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Properties

Label 60.96.1.b.3
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 $2^{8}$
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: 60.96.1.379

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}11&40\\12&43\end{bmatrix}$, $\begin{bmatrix}13&14\\18&25\end{bmatrix}$, $\begin{bmatrix}13&46\\54&41\end{bmatrix}$, $\begin{bmatrix}55&16\\6&49\end{bmatrix}$, $\begin{bmatrix}59&26\\36&53\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.b.3.1, 60.192.1-60.b.3.2, 60.192.1-60.b.3.3, 60.192.1-60.b.3.4, 60.192.1-60.b.3.5, 60.192.1-60.b.3.6, 60.192.1-60.b.3.7, 60.192.1-60.b.3.8, 60.192.1-60.b.3.9, 60.192.1-60.b.3.10, 60.192.1-60.b.3.11, 60.192.1-60.b.3.12, 120.192.1-60.b.3.1, 120.192.1-60.b.3.2, 120.192.1-60.b.3.3, 120.192.1-60.b.3.4, 120.192.1-60.b.3.5, 120.192.1-60.b.3.6, 120.192.1-60.b.3.7, 120.192.1-60.b.3.8, 120.192.1-60.b.3.9, 120.192.1-60.b.3.10, 120.192.1-60.b.3.11, 120.192.1-60.b.3.12, 120.192.1-60.b.3.13, 120.192.1-60.b.3.14, 120.192.1-60.b.3.15, 120.192.1-60.b.3.16, 120.192.1-60.b.3.17, 120.192.1-60.b.3.18, 120.192.1-60.b.3.19, 120.192.1-60.b.3.20, 120.192.1-60.b.3.21, 120.192.1-60.b.3.22, 120.192.1-60.b.3.23, 120.192.1-60.b.3.24, 120.192.1-60.b.3.25, 120.192.1-60.b.3.26, 120.192.1-60.b.3.27, 120.192.1-60.b.3.28, 120.192.1-60.b.3.29, 120.192.1-60.b.3.30, 120.192.1-60.b.3.31, 120.192.1-60.b.3.32, 120.192.1-60.b.3.33, 120.192.1-60.b.3.34, 120.192.1-60.b.3.35, 120.192.1-60.b.3.36, 120.192.1-60.b.3.37, 120.192.1-60.b.3.38, 120.192.1-60.b.3.39, 120.192.1-60.b.3.40, 120.192.1-60.b.3.41, 120.192.1-60.b.3.42, 120.192.1-60.b.3.43, 120.192.1-60.b.3.44
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
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 $ $=$ $ 7 x^{2} - 15 x y + 2 x z - 2 z^{2} $
$=$ $14 x^{2} + 4 x z - 15 y^{2} - 4 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 5175 x^{4} - 30 x^{3} y + 2 x^{2} y^{2} - 240 x^{2} z^{2} + 2 x y z^{2} - 7 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 30z$
$\displaystyle Z$ $=$ $\displaystyle w$

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{2}{3^3\cdot5^2\cdot7^{12}\cdot23^4}\cdot\frac{97754953162519998156359844372415661813013444691200000000000xz^{23}-128814141589982523417375423021238990531524009621720000000000xz^{21}w^{2}+76195518187504097501048262084797579434743650640840000000000xz^{19}w^{4}-26560120953082770839671812350374391117489700384084000000000xz^{17}w^{6}+6012744571104512665444132180995019143101798260899840000000xz^{15}w^{8}-917241759246763174861203047450698885588689455607192000000xz^{13}w^{10}+94668001399937338779426598257963280723311127472548800000xz^{11}w^{12}-6470567983686624874971654447740931752307379006922400000xz^{9}w^{14}+277134158739325772893750685405112840972372085728960000xz^{7}w^{16}-5922757582680818821771845967878868237933679608633200xz^{5}w^{18}-48518790051448977109147149458409197806337325650160xz^{3}w^{20}+3769253364542126936328427843146995491894496713800xzw^{22}-115349644664012677765033729101670391591633114607600000000000y^{2}z^{22}+145270496124891409357045545700023608713734943257840000000000y^{2}z^{20}w^{2}-81461079473400282568293315849982377649604346184320000000000y^{2}z^{18}w^{4}+26647807801779456315565791560157366503400292922172000000000y^{2}z^{16}w^{6}-5588014642898444771687203507142919991153584757880400000000y^{2}z^{14}w^{8}+775774393252985122996051814493506969935977536363616000000y^{2}z^{12}w^{10}-71053069725547531138546970726629455898579965103748400000y^{2}z^{10}w^{12}+4080331793115638855349728009090442444910853347075200000y^{2}z^{8}w^{14}-103955984930038876842737279150624726460546625052604000y^{2}z^{6}w^{16}-7675178938706886303586246905846473130832199638308000y^{2}z^{4}w^{18}+1194323974367329110924178311299629003598870950248360y^{2}z^{2}w^{20}-56957393949496938520433515093102321242610569169296y^{2}w^{22}-28816728840701649571831052085476210327057905154400000000000yz^{23}+33619398490827945990340464323494076038210039366800000000000yz^{21}w^{2}-17296744350672385219036011524410960354399351555440000000000yz^{19}w^{4}+5130486756833644839331465228122115056329134564416000000000yz^{17}w^{6}-960272726543972840606542498020858546121111732998240000000yz^{15}w^{8}+116157585680894994918904935346296977281056819978864000000yz^{13}w^{10}-8743321782983002936528303348187765068448449578624000000yz^{11}w^{12}+300564251569328760715958198279845334248425254294400000yz^{9}w^{14}+19833434466614587558061086658739774647030960511528000yz^{7}w^{16}-3900200353763438374195919033434922881094673928562400yz^{5}w^{18}+215688779047003108971294545893441212067935451575840yz^{3}w^{20}-40121529744420769699391689179704619814421344951800000000000z^{24}+56838833233541022562693327188277521759504843505560000000000z^{22}w^{2}-36489307729150202455729624550592311382518441935272000000000z^{20}w^{4}+13959940663851200552135518712518198394836463692860000000000z^{18}w^{6}-3516743952847092314620971984081961483562399209834500000000z^{16}w^{8}+607719324617202321395196321999652293919565201777208000000z^{14}w^{10}-72752834124696186538863035365572426439962159957055200000z^{12}w^{12}+5968924889974416177678049829514585495651599562589680000z^{10}w^{14}-328551669986994199889867690034474032354288591381610000z^{8}w^{16}+12402109685069932220051475013282529424007475246895600z^{6}w^{18}-430851046251934312912005463858552893949391866164960z^{4}w^{20}+24453537867804153487706957683598343635481832390896z^{2}w^{22}-1172164342762368452308645485254327036885797368683w^{24}}{w^{4}(25233263498940356345896730250000xz^{17}w^{2}-55048800869924015495598024600000xz^{15}w^{4}+25415008223601761219470215532500xz^{13}w^{6}+121984533570577221805893049903500xz^{11}w^{8}-76740538890244332861839782921500xz^{9}w^{10}+17142783131611801425138653113740xz^{7}w^{12}-1651232608739856915030864082512xz^{5}w^{14}+61196475912374192736561169920xz^{3}w^{16}-484411539858281478406656000xzw^{18}-754334914027079854934990358750000y^{2}z^{18}+625666375932963418193728581000000y^{2}z^{16}w^{2}-31778293740339454735658119875000y^{2}z^{14}w^{4}-50659576652646080878344009480000y^{2}z^{12}w^{6}-144411390084111446549847251179125y^{2}z^{10}w^{8}+82763550469759701534310402267050y^{2}z^{8}w^{10}-15429851777724063951362974183665y^{2}z^{6}w^{12}+1107616797165339470673127136400y^{2}z^{4}w^{14}-23722722105107967031223750400y^{2}z^{2}w^{16}+50636049959018818635264000y^{2}w^{18}+216285115705203054393400545000000yz^{19}-534841256981995233931327842000000yz^{17}w^{2}+323124288201446740032410199450000yz^{15}w^{4}-57444467803080957971184652245000yz^{13}w^{6}-37293844931387249283312853185000yz^{11}w^{8}+17085397814817933558434215989600yz^{9}w^{10}-2331499512106745368923517305120yz^{7}w^{12}+106551031746492425698557772800yz^{5}w^{14}-941817186405086253541171200yz^{3}w^{16}+7209503856840101813113351500000z^{18}w^{2}-13215258545966429978172751931250z^{16}w^{4}+5958337524912719146824646207500z^{14}w^{6}-51696379810700526014775711042750z^{12}w^{8}+36509660906400281608325245982475z^{10}w^{10}-10174234021528049228173925583090z^{8}w^{12}+1343834249337856572382605732031z^{6}w^{14}-80305357414117456548533964400z^{4}w^{16}+1595017753662936153717653760z^{2}w^{18}-3375736663934587909017600w^{20})}$

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.b.1 $12$ $2$ $2$ $1$ $0$ dimension zero
60.48.0.a.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0.c.3 $60$ $2$ $2$ $0$ $0$ 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
60.192.5.b.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.e.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.r.3 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.192.5.u.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.288.9.b.1 $60$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
60.480.33.f.1 $60$ $5$ $5$ $33$ $1$ $1^{16}\cdot8^{2}$
60.576.33.f.1 $60$ $6$ $6$ $33$ $0$ $1^{16}\cdot8^{2}$
60.960.65.f.4 $60$ $10$ $10$ $65$ $2$ $1^{32}\cdot8^{4}$
120.192.5.f.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ih.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.lx.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ma.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mt.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mw.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nc.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ne.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nk.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nl.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nr.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ns.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nt.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nu.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ok.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qo.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qw.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qx.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.9.cb.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.co.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.fh.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.fl.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.lp.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.lu.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.mr.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.mw.4 $120$ $2$ $2$ $9$ $?$ not computed
180.288.9.b.2 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.b.2 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.j.2 $180$ $3$ $3$ $17$ $?$ not computed