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Properties

Label 60.72.1.fg.2
Level $60$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $60$ $\SL_2$-level: $30$ Newform level: $400$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $6^{2}\cdot30^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $16$ 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: 30D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.72.1.266

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}9&40\\40&3\end{bmatrix}$, $\begin{bmatrix}16&15\\21&11\end{bmatrix}$, $\begin{bmatrix}23&35\\19&32\end{bmatrix}$, $\begin{bmatrix}31&30\\21&59\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $24$
Cyclic 60-torsion field degree: $384$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{4}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 400.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 33x - 62 $
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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
$(-2:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{5^3}\cdot\frac{30x^{2}y^{22}-1844375x^{2}y^{20}z^{2}+29892187500x^{2}y^{18}z^{4}+42802734375x^{2}y^{16}z^{6}-1502954150390625000x^{2}y^{14}z^{8}+4000991816253662109375x^{2}y^{12}z^{10}+3632157664833068847656250x^{2}y^{10}z^{12}-11554220673596858978271484375x^{2}y^{8}z^{14}-11847441783452630043029785156250x^{2}y^{6}z^{16}-4004050424551963806152343750000000x^{2}y^{4}z^{18}-564170932193594053387641906738281250x^{2}y^{2}z^{20}-28368418725221417844295501708984375000x^{2}z^{22}-255xy^{22}z+6488125xy^{20}z^{3}+43838281250xy^{18}z^{5}+27711738281250xy^{16}z^{7}-17273679089355468750xy^{14}z^{9}+70475022997741699218750xy^{12}z^{11}-4679896195564270019531250xy^{10}z^{13}-153828295886301994323730468750xy^{8}z^{15}-116638693725527226924896240234375xy^{6}z^{17}-33887772588059864938259124755859375xy^{4}z^{19}-4313704948799517005681991577148437500xy^{2}z^{21}-201136884925290942192077636718750000000xz^{23}-y^{24}+92245y^{22}z^{2}-2940380625y^{20}z^{4}+37777697656250y^{18}z^{6}-150242030449218750y^{16}z^{8}+138683951770019531250y^{14}z^{10}+623202871889953613281250y^{12}z^{12}-721798265226898193359375000y^{10}z^{14}-1317908755149388313293457031250y^{8}z^{16}-603345688423604667186737060546875y^{6}z^{18}-116702949526481889188289642333984375y^{4}z^{20}-9877254569798773154616355895996093750y^{2}z^{22}-288800094949710764922201633453369140625z^{24}}{z^{2}y^{6}(20x^{2}y^{14}+9375x^{2}y^{12}z^{2}-468750x^{2}y^{10}z^{4}-478515625x^{2}y^{8}z^{6}+4119873046875x^{2}y^{4}z^{10}-190734863281250x^{2}y^{2}z^{12}+2384185791015625x^{2}z^{14}-70xy^{14}z+25000xy^{12}z^{3}+12187500xy^{10}z^{5}+136718750xy^{8}z^{7}-122070312500xy^{6}z^{9}+2746582031250xy^{4}z^{11}+95367431640625xy^{2}z^{13}-2384185791015625xz^{15}-y^{16}-1470y^{14}z^{2}-628125y^{12}z^{4}-1093750y^{10}z^{6}+19277343750y^{8}z^{8}+122070312500y^{6}z^{10}-144500732421875y^{4}z^{12}+6198883056640625y^{2}z^{14}-73909759521484375z^{16})}$

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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
15.36.0.a.2 $15$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.ch.2 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.36.1.fy.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
60.144.9.by.2 $60$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
60.144.9.bz.2 $60$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
60.144.9.dm.2 $60$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
60.144.9.do.1 $60$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
60.144.9.gu.2 $60$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
60.144.9.gw.2 $60$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
60.144.9.hc.2 $60$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
60.144.9.hd.1 $60$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
60.216.9.m.1 $60$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
60.288.13.ry.1 $60$ $4$ $4$ $13$ $1$ $1^{6}\cdot2^{3}$
60.360.21.cv.1 $60$ $5$ $5$ $21$ $6$ $1^{8}\cdot2^{6}$
120.144.9.jdz.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.jeg.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.kzc.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.kzq.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.sco.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.sdc.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.sez.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.sfg.2 $120$ $2$ $2$ $9$ $?$ not computed
180.216.13.if.2 $180$ $3$ $3$ $13$ $?$ not computed
300.360.21.j.2 $300$ $5$ $5$ $21$ $?$ not computed