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Properties

Label 60.72.1.dv.1
Level $60$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $720$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $8$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20I1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.72.1.294

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}3&35\\50&51\end{bmatrix}$, $\begin{bmatrix}21&55\\44&7\end{bmatrix}$, $\begin{bmatrix}29&10\\28&13\end{bmatrix}$, $\begin{bmatrix}49&30\\4&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $128$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{4}\cdot3^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 720.2.a.e

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 18x + 27 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(3:0:1)$

Maps to other modular curves

$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3}\cdot\frac{2124x^{2}y^{22}+16840699956x^{2}y^{20}z^{2}+27152850496500x^{2}y^{18}z^{4}+4184067310990092x^{2}y^{16}z^{6}-6338716520801433480x^{2}y^{14}z^{8}+5902929446452733733768x^{2}y^{12}z^{10}-2459861683611625501561368x^{2}y^{10}z^{12}+552612867744605808833469720x^{2}y^{8}z^{14}-70352172886997686613512936068x^{2}y^{6}z^{16}+5050264589191057777282039732740x^{2}y^{4}z^{18}-189903467028703978085561374478364x^{2}y^{2}z^{20}+2902468344472240638158596494588444x^{2}z^{22}+1536516xy^{22}z+287519331132xy^{20}z^{3}+152000584834860xy^{18}z^{5}-15408383957602092xy^{16}z^{7}+73569220381536514920xy^{14}z^{9}-52149166424015604294888xy^{12}z^{11}+18463433147907951774603672xy^{10}z^{13}-3658734116744574876908048280xy^{8}z^{15}+421097904623456199132174144468xy^{6}z^{17}-27829881082219617421927347690900xy^{4}z^{19}+976899768701435208860343633776124xy^{2}z^{21}-14088877297880289603067667547484668xz^{23}+y^{24}+401268708y^{22}z^{2}+2895521149242y^{20}z^{4}+649945001634660y^{18}z^{6}+507788953139421783y^{16}z^{8}-645757418560642585848y^{14}z^{10}+333407415987497442773196y^{12}z^{12}-90079143378492551218466904y^{10}z^{14}+13783001579312442126384540735y^{8}z^{16}-1221546769790968990315098411372y^{6}z^{18}+61323110409633713297849912618394y^{4}z^{20}-1591116188285205046301462727052428y^{2}z^{22}+16144416793390703215870269488157129z^{24}}{y^{2}(x^{2}y^{20}-186246x^{2}y^{18}z^{2}+1212674733x^{2}y^{16}z^{4}-1338840179424x^{2}y^{14}z^{6}+412279697216826x^{2}y^{12}z^{8}-43439632012170108x^{2}y^{10}z^{10}+1426344620217056442x^{2}y^{8}z^{12}-5188335188688x^{2}y^{6}z^{14}+19487638017189x^{2}y^{4}z^{16}-76255974849870x^{2}y^{2}z^{18}+205891132094649x^{2}z^{20}-108xy^{20}z+4495095xy^{18}z^{3}-15307458165xy^{16}z^{5}+11303367668676xy^{14}z^{7}-2683712675488788xy^{12}z^{9}+237932735802890778xy^{10}z^{11}-6923624206843608030xy^{8}z^{13}-11924802651420xy^{6}z^{15}+47448162128808xy^{4}z^{17}-205891132094649xy^{2}z^{19}+617673396283947xz^{21}+5598y^{20}z^{2}-81133812y^{18}z^{4}+146204215191y^{16}z^{6}-64349740392696y^{14}z^{8}+9432565751237544y^{12}z^{10}-504446641446722832y^{10}z^{12}+7933775731789119858y^{8}z^{14}+67029943324824y^{6}z^{16}-238935387862926y^{4}z^{18}+823564528378596y^{2}z^{20}-1853020188851841z^{22})}$

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
20.36.0.b.2 $20$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.e.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.36.1.dr.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
60.144.5.bi.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.de.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.eb.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.eg.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.ov.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.ox.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.pf.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.pg.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.216.9.cx.1 $60$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
60.288.17.kk.2 $60$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
60.360.17.bb.1 $60$ $5$ $5$ $17$ $3$ $1^{8}\cdot2^{4}$
120.144.5.lq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.wi.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bcn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bdw.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.efk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eif.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eim.1 $120$ $2$ $2$ $5$ $?$ not computed
300.360.17.p.1 $300$ $5$ $5$ $17$ $?$ not computed