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Properties

Label 60.288.5-60.od.1.7
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: 60.288.5.3053

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}29&5\\48&41\end{bmatrix}$, $\begin{bmatrix}37&15\\48&17\end{bmatrix}$, $\begin{bmatrix}49&30\\36&37\end{bmatrix}$, $\begin{bmatrix}53&45\\30&17\end{bmatrix}$, $\begin{bmatrix}59&10\\6&59\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.5.od.1 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 $
$=$ $20 x^{2} + 25 x y + 15 y^{2} - 5 y z + 15 z^{2} - w^{2} - t^{2}$
$=$ $10 x^{2} + 5 x y + 30 x z - 15 y^{2} + 5 y z + w^{2} - 2 w t$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{8} + 10 x^{6} y^{2} + 20 x^{5} y^{3} + 5 x^{4} y^{4} - 18 x^{4} y^{2} z^{2} + 20 x^{3} y^{5} + \cdots + 9 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{9299850xzw^{16}-49316400xzw^{15}t-65144400xzw^{14}t^{2}+310312800xzw^{13}t^{3}+800766600xzw^{12}t^{4}+115441200xzw^{11}t^{5}-1913115600xzw^{10}t^{6}-3067214400xzw^{9}t^{7}-2240111700xzw^{8}t^{8}-432296400xzw^{7}t^{9}+616410000xzw^{6}t^{10}+582319200xzw^{5}t^{11}+293831400xzw^{4}t^{12}+89077200xzw^{3}t^{13}+14816400xzw^{2}t^{14}+1526400xzwt^{15}+111450xzt^{16}+3377550yzw^{16}-5020800yzw^{15}t-61011600yzw^{14}t^{2}-62763600yzw^{13}t^{3}+334149000yzw^{12}t^{4}+876261600yzw^{11}t^{5}+667136400yzw^{10}t^{6}-601935600yzw^{9}t^{7}-1775825100yzw^{8}t^{8}-1702728000yzw^{7}t^{9}-961446000yzw^{6}t^{10}-298302000yzw^{5}t^{11}-15737400yzw^{4}t^{12}+13029600yzw^{3}t^{13}+5190000yzw^{2}t^{14}+1465200yzwt^{15}+173550yzt^{16}-3337275z^{2}w^{16}+12681600z^{2}w^{15}t+23794800z^{2}w^{14}t^{2}-55407600z^{2}w^{13}t^{3}-145411800z^{2}w^{12}t^{4}+16147200z^{2}w^{11}t^{5}+264711000z^{2}w^{10}t^{6}+338950800z^{2}w^{9}t^{7}+74473650z^{2}w^{8}t^{8}-169036800z^{2}w^{7}t^{9}-138984000z^{2}w^{6}t^{10}-86773200z^{2}w^{5}t^{11}-28690800z^{2}w^{4}t^{12}-374400z^{2}w^{3}t^{13}+55800z^{2}w^{2}t^{14}-507600z^{2}wt^{15}-89775z^{2}t^{16}+419904w^{18}-2294254w^{17}t-977127w^{16}t^{2}+13913568w^{15}t^{3}+14228780w^{14}t^{4}-32690864w^{13}t^{5}-72242196w^{12}t^{6}-16631088w^{11}t^{7}+85085812w^{10}t^{8}+114602580w^{9}t^{9}+68298558w^{8}t^{10}+10064832w^{7}t^{11}-13046124w^{6}t^{12}-8896864w^{5}t^{13}-3207780w^{4}t^{14}-961392w^{3}t^{15}-111348w^{2}t^{16}+29866wt^{17}+6561t^{18}}{480xzw^{16}-2880xzw^{15}t-4080xzw^{14}t^{2}+72240xzw^{13}t^{3}-246930xzw^{12}t^{4}+365520xzw^{11}t^{5}-169290xzw^{10}t^{6}-64020xzw^{9}t^{7}+346680xzw^{8}t^{8}-267420xzw^{7}t^{9}-208980xzw^{6}t^{10}-288420xzw^{5}t^{11}-57990xzw^{4}t^{12}+40380xzw^{3}t^{13}+44310xzw^{2}t^{14}+14040xzwt^{15}+2040xzt^{16}+720yzw^{16}-10680yzw^{15}t+65220yzw^{14}t^{2}-231330yzw^{13}t^{3}+532350yzw^{12}t^{4}-828420yzw^{11}t^{5}+957270yzw^{10}t^{6}-729990yzw^{9}t^{7}+313740yzw^{8}t^{8}-348180yzw^{7}t^{9}-15360yzw^{6}t^{10}+36750yzw^{5}t^{11}+133050yzw^{4}t^{12}+83040yzw^{3}t^{13}+31710yzw^{2}t^{14}+6090yzwt^{15}+660yzt^{16}-120z^{2}w^{16}+4020z^{2}w^{15}t-38190z^{2}w^{14}t^{2}+175575z^{2}w^{13}t^{3}-463215z^{2}w^{12}t^{4}+731715z^{2}w^{11}t^{5}-659685z^{2}w^{10}t^{6}+308880z^{2}w^{9}t^{7}-80460z^{2}w^{8}t^{8}-22530z^{2}w^{7}t^{9}+28320z^{2}w^{6}t^{10}+31515z^{2}w^{5}t^{11}+21375z^{2}w^{4}t^{12}+795z^{2}w^{3}t^{13}-1905z^{2}w^{2}t^{14}-690z^{2}wt^{15}+48w^{17}t-720w^{16}t^{2}+4488w^{15}t^{3}-15856w^{14}t^{4}+34779w^{13}t^{5}-47421w^{12}t^{6}+40419w^{11}t^{7}-19037w^{10}t^{8}-5472w^{9}t^{9}+2528w^{8}t^{10}+8958w^{7}t^{11}+10878w^{6}t^{12}+1383w^{5}t^{13}-3167w^{4}t^{14}-2481w^{3}t^{15}-717w^{2}t^{16}-90wt^{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.1 :

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

Equation of the image curve:

$0$ $=$ $ 5X^{8}+10X^{6}Y^{2}+20X^{5}Y^{3}+5X^{4}Y^{4}-18X^{4}Y^{2}Z^{2}+20X^{3}Y^{5}+24X^{3}Y^{3}Z^{2}+20X^{2}Y^{6}-30X^{2}Y^{4}Z^{2}-36XY^{5}Z^{2}-12Y^{6}Z^{2}+9Y^{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.1.45 $60$ $2$ $2$ $3$ $0$ $2$

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.2.5 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ge.2.16 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ih.1.5 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ij.1.11 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.mf.2.11 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.mh.2.12 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.mi.1.1 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.ml.1.2 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nd.1.15 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ng.1.12 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.nn.1.7 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.np.1.8 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nx.1.12 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2^{2}$
60.576.13-60.nz.1.12 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2^{2}$
60.576.13-60.oa.1.3 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2^{2}$
60.576.13-60.od.1.4 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2^{2}$
60.576.17-60.iq.1.13 $60$ $2$ $2$ $17$ $3$ $1^{6}\cdot2^{3}$
60.576.17-60.it.1.14 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{3}$
60.576.17-60.iz.1.9 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.576.17-60.jb.1.10 $60$ $2$ $2$ $17$ $4$ $1^{6}\cdot2^{3}$
60.576.17-60.jg.1.15 $60$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{3}$
60.576.17-60.jj.1.16 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.576.17-60.jp.1.11 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{3}$
60.576.17-60.jr.1.20 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.864.21-60.d.1.14 $60$ $3$ $3$ $21$ $0$ $1^{8}\cdot2^{4}$
60.1440.37-60.nl.1.31 $60$ $5$ $5$ $37$ $0$ $1^{16}\cdot2^{8}$