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Properties

Label 60.72.1.n.2
Level $60$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $20$
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 $2^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-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: 60.72.1.351

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&10\\24&59\end{bmatrix}$, $\begin{bmatrix}23&25\\28&13\end{bmatrix}$, $\begin{bmatrix}29&5\\0&31\end{bmatrix}$, $\begin{bmatrix}37&55\\4&39\end{bmatrix}$, $\begin{bmatrix}49&35\\16&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.144.1-60.n.2.1, 60.144.1-60.n.2.2, 60.144.1-60.n.2.3, 60.144.1-60.n.2.4, 60.144.1-60.n.2.5, 60.144.1-60.n.2.6, 60.144.1-60.n.2.7, 60.144.1-60.n.2.8, 120.144.1-60.n.2.1, 120.144.1-60.n.2.2, 120.144.1-60.n.2.3, 120.144.1-60.n.2.4, 120.144.1-60.n.2.5, 120.144.1-60.n.2.6, 120.144.1-60.n.2.7, 120.144.1-60.n.2.8, 120.144.1-60.n.2.9, 120.144.1-60.n.2.10, 120.144.1-60.n.2.11, 120.144.1-60.n.2.12, 120.144.1-60.n.2.13, 120.144.1-60.n.2.14, 120.144.1-60.n.2.15, 120.144.1-60.n.2.16, 120.144.1-60.n.2.17, 120.144.1-60.n.2.18, 120.144.1-60.n.2.19, 120.144.1-60.n.2.20, 120.144.1-60.n.2.21, 120.144.1-60.n.2.22, 120.144.1-60.n.2.23, 120.144.1-60.n.2.24
Cyclic 60-isogeny field degree: $4$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

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

$ 0 $ $=$ $ 15 x^{2} - z^{2} - 4 z w $
$=$ $15 x y + 15 y^{2} + z w - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 225 x^{4} - 15 x^{2} y^{2} - 30 x^{2} y z - 30 x^{2} z^{2} + y^{2} z^{2} - 2 y z^{3} + 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{61425y^{2}z^{16}+603450y^{2}z^{15}w-1066500y^{2}z^{14}w^{2}-276048000y^{2}z^{13}w^{3}-7729344000y^{2}z^{12}w^{4}-106676654400y^{2}z^{11}w^{5}-869051073600y^{2}z^{10}w^{6}-4593395952000y^{2}z^{9}w^{7}-16714521504000y^{2}z^{8}w^{8}-43357911552000y^{2}z^{7}w^{9}-80877759033600y^{2}z^{6}w^{10}-104969165414400y^{2}z^{5}w^{11}-85051962144000y^{2}z^{4}w^{12}-34026378048000y^{2}z^{3}w^{13}-4268536704000y^{2}z^{2}w^{14}-65072332800y^{2}zw^{15}-11059200y^{2}w^{16}+z^{18}+732z^{17}w+183153z^{16}w^{2}+18750934z^{15}w^{3}+554988060z^{14}w^{4}+8184696432z^{13}w^{5}+72788090064z^{12}w^{6}+431091034176z^{11}w^{7}+1810007491008z^{10}w^{8}+5610474218880z^{9}w^{9}+13137293515008z^{8}w^{10}+23333218354176z^{7}w^{11}+30807649400064z^{6}w^{12}+28541707410432z^{5}w^{13}+16348473250560z^{4}w^{14}+4503987229184z^{3}w^{15}+390028182528z^{2}w^{16}+4522463232zw^{17}+741376w^{18}}{w(z-w)^{5}(z+4w)^{2}(15y^{2}z^{8}-270y^{2}z^{7}w+2220y^{2}z^{6}w^{2}-72240y^{2}z^{5}w^{3}-151200y^{2}z^{4}w^{4}-489840y^{2}z^{3}w^{5}-397680y^{2}z^{2}w^{6}-61920y^{2}zw^{7}-960y^{2}w^{8}-z^{10}+16z^{9}w-113z^{8}w^{2}+438z^{7}w^{3}-836z^{6}w^{4}+3632z^{5}w^{5}+3904z^{4}w^{6}-14192z^{3}w^{7}+4432z^{2}w^{8}+2656zw^{9}+64w^{10})}$

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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
$X_0(20)$ $20$ $2$ $2$ $1$ $0$ dimension zero
60.36.0.b.2 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.d.1 $60$ $2$ $2$ $0$ $0$ full Jacobian

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.bf.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.dl.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.fc.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.ff.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.fl.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.fn.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.ga.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.gd.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.216.13.bf.1 $60$ $3$ $3$ $13$ $0$ $1^{6}\cdot2^{3}$
60.288.13.ij.2 $60$ $4$ $4$ $13$ $0$ $1^{6}\cdot2^{3}$
60.360.13.j.1 $60$ $5$ $5$ $13$ $0$ $1^{6}\cdot2^{3}$
120.144.5.fv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.yc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bkv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.blq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bnb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bnu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.brh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bsc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cam.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.can.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.caq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.car.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cbc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cbd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cbg.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cbh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cce.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ccf.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cci.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ccj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ccm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ccn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ccq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ccr.2 $120$ $2$ $2$ $5$ $?$ not computed
300.360.13.g.2 $300$ $5$ $5$ $13$ $?$ not computed