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Properties

Label 60.72.1.co.1
Level $60$
Index $72$
Genus $1$
Analytic rank $1$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $3600$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.72.1.283

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}7&20\\20&41\end{bmatrix}$, $\begin{bmatrix}51&35\\52&31\end{bmatrix}$, $\begin{bmatrix}51&35\\56&49\end{bmatrix}$, $\begin{bmatrix}53&40\\26&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{4}\cdot3^{2}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 3600.2.a.be

Models

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

$ 0 $ $=$ $ 3 y^{2} - 5 y z + 6 y w + 2 z^{2} - 6 z w + 2 w^{2} $
$=$ $15 x^{2} - 2 y^{2} + 6 y z - 5 z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 1700 x^{4} + 1080 x^{3} z - 315 x^{2} y^{2} + 244 x^{2} z^{2} - 162 x y^{2} z + 24 x z^{3} + 9 y^{4} + \cdots + 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 10x$
$\displaystyle Z$ $=$ $\displaystyle 10w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^4}\cdot\frac{526774833305yz^{17}+4335480430026yz^{16}w+6938705677596yz^{15}w^{2}-40795244704080yz^{14}w^{3}-184177857355020yz^{13}w^{4}-149061535125912yz^{12}w^{5}+701659842008544yz^{11}w^{6}+2106425180389248yz^{10}w^{7}+1752854024787120yz^{9}w^{8}-2338545827884320yz^{8}w^{9}-7909784051739264yz^{7}w^{10}-10111312852471296yz^{6}w^{11}-7577335908379008yz^{5}w^{12}-3532982925623040yz^{4}w^{13}-998660437770240yz^{3}w^{14}-158430813843456yz^{2}w^{15}-12798618492672yzw^{16}-408600184320yw^{17}-535039803737z^{18}-4335480430026z^{17}w-6083515301354z^{16}w^{2}+44335063306680z^{15}w^{3}+179864726803200z^{14}w^{4}+96298522392744z^{13}w^{5}-790884242039304z^{12}w^{6}-1980152287006464z^{11}w^{7}-1090105236873936z^{10}w^{8}+3147420968261280z^{9}w^{9}+7540767673700256z^{8}w^{10}+7709673814039296z^{7}w^{11}+4080224348978688z^{6}w^{12}+717065043787008z^{5}w^{13}-380029163132160z^{4}w^{14}-244928793876480z^{3}w^{15}-52620241206528z^{2}w^{16}-4950879791616zw^{17}-172694757888w^{18}}{(z+w)^{4}(32yz^{13}+1872yz^{12}w+47376yz^{11}w^{2}+677736yz^{10}w^{3}+5989050yz^{9}w^{4}+33237297yz^{8}w^{5}+108815076yz^{7}w^{6}+143017164yz^{6}w^{7}-352322055yz^{5}w^{8}-2038855050yz^{4}w^{9}-4308631056yz^{3}w^{10}-4867177248yz^{2}w^{11}-2892371652yzw^{12}-709375320yw^{13}-32z^{14}-1872z^{13}w-47312z^{12}w^{2}-674376z^{11}w^{3}-5914074z^{10}w^{4}-32313825z^{9}w^{5}-102032784z^{8}w^{6}-113672322z^{7}w^{7}+414717003z^{6}w^{8}+2014302006z^{5}w^{9}+3798051930z^{4}w^{10}+3517877952z^{3}w^{11}+1154024496z^{2}w^{12}-426534984zw^{13}-299817288w^{14})}$

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.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.b.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.36.1.bg.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.bu.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.cr.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.ng.1 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.nj.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.op.2 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.ov.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.qj.2 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.qk.2 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.216.13.hk.2 $60$ $3$ $3$ $13$ $1$ $1^{6}\cdot2^{3}$
60.288.13.oa.1 $60$ $4$ $4$ $13$ $4$ $1^{6}\cdot2^{3}$
60.360.13.cf.1 $60$ $5$ $5$ $13$ $3$ $1^{6}\cdot2^{3}$
120.144.5.iy.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.st.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dvl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dwg.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.edu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eey.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eqd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eqk.2 $120$ $2$ $2$ $5$ $?$ not computed
300.360.13.bm.1 $300$ $5$ $5$ $13$ $?$ not computed