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Properties

Label 60.144.5.pg.1
Level $60$
Index $144$
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: $144$ $\PSL_2$-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 $2^{6}\cdot4$
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: 60.144.5.498

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}17&0\\38&49\end{bmatrix}$, $\begin{bmatrix}17&25\\38&51\end{bmatrix}$, $\begin{bmatrix}27&10\\26&49\end{bmatrix}$, $\begin{bmatrix}53&5\\50&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.288.5-60.pg.1.1, 60.288.5-60.pg.1.2, 60.288.5-60.pg.1.3, 60.288.5-60.pg.1.4, 60.288.5-60.pg.1.5, 60.288.5-60.pg.1.6, 60.288.5-60.pg.1.7, 60.288.5-60.pg.1.8, 120.288.5-60.pg.1.1, 120.288.5-60.pg.1.2, 120.288.5-60.pg.1.3, 120.288.5-60.pg.1.4, 120.288.5-60.pg.1.5, 120.288.5-60.pg.1.6, 120.288.5-60.pg.1.7, 120.288.5-60.pg.1.8
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $15360$

Jacobian

Conductor: $2^{18}\cdot3^{8}\cdot5^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 100.2.a.a, 720.2.a.e, 720.2.f.e, 3600.2.a.be

Models

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

$ 0 $ $=$ $ x y - y^{2} + z^{2} $
$=$ $2 x y + 5 x z - 2 y^{2} - 3 z^{2} + t^{2}$
$=$ $5 x^{2} - 3 x y + 10 x z + 3 y^{2} + 7 z^{2} - 3 w^{2} - 2 t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 7290 x^{8} + 2430 x^{7} y - 19440 x^{7} z - 189 x^{6} y^{2} - 9720 x^{6} y z - 50220 x^{6} z^{2} + \cdots + 15665 z^{8} $
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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 canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x-\frac{4}{5}t$
$\displaystyle Y$ $=$ $\displaystyle 9y+9w$
$\displaystyle Z$ $=$ $\displaystyle 3z+\frac{3}{5}t$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3}\cdot\frac{512545320z^{2}w^{16}+3075271920z^{2}w^{14}t^{2}+6717414240z^{2}w^{12}t^{4}+6749256960z^{2}w^{10}t^{6}+3144614400z^{2}w^{8}t^{8}+446653440z^{2}w^{6}t^{10}-297768960z^{2}w^{4}t^{12}-269291520z^{2}w^{2}t^{14}-79994880z^{2}t^{16}-61509375w^{18}-492075000w^{16}t^{2}-1510394688w^{14}t^{4}-2287228752w^{12}t^{6}-1815478272w^{10}t^{8}-732810240w^{8}t^{10}-129358080w^{6}t^{12}+497664w^{4}t^{14}+8257536w^{2}t^{16}+3198976t^{18}}{t^{4}w^{2}(1215z^{2}w^{10}+4050z^{2}w^{8}t^{2}+1350z^{2}w^{6}t^{4}-900z^{2}w^{4}t^{6}+600z^{2}w^{2}t^{8}-320z^{2}t^{10}-81w^{8}t^{4}-108w^{6}t^{6}+81w^{4}t^{8}-72w^{2}t^{10}+64t^{12})}$

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
20.72.1.m.2 $20$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
60.72.1.cf.1 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.72.1.dv.1 $60$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
60.72.3.qt.1 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.72.3.rf.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.72.3.rt.1 $60$ $2$ $2$ $3$ $1$ $2$
60.72.3.yu.2 $60$ $2$ $2$ $3$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.432.29.dwo.2 $60$ $3$ $3$ $29$ $5$ $1^{12}\cdot2^{6}$
60.576.33.nt.2 $60$ $4$ $4$ $33$ $5$ $1^{14}\cdot2^{7}$
60.720.37.ny.1 $60$ $5$ $5$ $37$ $6$ $1^{16}\cdot2^{8}$