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Properties

Label 60.96.1.l.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: $600$
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 $2^{6}\cdot4$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.96.1.264

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}5&22\\12&59\end{bmatrix}$, $\begin{bmatrix}32&29\\33&20\end{bmatrix}$, $\begin{bmatrix}53&40\\6&43\end{bmatrix}$, $\begin{bmatrix}56&39\\21&46\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.l.4.1, 60.192.1-60.l.4.2, 60.192.1-60.l.4.3, 60.192.1-60.l.4.4, 60.192.1-60.l.4.5, 60.192.1-60.l.4.6, 60.192.1-60.l.4.7, 60.192.1-60.l.4.8, 120.192.1-60.l.4.1, 120.192.1-60.l.4.2, 120.192.1-60.l.4.3, 120.192.1-60.l.4.4, 120.192.1-60.l.4.5, 120.192.1-60.l.4.6, 120.192.1-60.l.4.7, 120.192.1-60.l.4.8, 120.192.1-60.l.4.9, 120.192.1-60.l.4.10, 120.192.1-60.l.4.11, 120.192.1-60.l.4.12, 120.192.1-60.l.4.13, 120.192.1-60.l.4.14, 120.192.1-60.l.4.15, 120.192.1-60.l.4.16, 120.192.1-60.l.4.17, 120.192.1-60.l.4.18, 120.192.1-60.l.4.19, 120.192.1-60.l.4.20, 120.192.1-60.l.4.21, 120.192.1-60.l.4.22, 120.192.1-60.l.4.23, 120.192.1-60.l.4.24
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{3}\cdot3\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 600.2.a.h

Models

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

$ 0 $ $=$ $ 3 x^{2} - 2 y^{2} - 2 y z + z^{2} $
$=$ $7 x^{2} - 32 x y - 2 x w + 7 y^{2} + 6 y z + 2 y w - 3 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 16 x^{4} - 128 x^{3} y + 744 x^{2} y^{2} + 48 x^{2} y z - 21 x^{2} z^{2} - 1952 x y^{3} - 192 x y^{2} z + \cdots + 9 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

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 2^5\,\frac{1318009427850856660393847786767183913260941975242056228301200854932639878662806568960xz^{23}+880985639491249572171181493477761283832369309372718234075273911176185973180780371968xz^{22}w-94348465142167383900652246772328866984587955427514357355658570028058725779378536448xz^{21}w^{2}-219189509401800143059571022887126142073721287059385975149805178733184854959271038976xz^{20}w^{3}-37660253597630922120069590318644454253691009096976562829237851841379950548118192128xz^{19}w^{4}+19701265899182386600734895717032500471551677203900082423518861532015600008048007168xz^{18}w^{5}+5219344920238809953138971464761412821468047908731932438873016598787777143681239040xz^{17}w^{6}-829504489139628136589123752642629921208896073859033055921756582465315976325389824xz^{16}w^{7}-326822388107338023269206793876984839097868749946927961340278090448664419591208960xz^{15}w^{8}+54594426421405727611975384515505234688817047416407010404934700963314907905041408xz^{14}w^{9}+3314128123963911409193231665716826308084361810995537034287080666148909487644672xz^{13}w^{10}-2908528810240895564792319866109103935257504601251435383008890145689530799503872xz^{12}w^{11}-5071240490675469152127948724583407120635504010966257595137979817157704945664xz^{11}w^{12}+235624247582077569344518471166391892291783113124242258447684674783138674288128xz^{10}w^{13}-35894145788798711725052756393136972810160747210491798127393370963681701824768xz^{9}w^{14}+307085422791497278123211241169310311777529868204956374199908696997488703040xz^{8}w^{15}+425989026939538254037908883744413279670023646274385462628744520743246998528xz^{7}w^{16}-75657480983415714734521819930042470229279064331055117915765974003409185696xz^{6}w^{17}-24141341797035983775799253039052773508333889403896443829318214321536290432xz^{5}w^{18}+2818170291570766175581172206079411972477910479709201033811352103092778960xz^{4}w^{19}-399652567125129514948197515156998589434534969567477193427576481188058176xz^{3}w^{20}-4893073334029106874525269924563489441126638799164954507929731850449656xz^{2}w^{21}+1623324847418877036576052409243806711080836660927181621261838467058728xzw^{22}+75540605952869003938355431059480230381299593554428933192635373450098xw^{23}+5772731972527431158888361038585541569459301754804422293341022052006211364697121165312y^{2}z^{22}+2853366400636380371679089460154796071253376241252398100355118724424898516081066639360y^{2}z^{21}w+58234337539122630876465265590681470665806831682048812128666793155989966221329000448y^{2}z^{20}w^{2}-450302137539579972569674335538635963043528526215694195130535199977951408905350914048y^{2}z^{19}w^{3}-57961647299864489349178538147896565300120140065097898664693094701463965325915376640y^{2}z^{18}w^{4}+22273560688015922941469054896556918124209117975302488385496790978733905089768365056y^{2}z^{17}w^{5}+6766683211264941361415212415927920332672567540012233631790036655778645164732944128y^{2}z^{16}w^{6}-1444068331258125147037219900625335143535749253430036184223490500523152418274631680y^{2}z^{15}w^{7}-14999046002633462855977054877300478579316532929551658476068420443553746448896512y^{2}z^{14}w^{8}+35970578330660644992970268932470502532965160239828715087427687423160668981268480y^{2}z^{13}w^{9}+2440452454853196522740462162336034179188010717633073357586884321514173577100544y^{2}z^{12}w^{10}-5697136013140055850144614893757985547555664470547875616899445416113173123217408y^{2}z^{11}w^{11}+1040248344859505390681117606358940507153783867420201453950726217326596471634176y^{2}z^{10}w^{12}-42479781758169806965568468174353856047881686249711024104904501093657734900608y^{2}z^{9}w^{13}+1016493665546769271516172275954950701578321314321561504651895682874222924000y^{2}z^{8}w^{14}-709154059375683606613546787082094656928430636370840839994784378506648749568y^{2}z^{7}w^{15}+859346665317071626959371708690845888062776214465064934635315682269228415952y^{2}z^{6}w^{16}-136145709738887494884255249417315044890023152974103358608593144964977848000y^{2}z^{5}w^{17}+9956033981483844453506077918747730895234471959230235878618664417245906840y^{2}z^{4}w^{18}-428197121637190085527182298763608659915741316516957558296696786768131040y^{2}z^{3}w^{19}-59841574914995087823180353507424681649926870174687686996315125655088612y^{2}z^{2}w^{20}-5996912305432463019758952643587764912098536043700596556232740245713172y^{2}zw^{21}+497853048654003274382274901795458429976401383076276252891897794887335y^{2}w^{22}-3794072401321452808880813105670844230870683458080733990208532617698904205862750257152yz^{23}-1884269459672859293390221906509447587803268534990819576381133025737529826579747954688yz^{22}w+8751937647549683048173801647130307257466092822961273896593088353313293482920968192yz^{21}w^{2}+338496719337144856063497897485012176362365346750510524884933669537506772323850440704yz^{20}w^{3}+45314731813388083055501832605423366820884103847274079626205974219632868277480210432yz^{19}w^{4}-19512506558123525895415252096321172248910157988158813247089414674585314497458706432yz^{18}w^{5}-628868339607405684494735335594251029705059368740470061938936295233491521262255513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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
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0.c.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.1.k.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.bc.2 $60$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
60.480.33.cc.3 $60$ $5$ $5$ $33$ $1$ $1^{16}\cdot8^{2}$
60.576.33.fd.4 $60$ $6$ $6$ $33$ $1$ $1^{16}\cdot8^{2}$
60.960.65.jk.1 $60$ $10$ $10$ $65$ $1$ $1^{32}\cdot8^{4}$
120.192.5.zb.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zg.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zr.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zw.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bai.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bal.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.baq.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bat.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bfw.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bfz.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bgm.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bgp.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bhb.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bhg.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bhj.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bho.4 $120$ $2$ $2$ $5$ $?$ not computed
180.288.9.l.4 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.w.3 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.ba.4 $180$ $3$ $3$ $17$ $?$ not computed