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Properties

Label 60.144.5.op.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 $2^{2}\cdot4^{3}$
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: 60.144.5.858

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}33&5\\32&1\end{bmatrix}$, $\begin{bmatrix}39&40\\34&57\end{bmatrix}$, $\begin{bmatrix}43&50\\58&43\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $15360$

Jacobian

Conductor: $2^{19}\cdot3^{4}\cdot5^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 80.2.c.a, 360.2.a.a, 400.2.a.c, 3600.2.a.be

Models

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

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

$ 0 $ $=$ $ 49 x^{8} - 196 x^{7} z - 170 x^{6} y^{2} + 280 x^{6} z^{2} + 680 x^{5} y^{2} z - 154 x^{5} z^{3} + \cdots + 361 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:

$\displaystyle X$ $=$ $\displaystyle x-3y$
$\displaystyle Y$ $=$ $\displaystyle 3z+3w$
$\displaystyle Z$ $=$ $\displaystyle t$

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}{3^2}\cdot\frac{936148743481917209765625xw^{16}t+3260364029141252249531250xw^{14}t^{3}-3084323406126252307875000xw^{12}t^{5}+40673974793996892851100000xw^{10}t^{7}-20880874275647036157000000xw^{8}t^{9}-26951438982916868944680000xw^{6}t^{11}+17358535499988547236902400xw^{4}t^{13}-2253402742246920815339520xw^{2}t^{15}-67580278634678440527360xt^{17}+807184016043582827812500yw^{14}t^{3}+1148329478057093382375000yw^{12}t^{5}+5373627484457506212900000yw^{10}t^{7}+9079320318593047356600000yw^{8}t^{9}-11049937746418569220560000yw^{6}t^{11}+1389124434660617533881600yw^{4}t^{13}+751348666710297258670080yw^{2}t^{15}-129628670891871208273920yt^{17}+3496031023949570605078125z^{2}w^{16}+9823423879308439676250000z^{2}w^{14}t^{2}+10390893697283024023312500z^{2}w^{12}t^{4}+151922388504017267405100000z^{2}w^{10}t^{6}-139089018762076687322850000z^{2}w^{8}t^{8}-63912330657260389721664000z^{2}w^{6}t^{10}+63146262492184191895785600z^{2}w^{4}t^{12}-10465725323404298023464960z^{2}w^{2}t^{14}+5882143227557031154944z^{2}t^{16}-1781590047015045617187500w^{18}-6038222046581578942968750w^{16}t^{2}-8314664292108976111171875w^{14}t^{4}-67030701291866272683937500w^{12}t^{6}+5992506717334095865725000w^{10}t^{8}+66964059589357163174760000w^{8}t^{10}-1623220789540447565676000w^{6}t^{12}-15722348992012012209724800w^{4}t^{14}+2865543434983169874097920w^{2}t^{16}+56968499408424358832640t^{18}}{4094340734250000000xw^{16}t-22757221866795000000xw^{14}t^{3}-310941615769921500000xw^{12}t^{5}-557651111863973400000xw^{10}t^{7}+181106041735686600000xw^{8}t^{9}+1006526522993656070250xw^{6}t^{11}+700598604591706585050xw^{4}t^{13}+152494459989932858760xw^{2}t^{15}+4345439727024076680xt^{17}-5624777063880000000yw^{14}t^{3}-69454151749632000000yw^{12}t^{5}-220017214942621200000yw^{10}t^{7}-226907293397148000000yw^{8}t^{9}+7608491734885835625yw^{6}t^{11}+132365015578766218950yw^{4}t^{13}+64188306752752596960yw^{2}t^{15}+8335176883479372960yt^{17}-8809891786800000000z^{2}w^{16}-316300528295505000000z^{2}w^{14}t^{2}-1564680164694972000000z^{2}w^{12}t^{4}-1678304821500637200000z^{2}w^{10}t^{6}+2023598520647613000000z^{2}w^{8}t^{8}+4598891081251984076625z^{2}w^{6}t^{10}+2634785645156394090075z^{2}w^{4}t^{12}+470205432365528274480z^{2}w^{2}t^{14}-378224230166990172z^{2}t^{16}+3670788244500000000w^{18}+134728517385393750000w^{16}t^{2}+759454279115740000000w^{14}t^{4}+1252190595734266500000w^{12}t^{6}-155758012605532350000w^{10}t^{8}-2390547589039224517500w^{8}t^{10}-2510131775040457410375w^{6}t^{12}-1068407300791290935475w^{4}t^{14}-173744836084379183085w^{2}t^{16}-3663097955788603320t^{18}}$

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
20.72.3.w.2 $20$ $2$ $2$ $3$ $0$ $1^{2}$
60.72.1.bz.2 $60$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
60.72.1.co.1 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.72.1.dt.2 $60$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
60.72.3.qz.2 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.72.3.rn.1 $60$ $2$ $2$ $3$ $2$ $2$
60.72.3.yw.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.dvl.2 $60$ $3$ $3$ $29$ $4$ $1^{12}\cdot2^{6}$
60.576.33.nc.1 $60$ $4$ $4$ $33$ $7$ $1^{14}\cdot2^{7}$
60.720.37.od.1 $60$ $5$ $5$ $37$ $7$ $1^{16}\cdot2^{8}$