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Properties

Label 60.96.1.j.1
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: $72$
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 $2^{4}\cdot4^{2}$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.96.1.479

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}8&55\\15&28\end{bmatrix}$, $\begin{bmatrix}37&22\\6&25\end{bmatrix}$, $\begin{bmatrix}52&45\\39&14\end{bmatrix}$, $\begin{bmatrix}53&20\\48&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.j.1.1, 60.192.1-60.j.1.2, 60.192.1-60.j.1.3, 60.192.1-60.j.1.4, 60.192.1-60.j.1.5, 60.192.1-60.j.1.6, 60.192.1-60.j.1.7, 60.192.1-60.j.1.8, 120.192.1-60.j.1.1, 120.192.1-60.j.1.2, 120.192.1-60.j.1.3, 120.192.1-60.j.1.4, 120.192.1-60.j.1.5, 120.192.1-60.j.1.6, 120.192.1-60.j.1.7, 120.192.1-60.j.1.8, 120.192.1-60.j.1.9, 120.192.1-60.j.1.10, 120.192.1-60.j.1.11, 120.192.1-60.j.1.12, 120.192.1-60.j.1.13, 120.192.1-60.j.1.14, 120.192.1-60.j.1.15, 120.192.1-60.j.1.16, 120.192.1-60.j.1.17, 120.192.1-60.j.1.18, 120.192.1-60.j.1.19, 120.192.1-60.j.1.20, 120.192.1-60.j.1.21, 120.192.1-60.j.1.22, 120.192.1-60.j.1.23, 120.192.1-60.j.1.24
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{3}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 72.2.a.a

Models

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

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

$ 0 $ $=$ $ 236 x^{4} + 4 x^{3} y - 86 x^{3} z - x^{2} y^{2} - 2 x^{2} y z - 59 x^{2} z^{2} - x y z^{2} + \cdots - 4 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 10w$
$\displaystyle Z$ $=$ $\displaystyle 2z$

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}{2^{12}}\cdot\frac{3022289782613194086811927309883170267830108285099143331840xz^{23}-249078066623709319711339256185945690890272617325081395200xz^{22}w+4060385976026113190691597985585162171566825985683121766400xz^{21}w^{2}-634882141359171143394761121626077956433826688913144545280xz^{20}w^{3}+2453439372184497189413207157695384052579507644254440652800xz^{19}w^{4}-536103254490267607275097684781374280225781856334095319040xz^{18}w^{5}+848003536470148581615264208966421436051154882941859921920xz^{17}w^{6}-219175999592021001741689642767496465691873247029033369600xz^{16}w^{7}+175418422197873670295311685117766709794328251503611084800xz^{15}w^{8}-46976880796333500112344698761207417929558742090756915200xz^{14}w^{9}+18539744526550673973126722077763198167700166812884008960xz^{13}w^{10}-3472005161820895301810912475876594379560474727844904960xz^{12}w^{11}+19992244127215454179990262681609005977822077523558400xz^{11}w^{12}+280688467659905784536491004022062190165427326449295360xz^{10}w^{13}-231454994495213812523124836460802973221133207285760000xz^{9}w^{14}+74508412896140913012837323822096991758019171906800640xz^{8}w^{15}-4945939008643988869703447897924554882073759345902080xz^{7}w^{16}-4443799369208870482525130194395396794691646178284800xz^{6}w^{17}+1244038236916494280174925367588008641696686095710720xz^{5}w^{18}-17414719353595083449470217536653623337786196720800xz^{4}w^{19}-31080568828733178461189182600940507300952818577120xz^{3}w^{20}+2818403005306382057416273525342111019805488394600xz^{2}w^{21}-8552846449524608883683662288188146927224201365xw^{23}-7212990802779543793897111409027331276453465294199836377088y^{2}z^{22}+594393950523625604085463648519971861764819782537083617280y^{2}z^{21}w-10355446148605827709636222020732946608766569935598734278656y^{2}z^{20}w^{2}+1625177895246958353610112891047601073681619456882974392320y^{2}z^{19}w^{3}-6729048272823956997363114260523383495131156258242246737920y^{2}z^{18}w^{4}+1482133881149651736698048735757316258520897219920994500608y^{2}z^{17}w^{5}-2546997321262072720234371376136322454451926431535520612352y^{2}z^{16}w^{6}+673105622709227694513109744229191576844278869528498143232y^{2}z^{15}w^{7}-594358246718644331264324760044222441170038933116440166400y^{2}z^{14}w^{8}+165764988172486014040667325918294108431875667254246932480y^{2}z^{13}w^{9}-80311285089696047995028924062375762946668656944079224832y^{2}z^{12}w^{10}+18942594909581861726045886038614258642983191966549016576y^{2}z^{11}w^{11}-3166131025673922587414535516071705662797700258626260992y^{2}z^{10}w^{12}-174630127216532627919548788866577191869828632357560320y^{2}z^{9}w^{13}+574121492251047906562602926311647062174321144499840000y^{2}z^{8}w^{14}-266799936403325910839413632536473566621946302798594048y^{2}z^{7}w^{15}+78030112412136777293915760984755736763178143602777408y^{2}z^{6}w^{16}-2842921379883417512356198040483414970373581524299392y^{2}z^{5}w^{17}-5535277030122877828147311527668239687770372734301920y^{2}z^{4}w^{18}+1329019077403109491892742047443628704680735949515520y^{2}z^{3}w^{19}+421720727717459885222687375357617997938188184164y^{2}z^{2}w^{20}-34351542879724443623611031858077357096137227753640y^{2}zw^{21}+3324619171772268009193820143733602929148715530677y^{2}w^{22}-4600569111108946679866565078234108624947633880955598405632yz^{23}-843472017147334923796802904705917552655092399862165012480yz^{22}w-7746386067918955666757989204145075503583056125104744300544yz^{21}w^{2}-513216374944397957514676760046429908683757660421310382080yz^{20}w^{3}-5752632294532805582475548473269563453024851732564618772480yz^{19}w^{4}+208910568785252876902582108632560101136929096795430060032yz^{18}w^{5}-2474689994642234566815440248926292761186216298236059516928yz^{17}w^{6}+315810122863899783431990001172542946815456201083374338048yz^{16}w^{7}-672786278882388935439143898223102263260047525320791408640yz^{15}w^{8}+132439456903461098697512696152291047884453957924900536320yz^{14}w^{9}-114197514592250386287986588284631255860573183607722426368yz^{13}w^{10}+27046592972011613786116844767612153139367735925635416064yz^{12}w^{11}-10194063215482162906662111694609607753665271639521538048yz^{11}w^{12}+1744761750370404198067026683076199068558400912919429120yz^{10}w^{13}+150828237362192733225599290719411750932993113093299200yz^{9}w^{14}-162552428667132892607350280026103250155424427100378112yz^{8}w^{15}+121890440416852534520463564736445120864607102506066112yz^{7}w^{16}-41347946087581618954346454101502556183680546805696128yz^{6}w^{17}+3477588317932708782633810661444412407197665116384480yz^{5}w^{18}+2444566715252330486741630994642299630546057198086880yz^{4}w^{19}-758676230904018693180113357742988532930739878433604yz^{3}w^{20}+16263250357024466819771335427564936133891571677840yz^{2}w^{21}+18344509268903127059978954198334254576780880963723yzw^{22}-1864136035710784092489374900797077150505120663005yw^{23}-1956068564729272080249176505369949037980168077705689956352z^{24}-594393950523625604085463648519971861764819782537083617280z^{23}w-24968032472334495168289181177854761110808210873872484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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
12.48.1.k.1 $12$ $2$ $2$ $1$ $0$ dimension zero
60.48.0.c.3 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0.c.4 $60$ $2$ $2$ $0$ $0$ full Jacobian

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.bm.1 $60$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
60.480.33.bv.1 $60$ $5$ $5$ $33$ $2$ $1^{16}\cdot8^{2}$
60.576.33.ei.1 $60$ $6$ $6$ $33$ $2$ $1^{16}\cdot8^{2}$
60.960.65.hh.2 $60$ $10$ $10$ $65$ $6$ $1^{32}\cdot8^{4}$
120.192.5.vk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.vk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.vl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.vl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wc.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wp.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ww.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ww.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xf.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xi.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xi.3 $120$ $2$ $2$ $5$ $?$ not computed
180.288.9.j.2 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.s.2 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.t.2 $180$ $3$ $3$ $17$ $?$ not computed