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Properties

Label 60.288.5-60.oc.2.7
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: $240$
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: 60.288.5.3052

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&20\\0&47\end{bmatrix}$, $\begin{bmatrix}37&25\\12&29\end{bmatrix}$, $\begin{bmatrix}41&10\\24&1\end{bmatrix}$, $\begin{bmatrix}41&40\\6&29\end{bmatrix}$, $\begin{bmatrix}59&5\\18&1\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.5.oc.2 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: $2^{9}\cdot3^{5}\cdot5^{5}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 15.2.a.a$^{2}$, 30.2.a.a, 240.2.f.a

Models

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

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

$ 0 $ $=$ $ x^{8} + 2 x^{6} y^{2} + 4 x^{5} y^{3} + x^{4} y^{4} + 6 x^{4} y^{2} z^{2} + 4 x^{3} y^{5} + \cdots + 5 y^{4} z^{4} $
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Rational points

This modular curve has no real points, 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{2}{3^2}\cdot\frac{19804183776668672xzw^{15}t-42102396565410816xzw^{13}t^{3}-110348293639802880xzw^{11}t^{5}-45319284326466304xzw^{9}t^{7}+41476519113428160xzw^{7}t^{9}+20774088197111040xzw^{5}t^{11}-1627304505374576xzw^{3}t^{13}-39946668333264xzwt^{15}-18342876511557120xw^{17}-33895350433680384xw^{15}t^{2}+71500648827383808xw^{13}t^{4}+169067685415497984xw^{11}t^{6}+70846234409523168xw^{9}t^{8}-46625897707879104xw^{7}t^{10}-23637089442700128xw^{5}t^{12}+1625081211919248xw^{3}t^{14}+29058516826566xwt^{16}+15852663452922880yzw^{15}t+22971058769636352yzw^{13}t^{3}-20317236237960192yzw^{11}t^{5}-30109631098057472yzw^{9}t^{7}-4997899035218112yzw^{7}t^{9}+3134181940728768yzw^{5}t^{11}+11672205522512yzw^{3}t^{13}+29151928211136yzwt^{15}+24007992757263872yw^{17}-28319753579013120yw^{15}t^{2}-91557307320098304yw^{13}t^{4}-27161117987388160yw^{11}t^{6}+52471266057837216yw^{9}t^{8}+17180731304706432yw^{7}t^{10}-4055719476844592yw^{5}t^{12}+7339296806016yw^{3}t^{14}-21055614832014ywt^{16}-39094192895394816z^{3}w^{14}t-70757460177865728z^{3}w^{12}t^{3}+31688770588114176z^{3}w^{10}t^{5}+62716360815427968z^{3}w^{8}t^{7}+11610080396853216z^{3}w^{6}t^{9}-7023535360111728z^{3}w^{4}t^{11}+98237435487552z^{3}w^{2}t^{13}-3058039057440z^{3}t^{15}-70645795278953216z^{2}w^{16}+96842424334181376z^{2}w^{14}t^{2}+313222263588604800z^{2}w^{12}t^{4}+86390972145827584z^{2}w^{10}t^{6}-163789465627028880z^{2}w^{8}t^{8}-53557546660149600z^{2}w^{6}t^{10}+14651774380690556z^{2}w^{4}t^{12}-43602884637720z^{2}w^{2}t^{14}+1990006865109z^{2}t^{16}+52556365944934912zw^{16}t-103342737718496256zw^{14}t^{3}-261000866160375552zw^{12}t^{5}-87514875892143488zw^{10}t^{7}+122851105792916928zw^{8}t^{9}+45353601011944752zw^{6}t^{11}-8944282125817504zw^{4}t^{13}-135924323167200zw^{2}t^{15}+3045038947698zt^{17}-34981932989305088w^{18}-6676364529587200w^{16}t^{2}+139813715528695680w^{14}t^{4}+171726352837800064w^{12}t^{6}+16711293665214416w^{10}t^{8}-58372164531452928w^{8}t^{10}-17272577960040004w^{6}t^{12}+2149742050470940w^{4}t^{14}+98299210599387w^{2}t^{16}-2008387814976t^{18}}{w(2672291047936xzw^{14}t+66138552012912xzw^{12}t^{3}-242080176594920xzw^{10}t^{5}+206388960393732xzw^{8}t^{7}-55532587894164xzw^{6}t^{9}+4694024665852xzw^{4}t^{11}-100490182850xzw^{2}t^{13}+239148450xzt^{15}+3479289197952xw^{16}-51726682793328xw^{14}t^{2}+407651836764xw^{12}t^{4}+258509497667004xw^{10}t^{6}-237065333116824xw^{8}t^{8}+61996513453518xw^{6}t^{10}-5027790081570xw^{4}t^{12}+103646937270xw^{2}t^{14}-239148450xt^{16}-11446132040128yzw^{14}t+70462475742096yzw^{12}t^{3}-101032979729224yzw^{10}t^{5}+46871210612028yzw^{8}t^{7}-7340959959888yzw^{6}t^{9}+342805033208yzw^{4}t^{11}-3156759220yzw^{2}t^{13}-4551866600864yw^{16}+91461176494320yw^{14}t^{2}-269124325838732yw^{12}t^{4}+245483987110260yw^{10}t^{6}-82568109510540yw^{8}t^{8}+10224099978562yw^{6}t^{10}-403299897824yw^{4}t^{12}+3300249570yw^{2}t^{14}+32217785243472z^{3}w^{13}t-181232009020104z^{3}w^{11}t^{3}+236309911651296z^{3}w^{9}t^{5}-99186580289448z^{3}w^{7}t^{7}+14015690648400z^{3}w^{5}t^{9}-591390245781z^{3}w^{3}t^{11}+4950371955z^{3}wt^{13}+13392711074216z^{2}w^{15}-286720013398272z^{2}w^{13}t^{2}+870897750891170z^{2}w^{11}t^{4}-798517403912364z^{2}w^{9}t^{6}+262755416868894z^{2}w^{7}t^{8}-31007394804508z^{2}w^{5}t^{10}+1141407724400z^{2}w^{3}t^{12}-8609344200z^{2}wt^{14}+2324920806512zw^{15}t+182180285093352zw^{13}t^{3}-643352034517540zw^{11}t^{5}+582339584501652zw^{9}t^{7}-177693245277396zw^{7}t^{9}+18678978077651zw^{5}t^{11}-594020787979zw^{3}t^{13}+3778546470zwt^{15}+6631274342696w^{17}-119075930945768w^{15}t^{2}+217451243883950w^{13}t^{4}+37150983604312w^{11}t^{6}-152849855257194w^{9}t^{8}+50188244918930w^{7}t^{10}-4504829893848w^{5}t^{12}+98911797640w^{3}t^{14}-239148450wt^{16})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.oc.2 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

Equation of the image curve:

$0$ $=$ $ X^{8}+2X^{6}Y^{2}+4X^{5}Y^{3}+X^{4}Y^{4}+6X^{4}Y^{2}Z^{2}+4X^{3}Y^{5}-8X^{3}Y^{3}Z^{2}+4X^{2}Y^{6}+10X^{2}Y^{4}Z^{2}+12XY^{5}Z^{2}+4Y^{6}Z^{2}+5Y^{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.1.42 $60$ $2$ $2$ $3$ $0$ $2$

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.cg.1.9 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.gf.1.19 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.id.2.9 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.if.2.19 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.ju.1.1 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.jw.1.16 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.jy.2.1 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.ka.2.3 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nc.2.12 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nh.2.14 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.nj.2.10 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2^{2}$
60.576.13-60.nl.2.12 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2^{2}$
60.576.13-60.nq.2.2 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2^{2}$
60.576.13-60.ns.2.16 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2^{2}$
60.576.13-60.nu.2.2 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2^{2}$
60.576.13-60.nw.2.4 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2^{2}$
60.576.17-60.in.2.13 $60$ $2$ $2$ $17$ $3$ $1^{6}\cdot2^{3}$
60.576.17-60.ip.2.15 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{3}$
60.576.17-60.iv.2.9 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.576.17-60.ix.2.11 $60$ $2$ $2$ $17$ $4$ $1^{6}\cdot2^{3}$
60.576.17-60.jd.2.14 $60$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{3}$
60.576.17-60.jf.2.16 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.576.17-60.jl.2.6 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{3}$
60.576.17-60.jn.2.16 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{3}$
60.864.21-60.a.1.13 $60$ $3$ $3$ $21$ $0$ $1^{8}\cdot2^{4}$
60.1440.37-60.nk.1.27 $60$ $5$ $5$ $37$ $0$ $1^{16}\cdot2^{8}$