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Properties

Label 60.96.1.p.4
Level $60$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $24$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.96.1.769

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}17&22\\21&5\end{bmatrix}$, $\begin{bmatrix}19&12\\0&29\end{bmatrix}$, $\begin{bmatrix}19&24\\24&25\end{bmatrix}$, $\begin{bmatrix}53&24\\12&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.p.4.1, 60.192.1-60.p.4.2, 60.192.1-60.p.4.3, 60.192.1-60.p.4.4, 60.192.1-60.p.4.5, 60.192.1-60.p.4.6, 60.192.1-60.p.4.7, 60.192.1-60.p.4.8, 120.192.1-60.p.4.1, 120.192.1-60.p.4.2, 120.192.1-60.p.4.3, 120.192.1-60.p.4.4, 120.192.1-60.p.4.5, 120.192.1-60.p.4.6, 120.192.1-60.p.4.7, 120.192.1-60.p.4.8, 120.192.1-60.p.4.9, 120.192.1-60.p.4.10, 120.192.1-60.p.4.11, 120.192.1-60.p.4.12, 120.192.1-60.p.4.13, 120.192.1-60.p.4.14, 120.192.1-60.p.4.15, 120.192.1-60.p.4.16, 120.192.1-60.p.4.17, 120.192.1-60.p.4.18, 120.192.1-60.p.4.19, 120.192.1-60.p.4.20, 120.192.1-60.p.4.21, 120.192.1-60.p.4.22, 120.192.1-60.p.4.23, 120.192.1-60.p.4.24
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

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

$ 0 $ $=$ $ 9 x^{4} - 9 x^{3} y + 18 x^{3} z + 234 x^{2} y^{2} + 144 x^{2} y z + 36 x^{2} z^{2} - 54 x y^{3} + \cdots + 29 z^{4} $
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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.

Maps between models of this curve

Birational map from embedded model to plane model:

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

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}{5}\cdot\frac{9568742783515665602301245572257272407002074281975211548857xz^{23}+95261655297866591208859775816726191121097225970226840669382xz^{22}w+745843155406068284059020065428736656598843979199716221097558xz^{21}w^{2}+4187901385980472705372828393724829654478165638674294792583738xz^{20}w^{3}+12983403737973431307124662860959899688656128335938894124164525xz^{19}w^{4}+40184822721760011413360329580311403774128913720785520818387443xz^{18}w^{5}+131804596012027048730794003316499146796236082576576094964801823xz^{17}w^{6}+355074093331932001905260724013367237761770257414405667497579367xz^{16}w^{7}+822584880549382772931639956238015409496681965932066679974439632xz^{15}w^{8}+1859256900299091430507646450157547115039518584655898455545237750xz^{14}w^{9}+3930911464654069152057552849401243057879935804363691900019921312xz^{13}w^{10}+7328838014116289033232405883175509558857980218064163893691105102xz^{12}w^{11}+12301754914350085340724645536821329724372801356140958925302052428xz^{11}w^{12}+19210231877398044836151460775017467852006021341898557084463783918xz^{10}w^{13}+27426757884404882411666962470258825023141509088143161245539666800xz^{9}w^{14}+33869773982413771319329979248789453005252576853844898235830227598xz^{8}w^{15}+34408081855117058218367686524117484590557776668144026809707366138xz^{7}w^{16}+27824653172444229547923470472304518009022769171563369480307926737xz^{6}w^{17}+17469660129151560922579932225462445728378852036376156871537563017xz^{5}w^{18}+8296298328785704144851063066859843480595659322521658949159148575xz^{4}w^{19}+2866262889046375265462449725927649214374698051895825059737549247xz^{3}w^{20}+677915105001292693040995275273328662218591926096353625964907852xz^{2}w^{21}+97695984562873717453880395305497514716461992880897516938401118xzw^{22}+6479776584070509085569872117226116679676555544506722782749368xw^{23}+31599837787301651104235133320253494977268899494843116867193y^{2}z^{22}+315504972847151324122730707969579030538512357879015387863060y^{2}z^{21}w+2378312596930899210572021165661605225179940547500406662075547y^{2}z^{20}w^{2}+6053856066347686631115810341122145167422797836846335715664340y^{2}z^{19}w^{3}+24572221904372404284277107959945828348879271089981977895118285y^{2}z^{18}w^{4}+80962586359861275281491501895479786418809420413444706586493252y^{2}z^{17}w^{5}+234714241591573875355312179445537483901756029365234419141349870y^{2}z^{16}w^{6}+566717936369767382238774281321390678320684532211284155203318288y^{2}z^{15}w^{7}+1299184093152467794946231297949404816681613346516194570834136170y^{2}z^{14}w^{8}+2858025465798201918209968665300077600978727844987544522842716180y^{2}z^{13}w^{9}+5583644438149650717447827825274338490143625005059657429261122498y^{2}z^{12}w^{10}+9644133980416881497042882841494824340076832173797663076496416000y^{2}z^{11}w^{11}+15441795603164937879920921323239447720361018189818204727858556982y^{2}z^{10}w^{12}+22912905291396821597871420667341149837302355903363210751784112820y^{2}z^{9}w^{13}+29784420285641113222951879460710163289112398102727432008551463630y^{2}z^{8}w^{14}+32006833953970419683319048773248568125208850387041449835781850512y^{2}z^{7}w^{15}+27415197083422585614449986344175931437734679037109131778441552630y^{2}z^{6}w^{16}+18234599484849173159518131517683172978832446874747553538053589788y^{2}z^{5}w^{17}+9166128666114780932170029121548079818182996944272858433114626615y^{2}z^{4}w^{18}+3349187380502326962972114881795147094434475464381135263454233860y^{2}z^{3}w^{19}+836638080462733569862171110360804110448629763387337986324816573y^{2}z^{2}w^{20}+127763527758502811361430733175966806272673787781332081688145940y^{2}zw^{21}+8913836185146173963475463116710441285012794099053540703843047y^{2}w^{22}+24165940449151081350217725747980744365990760465148096893268yz^{23}+297337235982146396174753406434045448056916368652984855712485yz^{22}w+1257388353080312346191322607511637322066677900099353709185412yz^{21}w^{2}+5647679741690674028992040326917666008866863176924281799779320yz^{20}w^{3}+16228131827153200605792777633482272455207327395573955268670685yz^{19}w^{4}+63110765193194927145526803916716877726494759274976981844467712yz^{18}w^{5}+174868055042571723940758268262542294458312596596766593171575615yz^{17}w^{6}+416258411082049136045956088904713199876897932891067266528068308yz^{16}w^{7}+983249500732041791256547188150560365930140769924420006946647890yz^{15}w^{8}+2156340371694288327257769612503558767708714007771486467566359430yz^{14}w^{9}+4199930122785970919171860449958107121742480654314118160797774478yz^{13}w^{10}+7396367170447681204029570254030870922171262705996127109986122410yz^{12}w^{11}+12032125325198343531382313461780300809090455669948925192172435602yz^{11}w^{12}+17968554759296213312249662904840712361442884130701716364727246250yz^{10}w^{13}+23771492068130513384982874916338555539148938094466003138308613430yz^{9}w^{14}+26691357683073682392318879450251290314158542537556678358188358642yz^{8}w^{15}+24622490633273658282970484062989731042205758216865246558695760165yz^{7}w^{16}+1826741766097887636550579700771311793023962820706914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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
60.48.1.bo.1 $60$ $2$ $2$ $1$ $0$ 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
60.288.9.bv.1 $60$ $3$ $3$ $9$ $0$ $1^{4}\cdot4$
60.480.33.od.4 $60$ $5$ $5$ $33$ $1$ $1^{16}\cdot16$
60.576.33.od.4 $60$ $6$ $6$ $33$ $0$ $1^{16}\cdot8^{2}$
60.960.65.ui.3 $60$ $10$ $10$ $65$ $2$ $1^{32}\cdot8^{2}\cdot16$
120.192.9.bqa.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.bqd.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.bqn.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.bqo.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.cjn.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.cjq.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.cjs.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.cjt.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.dee.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.def.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.deh.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.dek.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.doi.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.doj.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.dov.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.doy.3 $120$ $2$ $2$ $9$ $?$ not computed
180.288.9.p.4 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.dc.3 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.ef.3 $180$ $3$ $3$ $17$ $?$ not computed