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Properties

Label 60.72.1.cj.2
Level $60$
Index $72$
Genus $1$
Analytic rank $1$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $900$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.72.1.30

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&30\\58&19\end{bmatrix}$, $\begin{bmatrix}27&20\\32&49\end{bmatrix}$, $\begin{bmatrix}27&40\\28&21\end{bmatrix}$, $\begin{bmatrix}31&35\\58&1\end{bmatrix}$, $\begin{bmatrix}37&50\\42&49\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.144.1-60.cj.2.1, 60.144.1-60.cj.2.2, 60.144.1-60.cj.2.3, 60.144.1-60.cj.2.4, 60.144.1-60.cj.2.5, 60.144.1-60.cj.2.6, 60.144.1-60.cj.2.7, 60.144.1-60.cj.2.8, 60.144.1-60.cj.2.9, 60.144.1-60.cj.2.10, 60.144.1-60.cj.2.11, 60.144.1-60.cj.2.12, 60.144.1-60.cj.2.13, 60.144.1-60.cj.2.14, 60.144.1-60.cj.2.15, 60.144.1-60.cj.2.16, 120.144.1-60.cj.2.1, 120.144.1-60.cj.2.2, 120.144.1-60.cj.2.3, 120.144.1-60.cj.2.4, 120.144.1-60.cj.2.5, 120.144.1-60.cj.2.6, 120.144.1-60.cj.2.7, 120.144.1-60.cj.2.8, 120.144.1-60.cj.2.9, 120.144.1-60.cj.2.10, 120.144.1-60.cj.2.11, 120.144.1-60.cj.2.12, 120.144.1-60.cj.2.13, 120.144.1-60.cj.2.14, 120.144.1-60.cj.2.15, 120.144.1-60.cj.2.16
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{2}\cdot3^{2}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 900.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 300x - 1375 $
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Rational points

This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^{10}\cdot5^{10}}\cdot\frac{360x^{2}y^{22}+359235000x^{2}y^{20}z^{2}+25047984375000x^{2}y^{18}z^{4}+535634401640625000x^{2}y^{16}z^{6}+5578106457128906250000x^{2}y^{14}z^{8}+35921626899323730468750000x^{2}y^{12}z^{10}+162803505422546081542968750000x^{2}y^{10}z^{12}+538027203978233528137207031250000x^{2}y^{8}z^{14}+1311119441471756165027618408203125000x^{2}y^{6}z^{16}+2328113527166189990937709808349609375000x^{2}y^{4}z^{18}+2743198323428009059242904186248779296875000x^{2}y^{2}z^{20}+1937829493320453428928740322589874267578125000x^{2}z^{22}+57600xy^{22}z+18391050000xy^{20}z^{3}+848761031250000xy^{18}z^{5}+14924357957812500000xy^{16}z^{7}+140122931009765625000000xy^{14}z^{9}+850238949306225585937500000xy^{12}z^{11}+3689077059225741577148437500000xy^{10}z^{13}+11768615608828697204589843750000000xy^{8}z^{15}+27865250039580153751373291015625000000xy^{6}z^{17}+48027759972020988598465919494628906250000xy^{4}z^{19}+55244323869654530059173703193664550781250000xy^{2}z^{21}+37343192808803479975741356611251831054687500000xz^{23}+y^{24}+5449500y^{22}z^{2}+690800906250y^{20}z^{4}+19566217617187500y^{18}z^{6}+244849893325927734375y^{16}z^{8}+1807439534004638671875000y^{14}z^{10}+9219976939629684448242187500y^{12}z^{12}+34352974558941387176513671875000y^{10}z^{14}+95245863678176437318325042724609375y^{8}z^{16}+197314977408298259943723678588867187500y^{6}z^{18}+290933783284935426140204071998596191406250y^{4}z^{20}+287493417738688892004545778036117553710937500y^{2}z^{22}+138272410875415138725787983275949954986572265625z^{24}}{z^{6}y^{4}(y^{2}+3375z^{2})^{2}(x^{2}y^{8}+499500x^{2}y^{6}z^{2}+12324656250x^{2}y^{4}z^{4}+80115960937500x^{2}y^{2}z^{6}+151154483642578125x^{2}z^{8}+160xy^{8}z+18663750xy^{6}z^{3}+319734843750xy^{4}z^{5}+1729566738281250xy^{2}z^{7}+2912805285644531250xz^{9}+10900y^{8}z^{2}+465075000y^{6}z^{4}+4276325390625y^{4}z^{6}+12872758886718750y^{2}z^{8}+10785164337158203125z^{10})}$

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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
$X_{\pm1}(10)$ $10$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.c.2 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.36.1.bh.1 $60$ $2$ $2$ $1$ $1$ 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.bl.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.da.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.kh.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.kn.1 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.ok.1 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.ow.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.py.1 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.qe.1 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.216.13.gr.2 $60$ $3$ $3$ $13$ $2$ $1^{6}\cdot2^{3}$
60.288.13.nv.2 $60$ $4$ $4$ $13$ $2$ $1^{6}\cdot2^{3}$
60.360.13.bq.1 $60$ $5$ $5$ $13$ $2$ $1^{6}\cdot2^{3}$
120.144.5.is.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.vd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dbb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dcr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ecq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.efq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ene.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eou.1 $120$ $2$ $2$ $5$ $?$ not computed
300.360.13.bh.2 $300$ $5$ $5$ $13$ $?$ not computed