763225928568988323913271765681442022031z^{13}w^{5}-1363523461992311110967305254215187939387385188z^{12}w^{6}-1283310715495364335018485852421733955109326132z^{11}w^{7}-986249943829443908607948654235098347553514461z^{10}w^{8}-553481862718471377097484704620668503887932661z^{9}w^{9}-110978450193983693106156590661955001875031346z^{8}w^{10}+139442002031968955533232402665907112466265928z^{7}w^{11}+154129399798665678449143947490338268779908577z^{6}w^{12}+76140003563800167181461944381633400696110502z^{5}w^{13}+22311859019009650074497302498536554371285134z^{4}w^{14}+4668511857517546101442994612203041498797841z^{3}w^{15}+992111364091115396328508679778510048626664z^{2}w^{16}+201616987099136442894011260310062551930037zw^{17}+20429797792167773013875156452530997944694w^{18}}$

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
10.36.0.b.1 $10$ $2$ $2$ $0$ $0$ full Jacobian
30.24.1.g.2 $30$ $3$ $3$ $1$ $0$ dimension zero
30.36.0.b.1 $30$ $2$ $2$ $0$ $0$ full Jacobian
30.36.1.d.1 $30$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
30.216.13.n.2 $30$ $3$ $3$ $13$ $0$ $1^{6}\cdot2^{3}$
30.288.13.d.1 $30$ $4$ $4$ $13$ $0$ $1^{6}\cdot2^{3}$
30.360.13.e.1 $30$ $5$ $5$ $13$ $1$ $1^{6}\cdot2^{3}$
60.144.5.ez.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.fd.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.ht.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.hv.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.nf.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.nh.2 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.nv.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.nz.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
120.144.5.bkd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.blf.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cjp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ckd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dvc.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dvq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dzk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eam.2 $120$ $2$ $2$ $5$ $?$ not computed
150.360.13.c.2 $150$ $5$ $5$ $13$ $?$ not computed