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Properties

Label 60.144.5.kv.2
Level $60$
Index $144$
Genus $5$
Analytic rank $2$
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 $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-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: 60.144.5.725

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}1&25\\24&37\end{bmatrix}$, $\begin{bmatrix}19&55\\22&21\end{bmatrix}$, $\begin{bmatrix}31&0\\0&13\end{bmatrix}$, $\begin{bmatrix}31&50\\30&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.288.5-60.kv.2.1, 60.288.5-60.kv.2.2, 60.288.5-60.kv.2.3, 60.288.5-60.kv.2.4, 60.288.5-60.kv.2.5, 60.288.5-60.kv.2.6, 60.288.5-60.kv.2.7, 60.288.5-60.kv.2.8, 120.288.5-60.kv.2.1, 120.288.5-60.kv.2.2, 120.288.5-60.kv.2.3, 120.288.5-60.kv.2.4, 120.288.5-60.kv.2.5, 120.288.5-60.kv.2.6, 120.288.5-60.kv.2.7, 120.288.5-60.kv.2.8
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $128$
Full 60-torsion field degree: $15360$

Jacobian

Conductor: $2^{18}\cdot3^{4}\cdot5^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 80.2.c.a, 180.2.a.a, 400.2.a.e, 3600.2.a.be

Models

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

$ 0 $ $=$ $ x^{2} + x z + 5 y^{2} - z^{2} $
$=$ $3 x y - 3 x z + 3 z^{2} - w^{2} + t^{2}$
$=$ $3 x^{2} - 9 x y - 3 x z + 3 z^{2} - 2 w^{2} + t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 50625 x^{8} + 27000 x^{7} y - 1575 x^{6} y^{2} + 135000 x^{6} z^{2} - 30 x^{5} y^{3} + 10800 x^{5} y z^{2} + \cdots + 66 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:

$\displaystyle X$ $=$ $\displaystyle x+y$
$\displaystyle Y$ $=$ $\displaystyle 15z+15w$
$\displaystyle Z$ $=$ $\displaystyle t$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{5}\cdot\frac{18309375000xzw^{16}-65913750000xzw^{14}t^{2}+86386500000xzw^{12}t^{4}-52077600000xzw^{10}t^{6}+14558400000xzw^{8}t^{8}-1240704000xzw^{6}t^{10}-496281600xzw^{4}t^{12}+269291520xzw^{2}t^{14}-47996928xzt^{16}-18309375000z^{2}w^{16}+65913750000z^{2}w^{14}t^{2}-86386500000z^{2}w^{12}t^{4}+52077600000z^{2}w^{10}t^{6}-14558400000z^{2}w^{8}t^{8}+1240704000z^{2}w^{6}t^{10}+496281600z^{2}w^{4}t^{12}-269291520z^{2}w^{2}t^{14}+47996928z^{2}t^{16}+7323828125w^{18}-32713125000w^{16}t^{2}+57163500000w^{14}t^{4}-50200250000w^{12}t^{6}+23409600000w^{10}t^{8}-5426688000w^{8}t^{10}+285203200w^{6}t^{12}+222382080w^{4}t^{14}-90562560w^{2}t^{16}+13438976t^{18}}{t^{4}w^{2}(9375xzw^{10}-18750xzw^{8}t^{2}+3750xzw^{6}t^{4}+1500xzw^{4}t^{6}+600xzw^{2}t^{8}+192xzt^{10}-9375z^{2}w^{10}+18750z^{2}w^{8}t^{2}-3750z^{2}w^{6}t^{4}-1500z^{2}w^{4}t^{6}-600z^{2}w^{2}t^{8}-192z^{2}t^{10}+3125w^{12}-8750w^{10}t^{2}+6375w^{8}t^{4}-600w^{6}t^{6}-245w^{4}t^{8}-120w^{2}t^{10}-64t^{12})}$

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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
20.72.3.bi.1 $20$ $2$ $2$ $3$ $1$ $1^{2}$
60.72.1.ba.1 $60$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
60.72.1.cf.2 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.72.1.dq.2 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.72.3.ne.2 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.72.3.ou.1 $60$ $2$ $2$ $3$ $2$ $2$
60.72.3.qx.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$

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.bza.1 $60$ $3$ $3$ $29$ $6$ $1^{12}\cdot2^{6}$
60.576.33.kj.1 $60$ $4$ $4$ $33$ $8$ $1^{14}\cdot2^{7}$
60.720.37.hw.1 $60$ $5$ $5$ $37$ $7$ $1^{16}\cdot2^{8}$