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Properties

Label 60.96.1.c.2
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: $144$
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: 60.96.1.424

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}29&44\\0&53\end{bmatrix}$, $\begin{bmatrix}31&8\\6&49\end{bmatrix}$, $\begin{bmatrix}37&12\\6&7\end{bmatrix}$, $\begin{bmatrix}41&26\\48&55\end{bmatrix}$, $\begin{bmatrix}43&0\\6&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.c.2.1, 60.192.1-60.c.2.2, 60.192.1-60.c.2.3, 60.192.1-60.c.2.4, 60.192.1-60.c.2.5, 60.192.1-60.c.2.6, 60.192.1-60.c.2.7, 60.192.1-60.c.2.8, 60.192.1-60.c.2.9, 60.192.1-60.c.2.10, 60.192.1-60.c.2.11, 60.192.1-60.c.2.12, 60.192.1-60.c.2.13, 60.192.1-60.c.2.14, 60.192.1-60.c.2.15, 60.192.1-60.c.2.16, 120.192.1-60.c.2.1, 120.192.1-60.c.2.2, 120.192.1-60.c.2.3, 120.192.1-60.c.2.4, 120.192.1-60.c.2.5, 120.192.1-60.c.2.6, 120.192.1-60.c.2.7, 120.192.1-60.c.2.8, 120.192.1-60.c.2.9, 120.192.1-60.c.2.10, 120.192.1-60.c.2.11, 120.192.1-60.c.2.12, 120.192.1-60.c.2.13, 120.192.1-60.c.2.14, 120.192.1-60.c.2.15, 120.192.1-60.c.2.16
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{4}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 144.2.a.b

Models

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

$ 0 $ $=$ $ 5 x^{2} - 5 y^{2} + w^{2} $
$=$ $5 x^{2} - 15 x y + 2 y^{2} - 2 y z - 2 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 44 x^{4} + 7 x^{3} z - 5 x^{2} y^{2} + 6 x^{2} z^{2} - 2 x z^{3} - 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{3}{2}w$
$\displaystyle Z$ $=$ $\displaystyle z$

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}{2^4\cdot5\cdot11^{12}}\cdot\frac{11545640165611713485603309315297143066406250xz^{23}-26559265262398173941506110231428572265625000xz^{21}w^{2}+29538094410544895590822474672197614355468750xz^{19}w^{4}-20974996585053346017933022732881454687500000xz^{17}w^{6}+10269651346452700625441482305615485971875000xz^{15}w^{8}-3502712340614403578950521644737536210000000xz^{13}w^{10}+731547167222624814257446463876072133281250xz^{11}w^{12}+44571111948186815188023079836155751750000xz^{9}w^{14}-251587572818559123105122035057922737316250xz^{7}w^{16}+156489135978265670571817230171871055061000xz^{5}w^{18}-36576174152792146691238561348583482798450xz^{3}w^{20}-23787078043936413977808413896563695117187500y^{2}z^{22}+47398734303628078531914593100398887148437500y^{2}z^{20}w^{2}-46100121695452778533868654323002418828125000y^{2}z^{18}w^{4}+29548747251371486435542287501743855742187500y^{2}z^{16}w^{6}-13649529945568721712865484761600434393750000y^{2}z^{14}w^{8}+4875287843598883573321440263912901950250000y^{2}z^{12}w^{10}-2409586402939339261607848591726570911200000y^{2}z^{10}w^{12}+2017339972286191347771371352566391065662500y^{2}z^{8}w^{14}-1316165188636123286759948472788747683390500y^{2}z^{6}w^{16}+439892597064736476490811566204222872890500y^{2}z^{4}w^{18}-533702168692924320782086391906331206520y^{2}z^{2}w^{20}-28986863256558923373299076672781438447476y^{2}w^{22}-17492297527237012750058074210184702832031250yz^{23}+33790199314881769568255145031694635166015625yz^{21}w^{2}-32303309109833128324178935718046537128906250yz^{19}w^{4}+20477741605343108697772200260218749404296875yz^{17}w^{6}-9021723759601708809163000671681834403125000yz^{15}w^{8}+2594628512953041810982101631095402595687500yz^{13}w^{10}-502426191903021626425070823874682240206250yz^{11}w^{12}+254172665722389129717497362911936989728125yz^{9}w^{14}-446942954395142233479106421772668152515750yz^{7}w^{16}+481675975211181415073823522796837775737375yz^{5}w^{18}-255842797508698441220986162510111001147730yz^{3}w^{20}+55001738251115284569318672577049489689131yzw^{22}-5946999596352088349837425927273845898437500z^{24}+15156306709818193422606809653176394687500000z^{22}w^{2}-18890335122715743134680100638049884570312500z^{20}w^{4}+15282234939744607178843539069363516875000000z^{18}w^{6}-8812170129096867638261691596823536981250000z^{16}w^{8}+3735793220899004930900897141664664284750000z^{14}w^{10}-1130765246941324345666951635420085965987500z^{12}w^{12}+214480981645256941351719744698654879700000z^{10}w^{14}-11886442465530683411514842156585731894500z^{8}w^{16}-8718180313640562068159420731836660848000z^{6}w^{18}+418305475187676325741409778422835561420z^{4}w^{20}+137477021927064532460830554174265491456z^{2}w^{22}-17184627740883066557603819271783186432w^{24}}{w^{4}(40593386635408019531250xz^{19}-129898837233305662500000xz^{17}w^{2}+163804745262057100781250xz^{15}w^{4}-98485448433515265375000xz^{13}w^{6}+276103306483525974843750xz^{11}w^{8}-276893939906589924270000xz^{9}w^{10}+96312436898594858221500xz^{7}w^{12}-10091864226837657990000xz^{5}w^{14}+207054865739519490xz^{3}w^{16}+10671993879307389062500y^{2}z^{18}-26511784552453581562500y^{2}z^{16}w^{2}+138789862168720950000000y^{2}z^{14}w^{4}-324228157517569221537500y^{2}z^{12}w^{6}-323660846678111115437500y^{2}z^{10}w^{8}+535941502717117102798500y^{2}z^{8}w^{10}-214949496187380281038800y^{2}z^{6}w^{12}+28677251190719300455000y^{2}z^{4}w^{14}-12049772436340864236y^{2}z^{2}w^{16}-14018700840091332y^{2}w^{18}+39623205373652802343750yz^{19}-172141400300213249296875yz^{17}w^{2}+301174314586853229843750yz^{15}w^{4}-269070085157612109696875yz^{13}w^{6}-318277752515943646356250yz^{11}w^{8}+375185091055404857988375yz^{9}w^{10}-104345523911387707505700yz^{7}w^{12}+6629748911639700304150yz^{5}w^{14}-72003770128563760674yz^{3}w^{16}-310582298609279235yzw^{18}-970181261755217187500z^{20}+5741864949265623750000z^{18}w^{2}-22612368348829617187500z^{16}w^{4}+31980905485980812350000z^{14}w^{6}-166860152716949095262500z^{12}w^{8}+237990790590232182624000z^{10}w^{10}-156007638743710985068200z^{8}w^{12}+53058039527517540604000z^{6}w^{14}-7604962171652247228684z^{4}w^{16})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.1.c.1 $12$ $2$ $2$ $1$ $0$ dimension zero
60.48.0.a.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0.a.2 $60$ $2$ $2$ $0$ $0$ 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.c.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.c.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.e.3 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.e.4 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.v.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.192.5.v.3 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.192.5.w.3 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.192.5.w.4 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.288.9.h.1 $60$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
60.480.33.g.1 $60$ $5$ $5$ $33$ $4$ $1^{16}\cdot8^{2}$
60.576.33.g.1 $60$ $6$ $6$ $33$ $2$ $1^{16}\cdot8^{2}$
60.960.65.g.2 $60$ $10$ $10$ $65$ $8$ $1^{32}\cdot8^{4}$
120.192.5.hy.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hy.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ii.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ii.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pi.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pi.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pp.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pp.4 $120$ $2$ $2$ $5$ $?$ not computed
180.288.9.c.1 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.c.1 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.k.1 $180$ $3$ $3$ $17$ $?$ not computed