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Properties

Label 60.144.7.lz.1
Level $60$
Index $144$
Genus $7$
Analytic rank $4$
Cusps $12$
$\Q$-cusps $4$

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Invariants

Level: $60$ $\SL_2$-level: $60$ Newform level: $3600$
Index: $144$ $\PSL_2$-index:$144$
Genus: $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $4$ are rational) Cusp widths $1^{2}\cdot3^{2}\cdot4\cdot5^{2}\cdot12\cdot15^{2}\cdot20\cdot60$ Cusp orbits $1^{4}\cdot2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $4$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 60P7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.144.7.540

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&5\\6&47\end{bmatrix}$, $\begin{bmatrix}19&0\\6&1\end{bmatrix}$, $\begin{bmatrix}31&5\\0&29\end{bmatrix}$, $\begin{bmatrix}41&30\\6&11\end{bmatrix}$, $\begin{bmatrix}41&35\\42&13\end{bmatrix}$, $\begin{bmatrix}47&10\\54&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.288.7-60.lz.1.1, 60.288.7-60.lz.1.2, 60.288.7-60.lz.1.3, 60.288.7-60.lz.1.4, 60.288.7-60.lz.1.5, 60.288.7-60.lz.1.6, 60.288.7-60.lz.1.7, 60.288.7-60.lz.1.8, 60.288.7-60.lz.1.9, 60.288.7-60.lz.1.10, 60.288.7-60.lz.1.11, 60.288.7-60.lz.1.12, 60.288.7-60.lz.1.13, 60.288.7-60.lz.1.14, 60.288.7-60.lz.1.15, 60.288.7-60.lz.1.16, 60.288.7-60.lz.1.17, 60.288.7-60.lz.1.18, 60.288.7-60.lz.1.19, 60.288.7-60.lz.1.20, 60.288.7-60.lz.1.21, 60.288.7-60.lz.1.22, 60.288.7-60.lz.1.23, 60.288.7-60.lz.1.24, 60.288.7-60.lz.1.25, 60.288.7-60.lz.1.26, 60.288.7-60.lz.1.27, 60.288.7-60.lz.1.28, 60.288.7-60.lz.1.29, 60.288.7-60.lz.1.30, 60.288.7-60.lz.1.31, 60.288.7-60.lz.1.32, 120.288.7-60.lz.1.1, 120.288.7-60.lz.1.2, 120.288.7-60.lz.1.3, 120.288.7-60.lz.1.4, 120.288.7-60.lz.1.5, 120.288.7-60.lz.1.6, 120.288.7-60.lz.1.7, 120.288.7-60.lz.1.8, 120.288.7-60.lz.1.9, 120.288.7-60.lz.1.10, 120.288.7-60.lz.1.11, 120.288.7-60.lz.1.12, 120.288.7-60.lz.1.13, 120.288.7-60.lz.1.14, 120.288.7-60.lz.1.15, 120.288.7-60.lz.1.16, 120.288.7-60.lz.1.17, 120.288.7-60.lz.1.18, 120.288.7-60.lz.1.19, 120.288.7-60.lz.1.20, 120.288.7-60.lz.1.21, 120.288.7-60.lz.1.22, 120.288.7-60.lz.1.23, 120.288.7-60.lz.1.24, 120.288.7-60.lz.1.25, 120.288.7-60.lz.1.26, 120.288.7-60.lz.1.27, 120.288.7-60.lz.1.28, 120.288.7-60.lz.1.29, 120.288.7-60.lz.1.30, 120.288.7-60.lz.1.31, 120.288.7-60.lz.1.32
Cyclic 60-isogeny field degree: $2$
Cyclic 60-torsion field degree: $32$
Full 60-torsion field degree: $15360$

Jacobian

Conductor: $2^{17}\cdot3^{11}\cdot5^{11}$
Simple: no
Squarefree: no
Decomposition: $1^{7}$
Newforms: 15.2.a.a$^{2}$, 30.2.a.a, 3600.2.a.be$^{2}$, 3600.2.a.f, 3600.2.a.u

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ x t - x u + z w + z u $
$=$ $x w + x v - z w - z t$
$=$ $x t - x u + y w + y v$
$=$ $x w + x t - y t + y v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 5 x^{6} z^{2} - 3 x^{5} y^{3} + 25 x^{5} y z^{2} - 25 x^{4} y^{2} z^{2} + 25 x^{2} y^{4} z^{2} + \cdots + 5 y^{6} z^{2} $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:0:0:1)$, $(0:0:0:0:0:1:0)$, $(0:0:0:1/2:-1/2:-1/2:1)$, $(0:0:0:-1:1:-2:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{5}w$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{118398891112560z^{2}u^{10}+7376952919338060z^{2}u^{9}v+5601163627708500z^{2}u^{8}v^{2}-65966581879654920z^{2}u^{7}v^{3}-156800821390664430z^{2}u^{6}v^{4}+71849727087374670z^{2}u^{5}v^{5}+325493515748835000z^{2}u^{4}v^{6}-211639013611466040z^{2}u^{3}v^{7}-305117489283394380z^{2}u^{2}v^{8}+237286918579732320z^{2}uv^{9}-67794224492696340z^{2}v^{10}-60477970049148wtu^{10}+313147307118916wtu^{9}v-6347137328149985wtu^{8}v^{2}-21501858938959120wtu^{7}v^{3}-1179725958690244wtu^{6}v^{4}+76696025313655069wtu^{5}v^{5}+62572520846522043wtu^{4}v^{6}-92840820976626584wtu^{3}v^{7}-56443435500160103wtu^{2}v^{8}+58753208416254370wtuv^{9}-18078415688777515wtv^{10}-6466168487972wu^{11}+1759171503571610wu^{10}v+7092987969660238wu^{9}v^{2}+753925356760159wu^{8}v^{3}-64982143537697127wu^{7}v^{4}-123685505028056363wu^{6}v^{5}+43730745656671857wu^{5}v^{6}+186905668349469261wu^{4}v^{7}-63239090745546938wu^{3}v^{8}-115252540086218015wu^{2}v^{9}+76833316876532632wuv^{10}-18078415653524751wv^{11}+56646589585332t^{2}u^{10}+1121171316563360t^{2}u^{9}v-3731963549688776t^{2}u^{8}v^{2}-32843298908114944t^{2}u^{7}v^{3}-47067532094890636t^{2}u^{6}v^{4}+49602874115293296t^{2}u^{5}v^{5}+115564849664321582t^{2}u^{4}v^{6}-49282471830989224t^{2}u^{3}v^{7}-79054376990764596t^{2}u^{2}v^{8}+56492373490614268t^{2}uv^{9}-13558760543665376t^{2}v^{10}-147825659537100tu^{11}-478125233117546tu^{10}v+4777158739562313tu^{9}v^{2}+33400306438366646tu^{8}v^{3}+47110749190741903tu^{7}v^{4}-62790912707102524tu^{6}v^{5}-136832693487052311tu^{5}v^{6}+52824628280837008tu^{4}v^{7}+104504379943657043tu^{3}v^{8}-51972472898699032tu^{2}v^{9}+2259283228605279tuv^{10}+4519613271328262tv^{11}+4413675765625u^{12}+174307714130850u^{11}v+635509439553689u^{10}v^{2}-5190414650281317u^{9}v^{3}-34510104392479538u^{8}v^{4}-46972901375283317u^{7}v^{5}+62905573460840026u^{6}v^{6}+134049528473178766u^{5}v^{7}-51488257559135475u^{4}v^{8}-85836102164304085u^{3}v^{9}+61011968083323332u^{2}v^{10}-13558763085942617uv^{11}+282475249v^{12}}{3618572175z^{2}u^{10}+745062667980z^{2}u^{9}v-5490067970955z^{2}u^{8}v^{2}+6919884978180z^{2}u^{7}v^{3}-2545968439770z^{2}u^{6}v^{4}+1340796604980z^{2}u^{5}v^{5}+3197118183000z^{2}u^{4}v^{6}-11500891590z^{2}u^{3}v^{7}+456615645375z^{2}u^{2}v^{8}+166422915000z^{2}uv^{9}-15212437500z^{2}v^{10}+7089267328wtu^{10}+183373312880wtu^{9}v-1355909301297wtu^{8}v^{2}+1617856038900wtu^{7}v^{3}-598115052367wtu^{6}v^{4}-357292130202wtu^{5}v^{5}+571297846714wtu^{4}v^{6}-220085997415wtu^{3}v^{7}-44736120700wtu^{2}v^{8}+32157614750wtuv^{9}-319192500wtv^{10}+2736441412wu^{11}+98334745174wu^{10}v-858827630674wu^{9}v^{2}+550790358585wu^{8}v^{3}-81138647611wu^{7}v^{4}-323776340398wu^{6}v^{5}+286406928122wu^{5}v^{6}+32649176782wu^{4}v^{7}-13349115926wu^{3}v^{8}+20389087125wu^{2}v^{9}+4090068750wuv^{10}+9142711234t^{2}u^{10}-1798891196t^{2}u^{9}v-442103941900t^{2}u^{8}v^{2}+302721224510t^{2}u^{7}v^{3}+77629897772t^{2}u^{6}v^{4}-438836422969t^{2}u^{5}v^{5}+230836877768t^{2}u^{4}v^{6}+60453393349t^{2}u^{3}v^{7}-26668401050t^{2}u^{2}v^{8}+15284855875t^{2}uv^{9}+4090068750t^{2}v^{10}+7316561564tu^{11}-50546581435tu^{10}v+143512388435tu^{9}v^{2}+456533963607tu^{8}v^{3}-379539894607tu^{7}v^{4}+71780838108tu^{6}v^{5}+82802858831tu^{5}v^{6}-204275298230tu^{4}v^{7}-25281380138tu^{3}v^{8}-30121850525tu^{2}v^{9}-11094861000tuv^{10}+1014162500tv^{11}-7316561564u^{11}v+34087308637u^{10}v^{2}-121590258133u^{9}v^{3}-65161537772u^{8}v^{4}-73084109399u^{7}v^{5}-132630418291u^{6}v^{6}+33601391369u^{5}v^{7}+14026472008u^{4}v^{8}-1233103301u^{3}v^{9}+15979825875u^{2}v^{10}+4090068750uv^{11}}$

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
$X_0(30)$ $30$ $2$ $2$ $3$ $0$ $1^{4}$
60.24.0.s.1 $60$ $6$ $6$ $0$ $0$ full Jacobian
60.36.1.bg.1 $60$ $4$ $4$ $1$ $1$ $1^{6}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.288.13.nt.1 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.nt.2 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.nu.1 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.nu.2 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.oa.1 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.oa.2 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.ob.1 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.13.ob.2 $60$ $2$ $2$ $13$ $4$ $2^{3}$
60.288.17.g.1 $60$ $2$ $2$ $17$ $4$ $1^{10}$
60.288.17.p.1 $60$ $2$ $2$ $17$ $7$ $1^{10}$
60.288.17.gb.1 $60$ $2$ $2$ $17$ $8$ $1^{10}$
60.288.17.gc.1 $60$ $2$ $2$ $17$ $6$ $1^{10}$
60.288.17.hg.1 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.hg.2 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.hh.1 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.hh.2 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.ho.1 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.ho.2 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.hp.1 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.hp.2 $60$ $2$ $2$ $17$ $4$ $2^{5}$
60.288.17.ht.1 $60$ $2$ $2$ $17$ $7$ $1^{10}$
60.288.17.hu.1 $60$ $2$ $2$ $17$ $5$ $1^{10}$
60.288.17.if.1 $60$ $2$ $2$ $17$ $8$ $1^{10}$
60.288.17.ig.1 $60$ $2$ $2$ $17$ $7$ $1^{10}$
60.432.25.jt.1 $60$ $3$ $3$ $25$ $8$ $1^{18}$
60.720.43.nx.1 $60$ $5$ $5$ $43$ $13$ $1^{36}$
120.288.13.ifw.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.ifw.2 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.ifz.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.ifz.2 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.igu.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.igu.2 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.igx.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.igx.2 $120$ $2$ $2$ $13$ $?$ not computed
120.288.17.nbx.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.rwk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cgeu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cgex.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmqy.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmqy.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmrb.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmrb.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmrw.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmrw.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmrz.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmrz.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmsm.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmsp.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmtw.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cmtz.1 $120$ $2$ $2$ $17$ $?$ not computed