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Properties

Label 60.288.9-60.fx.1.32
Level $60$
Index $288$
Genus $9$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $8$

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Invariants

Level: $60$ $\SL_2$-level: $60$ Newform level: $240$
Index: $288$ $\PSL_2$-index:$144$
Genus: $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (all of which are rational) Cusp widths $2\cdot4\cdot6\cdot10\cdot12\cdot20\cdot30\cdot60$ Cusp orbits $1^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $8$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 60H9
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.288.9.12

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}1&15\\48&23\end{bmatrix}$, $\begin{bmatrix}13&0\\58&7\end{bmatrix}$, $\begin{bmatrix}13&45\\46&43\end{bmatrix}$, $\begin{bmatrix}19&45\\2&29\end{bmatrix}$, $\begin{bmatrix}19&45\\48&29\end{bmatrix}$, $\begin{bmatrix}37&45\\28&59\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.9.fx.1 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: $2^{25}\cdot3^{7}\cdot5^{7}$
Simple: no
Squarefree: no
Decomposition: $1^{9}$
Newforms: 15.2.a.a$^{2}$, 30.2.a.a, 48.2.a.a$^{2}$, 80.2.a.a$^{2}$, 240.2.a.a, 240.2.a.c

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $ $=$ $ x^{2} + x y - y s $
$=$ $ - x u + y z + y t$
$=$ $x z + x t + x u - u s$
$=$ $x y - y^{2} - z u - u r$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{9} y + 2 x^{8} y^{2} + x^{8} z^{2} - 2 x^{7} y^{3} + 3 x^{7} y z^{2} - 4 x^{6} y^{4} + \cdots + y^{8} z^{2} $
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Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1/2:-1/5:-1/2:0:9/10:1:0)$, $(0:0:0:0:0:0:1/2:1:0)$, $(0:0:0:0:0:-1:1:0:0)$, $(0:0:0:-1/2:0:0:1:0:0)$, $(0:0:-1:1:1:0:0:1:0)$, $(0:0:-1:1:-2:-3:0:1:0)$, $(0:0:0:2:0:0:1:0:0)$, $(0:0:-1:-1/5:-2:3:-18/5:1:0)$

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 -3^2\,\frac{1075744328340688551786120xr^{8}s-6425837440231388668548480xr^{6}s^{3}+3634288319352665673537765xr^{4}s^{5}+22887439835320192437204243xr^{2}s^{7}+16059965005719167026230420xs^{9}-943954665808000000000yv^{8}s+17215184947910400000000yv^{6}s^{3}+1378841933741838175390000yv^{4}s^{5}+8093632562537126293427500yv^{2}s^{7}+515972811402306403578400yvr^{7}s+3249638892451168934335200yvr^{5}s^{3}+20903418845609761949122340yvr^{3}s^{5}-6988943354664455016874345yvrs^{7}+3321326742834961279334520yr^{8}s-1783115192899936023102280yr^{6}s^{3}-14867484064943718206698715yr^{4}s^{5}-12288840673378086480924889yr^{2}s^{7}+4749290520510756252663157ys^{9}+4914578687300100698642414zvs^{8}-870920782208805639979800zr^{9}-6100461217251772096536080zr^{7}s^{2}+8488210897794762016484240zr^{5}s^{4}+21055757827629370090031074zr^{3}s^{6}+19307185012179169992341032zrs^{8}+1505587138000000000000wv^{9}-959346024172800000000wv^{7}s^{2}+69688581495940000000wv^{5}s^{4}-276762657070761274358000wv^{3}s^{6}-1737967579195358240930700wvs^{8}-4559587297825282260535800wr^{9}+7768625613302833853723720wr^{7}s^{2}-21937292230322419529924240wr^{5}s^{4}-40166026158997471247576617wr^{3}s^{6}-41922801934268336204579425wrs^{8}-16263120611870518141067950tvs^{8}+2497016133404776937349000tr^{9}-8522868323484434027938840tr^{7}s^{2}+18886144537177158908718600tr^{5}s^{4}+38587698164656223935917638tr^{3}s^{6}+44494224912790286850431625trs^{8}-2602313753600000000000uv^{9}-11463282589862400000000uv^{7}s^{2}+33853237276695320000000uv^{5}s^{4}+6890923945654345588528000uv^{3}s^{6}+6004121957152596737746000uvr^{8}-8335850873423005709291600uvr^{6}s^{2}+25246291699418823266030200uvr^{4}s^{4}+4673693227030076932088850uvr^{2}s^{6}+6313354160603725563742797uvs^{8}-1598591408056120937349000ur^{9}+7564001195091087561997280ur^{7}s^{2}-16728554661732988820038360ur^{5}s^{4}-50515305489136885659736183ur^{3}s^{6}-46497127463150585714002430urs^{8}-2162139980000000000000v^{10}+317076671120000000000v^{9}r-8202681685686400000000v^{8}s^{2}+3067935996752000000000v^{7}rs^{2}-9187155212447080000000v^{6}s^{4}+178315912404889600000000v^{5}rs^{4}-142438873611132057179000v^{4}s^{6}+12333271755586798517169000v^{3}rs^{6}+4337836559171586442356000v^{2}r^{8}-7602824745571279369260000v^{2}r^{6}s^{2}+25090496228030210524164000v^{2}r^{4}s^{4}-35647405244935601505082950v^{2}r^{2}s^{6}-905869138041003184616950v^{2}s^{8}-4141071890476676087592000vr^{9}+16581620446476383848712760vr^{7}s^{2}-41884352430079138011439120vr^{5}s^{4}-41634699057617772204185256vr^{3}s^{6}-42862653842037362053430137vrs^{8}+986115808959427683207000r^{10}-5153086856351074483389960r^{8}s^{2}+12781655785710934461016280r^{6}s^{4}+18960740090080896481237973r^{4}s^{6}+8366809882105045574727966r^{2}s^{8}-5764401813838587461980072s^{10}}{6969754081394992800xr^{8}s-6001575862533777396000xr^{6}s^{3}+1121249779650737731154960xr^{4}s^{5}+47933770229764089610188588xr^{2}s^{7}+276640030019077912021422595xs^{9}-20626322942269440000yv^{4}s^{5}-67979766467743016085884600yv^{2}s^{7}-74847934699794432000yvr^{7}s+10602593387756753760000yvr^{5}s^{3}+248646568338599062442560yvr^{3}s^{5}-194730735135785762057617060yvrs^{7}+24565836796465456800yr^{8}s-13585073994894488400000yr^{6}s^{3}+2153480242398093348956080yr^{4}s^{5}+88093622279495394390630180yr^{2}s^{7}-279559790333240568413662387ys^{9}+477143681961814771832196301zvs^{8}-399219153721344000zr^{9}+448653330473273776800zr^{7}s^{2}-46294877045629970074800zr^{5}s^{4}+8852068630813570544972836zr^{3}s^{6}+574265636807383929147254649zrs^{8}+4128321111919488000wv^{3}s^{6}+13572929852851673788642520wvs^{8}+47507079292839936000wr^{9}+4272338954153012248800wr^{7}s^{2}-2624820491583704423403600wr^{5}s^{4}+152965781460439506525898572wr^{3}s^{6}+236995288821382214079459251wrs^{8}-136197701671057244925705875tvs^{8}-29941436529100800000tr^{9}-2464523683939693605600tr^{7}s^{2}+1624795674915072639488400tr^{5}s^{4}-92025721322569229545195588tr^{3}s^{6}+155774598606317385534613068trs^{8}+1854244506594362976000uv^{3}s^{6}-75186273950853120000uvr^{8}-6552817334935562304000uvr^{6}s^{2}+4111333824461783846564000uvr^{4}s^{4}-238675147452556722315162040uvr^{2}s^{6}-220989292810013341019716572uvs^{8}+29941436529100800000ur^{9}+2494176341813419147200ur^{7}s^{2}-1625993414925425128184400ur^{5}s^{4}+91244832438370525798770228ur^{3}s^{6}-56658284803955303859219636urs^{8}+2064160555959744000v^{4}s^{6}+1153026472766727979840000v^{3}rs^{6}-54559951008583680000v^{2}r^{8}-4535299379353860000000v^{2}r^{6}s^{2}+2952592144330132934440000v^{2}r^{4}s^{4}-171470043109316427144735600v^{2}r^{2}s^{6}+6786465752090059278218860v^{2}s^{8}+63209699339212800000vr^{9}+5017560476584834742400vr^{7}s^{2}-3398866412609984152908800vr^{5}s^{4}+191905240777718483665612176vr^{3}s^{6}-216021311277900483682433555vrs^{8}-17964861917460480000r^{10}-1353173652434224706400r^{8}s^{2}+954185113536442303880400r^{6}s^{4}-52406483422960378834320948r^{4}s^{6}+198913819722464279128836486r^{2}s^{8}-51617025386899838957555318s^{10}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.9.fx.1 :

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

Equation of the image curve:

$0$ $=$ $ X^{9}Y+2X^{8}Y^{2}+X^{8}Z^{2}-2X^{7}Y^{3}+3X^{7}YZ^{2}-4X^{6}Y^{4}-3X^{6}Y^{2}Z^{2}+2X^{5}Y^{5}-6X^{5}Y^{3}Z^{2}+2X^{4}Y^{6}+8X^{4}Y^{4}Z^{2}-X^{3}Y^{7}+6X^{3}Y^{5}Z^{2}+9X^{3}Y^{3}Z^{4}-3X^{2}Y^{6}Z^{2}-3XY^{7}Z^{2}-9XY^{5}Z^{4}+Y^{8}Z^{2} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
3.8.0-3.a.1.1 $3$ $36$ $36$ $0$ $0$ full Jacobian
4.6.0.e.1 $4$ $48$ $24$ $0$ $0$ full Jacobian
$X_0(5)$ $5$ $48$ $24$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.1-12.l.1.10 $12$ $6$ $6$ $1$ $0$ $1^{8}$
30.144.3-30.a.1.14 $30$ $2$ $2$ $3$ $0$ $1^{6}$
60.144.3-30.a.1.9 $60$ $2$ $2$ $3$ $0$ $1^{6}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.576.17-60.a.1.38 $60$ $2$ $2$ $17$ $1$ $1^{8}$
60.576.17-60.x.1.38 $60$ $2$ $2$ $17$ $1$ $1^{8}$
60.576.17-60.gk.1.24 $60$ $2$ $2$ $17$ $2$ $1^{8}$
60.576.17-60.gl.1.24 $60$ $2$ $2$ $17$ $2$ $1^{8}$
60.576.17-60.ic.1.20 $60$ $2$ $2$ $17$ $2$ $1^{8}$
60.576.17-60.id.1.20 $60$ $2$ $2$ $17$ $2$ $1^{8}$
60.576.17-60.ig.1.24 $60$ $2$ $2$ $17$ $7$ $1^{8}$
60.576.17-60.ih.1.24 $60$ $2$ $2$ $17$ $7$ $1^{8}$
60.576.17-60.jm.1.28 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jm.2.28 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jn.1.16 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jn.2.16 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jq.1.22 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jq.2.22 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jr.1.20 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.jr.2.20 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.ks.1.24 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.ks.2.24 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.kt.1.24 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.kt.2.24 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.kw.1.16 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.kw.2.16 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.kx.1.16 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.17-60.kx.2.16 $60$ $2$ $2$ $17$ $1$ $2^{4}$
60.576.19-60.g.1.16 $60$ $2$ $2$ $19$ $2$ $1^{10}$
60.576.19-60.h.1.16 $60$ $2$ $2$ $19$ $1$ $1^{10}$
60.576.19-60.i.1.30 $60$ $2$ $2$ $19$ $3$ $1^{10}$
60.576.19-60.j.1.31 $60$ $2$ $2$ $19$ $3$ $1^{10}$
60.576.19-60.k.1.16 $60$ $2$ $2$ $19$ $6$ $1^{10}$
60.576.19-60.l.1.16 $60$ $2$ $2$ $19$ $2$ $1^{10}$
60.576.19-60.m.1.28 $60$ $2$ $2$ $19$ $5$ $1^{10}$
60.576.19-60.n.1.30 $60$ $2$ $2$ $19$ $6$ $1^{10}$
60.576.19-60.o.1.26 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.o.2.26 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.p.1.29 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.p.2.29 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.q.1.14 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.q.2.14 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.r.1.15 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.576.19-60.r.2.15 $60$ $2$ $2$ $19$ $1$ $2^{5}$
60.864.29-60.ceo.1.1 $60$ $3$ $3$ $29$ $4$ $1^{20}$
60.1440.49-60.bxd.1.24 $60$ $5$ $5$ $49$ $8$ $1^{40}$