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Properties

Label 60.288.5-60.od.2.7
Level $60$
Index $288$
Genus $5$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $60$ Newform level: $720$
Index: $288$ $\PSL_2$-index:$144$
Genus: $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $1^{2}\cdot2^{2}\cdot3^{2}\cdot5^{2}\cdot6^{2}\cdot10^{2}\cdot15^{2}\cdot30^{2}$ Cusp orbits $2^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2 \le \gamma \le 4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 30S5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.288.5.3051

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&15\\2&23\end{bmatrix}$, $\begin{bmatrix}23&30\\58&43\end{bmatrix}$, $\begin{bmatrix}37&15\\44&59\end{bmatrix}$, $\begin{bmatrix}41&30\\20&47\end{bmatrix}$, $\begin{bmatrix}47&15\\6&23\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.5.od.2 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $2$
Cyclic 60-torsion field degree: $32$
Full 60-torsion field degree: $7680$

Jacobian

Conductor: $2^{9}\cdot3^{7}\cdot5^{5}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 15.2.a.a$^{2}$, 30.2.a.a, 720.2.f.f

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ x^{2} + x y + y z $
$=$ $2 x^{2} - x y + 6 x z - 3 y^{2} - y z - w^{2} + t^{2}$
$=$ $4 x^{2} - 5 x y + 3 y^{2} + y z + 3 z^{2} - 2 w^{2} + 2 w t - t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{8} + 2 x^{6} y^{2} - 4 x^{5} y^{3} + x^{4} y^{4} - 18 x^{4} y^{2} z^{2} - 4 x^{3} y^{5} + \cdots + 45 y^{4} 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 to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2\cdot3^3\,\frac{986657280xzw^{16}-9237826560xzw^{15}t+35556602880xzw^{14}t^{2}-73071682560xzw^{13}t^{3}+81392378880xzw^{12}t^{4}-29959756800xzw^{11}t^{5}-46644368640xzw^{10}t^{6}+82383279360xzw^{9}t^{7}-61114799520xzw^{8}t^{8}+22267162560xzw^{7}t^{9}-424659840xzw^{6}t^{10}-3295732800xzw^{5}t^{11}+1321308480xzw^{4}t^{12}-114402960xzw^{3}t^{13}-62218320xzw^{2}t^{14}+19896240xzwt^{15}-1859970xzt^{16}-716797440yzw^{16}+4037329920yzw^{15}t-6981012480yzw^{14}t^{2}-5146905600yzw^{13}t^{3}+42467566080yzw^{12}t^{4}-83447316480yzw^{11}t^{5}+90303417600yzw^{10}t^{6}-57105173760yzw^{9}t^{7}+15996088800yzw^{8}t^{8}+5146906560yzw^{7}t^{9}-6938639040yzw^{6}t^{10}+2863597440yzw^{5}t^{11}-434231280yzw^{4}t^{12}-89463600yzw^{3}t^{13}+53796480yzw^{2}t^{14}-9804000yzwt^{15}+675510yzt^{16}-20440320z^{2}w^{16}+381696000z^{2}w^{15}t-1917911040z^{2}w^{14}t^{2}+4423710720z^{2}w^{13}t^{3}-4767020160z^{2}w^{12}t^{4}+248428800z^{2}w^{11}t^{5}+5790689280z^{2}w^{10}t^{6}-7086257280z^{2}w^{9}t^{7}+3095191440z^{2}w^{8}t^{8}+1039459680z^{2}w^{7}t^{9}-2035023840z^{2}w^{6}t^{10}+1068777360z^{2}w^{5}t^{11}-199180740z^{2}w^{4}t^{12}-51917280z^{2}w^{3}t^{13}+37290840z^{2}w^{2}t^{14}-8142960z^{2}wt^{15}+667455z^{2}t^{16}-155591424w^{18}+1246453248w^{17}t-3929505792w^{16}t^{2}+5408526336w^{15}t^{3}+143144320w^{14}t^{4}-11740801792w^{13}t^{5}+17125458304w^{12}t^{6}-7976134016w^{11}t^{7}-6639481232w^{10}t^{8}+12371010240w^{9}t^{9}-7740699712w^{8}t^{10}+1439118064w^{7}t^{11}+1051848044w^{6}t^{12}-782341472w^{5}t^{13}+169509420w^{4}t^{14}+28902656w^{3}t^{15}-24265867w^{2}t^{16}+5264018wt^{17}-419904t^{18}}{84864xzw^{16}-433536xzw^{15}t+641616xzw^{14}t^{2}+245328xzw^{13}t^{3}-1911372xzw^{12}t^{4}+2616552xzw^{11}t^{5}-1709100xzw^{10}t^{6}+371844xzw^{9}t^{7}+280332xzw^{8}t^{8}-283536xzw^{7}t^{9}+134838xzw^{6}t^{10}-56520xzw^{5}t^{11}+23178xzw^{4}t^{12}-3696xzw^{3}t^{13}-2064xzw^{2}t^{14}+960xzwt^{15}-96xzt^{16}-672yzw^{16}+189888yzw^{15}t-817008yzw^{14}t^{2}+1367376yzw^{13}t^{3}-867804yzw^{12}t^{4}-453672yzw^{11}t^{5}+1281348yzw^{10}t^{6}-1076172yzw^{9}t^{7}+463896yzw^{8}t^{8}-69900yzw^{7}t^{9}-40338yzw^{6}t^{10}+35424yzw^{5}t^{11}-17784yzw^{4}t^{12}+7290yzw^{3}t^{13}-1716yzw^{2}t^{14}-168yzwt^{15}+144yzt^{16}-7080z^{2}w^{16}+17808z^{2}w^{15}t+24816z^{2}w^{14}t^{2}-128376z^{2}w^{13}t^{3}+136122z^{2}w^{12}t^{4}+28680z^{2}w^{11}t^{5}-166548z^{2}w^{10}t^{6}+121092z^{2}w^{9}t^{7}-11988z^{2}w^{8}t^{8}-24780z^{2}w^{7}t^{9}+17130z^{2}w^{6}t^{10}-14007z^{2}w^{5}t^{11}+9066z^{2}w^{4}t^{12}-837z^{2}w^{3}t^{13}-1542z^{2}w^{2}t^{14}+420z^{2}wt^{15}+24z^{2}t^{16}-8520w^{18}+56176w^{17}t-109576w^{16}t^{2}+10456w^{15}t^{3}+239870w^{14}t^{4}-327924w^{13}t^{5}+100552w^{12}t^{6}+168544w^{11}t^{7}-218120w^{10}t^{8}+110376w^{9}t^{9}-15932w^{8}t^{10}-12247w^{7}t^{11}+9526w^{6}t^{12}-4925w^{5}t^{13}+2296w^{4}t^{14}-504w^{3}t^{15}-96w^{2}t^{16}+48wt^{17}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.od.2 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Equation of the image curve:

$0$ $=$ $ X^{8}+2X^{6}Y^{2}-4X^{5}Y^{3}+X^{4}Y^{4}-18X^{4}Y^{2}Z^{2}-4X^{3}Y^{5}-24X^{3}Y^{3}Z^{2}+4X^{2}Y^{6}-30X^{2}Y^{4}Z^{2}+36XY^{5}Z^{2}-12Y^{6}Z^{2}+45Y^{4}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
60.144.3-30.a.1.9 $60$ $2$ $2$ $3$ $0$ $2$
60.144.3-30.a.1.48 $60$ $2$ $2$ $3$ $0$ $2$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.576.13-60.ch.1.5 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ge.1.16 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ih.2.5 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ij.2.13 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.mf.1.11 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.mh.1.12 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.mi.2.1 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.ml.2.3 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nd.2.16 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ng.2.12 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.nn.2.6 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.np.2.8 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nx.2.12 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2^{2}$
60.576.13-60.nz.2.12 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2^{2}$
60.576.13-60.oa.2.2 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2^{2}$
60.576.13-60.od.2.4 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2^{2}$
60.576.17-60.iq.2.13 $60$ $2$ $2$ $17$ $3$ $1^{6}\cdot2^{3}$
60.576.17-60.it.2.15 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{3}$
60.576.17-60.iz.2.9 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.576.17-60.jb.2.11 $60$ $2$ $2$ $17$ $4$ $1^{6}\cdot2^{3}$
60.576.17-60.jg.2.14 $60$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{3}$
60.576.17-60.jj.2.16 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.576.17-60.jp.2.10 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{3}$
60.576.17-60.jr.2.20 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.864.21-60.d.2.14 $60$ $3$ $3$ $21$ $0$ $1^{8}\cdot2^{4}$
60.1440.37-60.nl.1.31 $60$ $5$ $5$ $37$ $0$ $1^{16}\cdot2^{8}$