Properties

Label 60.288.5-60.cq.1.7
Level $60$
Index $288$
Genus $5$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $3600$
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 $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20I5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.288.5.964

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}7&30\\18&31\end{bmatrix}$, $\begin{bmatrix}17&5\\58&29\end{bmatrix}$, $\begin{bmatrix}27&40\\4&31\end{bmatrix}$, $\begin{bmatrix}33&40\\40&41\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.5.cq.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $7680$

Jacobian

Conductor: $2^{17}\cdot3^{8}\cdot5^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 80.2.a.b, 360.2.f.c, 1800.2.a.v, 3600.2.a.be

Models

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

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

$ 0 $ $=$ $ 900 x^{4} y^{4} - 3600 x^{3} y^{5} + 3000 x^{2} y^{6} - 1080 x^{2} y^{4} z^{2} - 480 x^{2} y^{2} z^{4} + \cdots + 49 z^{8} $
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Rational points

This modular curve has no $\Q_p$ points for $p=19$, and therefore no 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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{126976yw^{17}-1232896yw^{15}t^{2}+4208640yw^{13}t^{4}-5876224yw^{11}t^{6}+2836480yw^{9}t^{8}-598272yw^{7}t^{10}+57392yw^{5}t^{12}-2240yw^{3}t^{14}+24ywt^{16}+4096w^{18}-28672w^{16}t^{2}+307200w^{14}t^{4}-1032960w^{12}t^{6}+1405696w^{10}t^{8}-625152w^{8}t^{10}+115504w^{6}t^{12}-8960w^{4}t^{14}+240w^{2}t^{16}-t^{18}}{t^{2}w^{10}(1600yw^{5}-360yw^{3}t^{2}+14ywt^{4}-400w^{4}t^{2}+65w^{2}t^{4}-t^{6})}$

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

$\displaystyle X$ $=$ $\displaystyle x+z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}t$

Equation of the image curve:

$0$ $=$ $ 900X^{4}Y^{4}-3600X^{3}Y^{5}+3000X^{2}Y^{6}-1080X^{2}Y^{4}Z^{2}-480X^{2}Y^{2}Z^{4}+1200XY^{7}+2160XY^{5}Z^{2}+960XY^{3}Z^{4}-275Y^{8}-240Y^{6}Z^{2}+154Y^{4}Z^{4}+168Y^{2}Z^{6}+49Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
20.144.1-20.b.1.3 $20$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
60.144.1-20.b.1.12 $60$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
60.144.1-60.ch.2.8 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.144.1-60.ch.2.13 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.144.3-60.xu.2.4 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.144.3-60.xu.2.15 $60$ $2$ $2$ $3$ $0$ $1^{2}$

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.gi.2.2 $60$ $2$ $2$ $13$ $1$ $2^{2}\cdot4$
60.576.13-60.gj.2.2 $60$ $2$ $2$ $13$ $1$ $2^{2}\cdot4$
60.576.13-60.gk.1.2 $60$ $2$ $2$ $13$ $1$ $2^{2}\cdot4$
60.576.13-60.gl.1.2 $60$ $2$ $2$ $13$ $1$ $2^{2}\cdot4$
60.864.29-60.ku.2.17 $60$ $3$ $3$ $29$ $3$ $1^{12}\cdot2^{6}$
60.1152.33-60.cy.2.11 $60$ $4$ $4$ $33$ $7$ $1^{14}\cdot2^{7}$
60.1440.37-60.co.1.2 $60$ $5$ $5$ $37$ $8$ $1^{16}\cdot2^{8}$