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Properties

Label 20.0.69444824657...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 61^{16}$
Root discriminant $310.51$
Ramified primes $2, 3, 5, 61$
Class number $648576100$ (GRH)
Class group $[10, 64857610]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216476277061, -6996693352, 113274078335, -21357492094, 25947045609, -5663152336, 3789497873, -545513248, 266289852, -23278548, 6370066, 133364, 120164, -37616, 15256, -3668, 939, 82, -15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 15*x^18 + 82*x^17 + 939*x^16 - 3668*x^15 + 15256*x^14 - 37616*x^13 + 120164*x^12 + 133364*x^11 + 6370066*x^10 - 23278548*x^9 + 266289852*x^8 - 545513248*x^7 + 3789497873*x^6 - 5663152336*x^5 + 25947045609*x^4 - 21357492094*x^3 + 113274078335*x^2 - 6996693352*x + 216476277061)
 
gp: K = bnfinit(x^20 - 4*x^19 - 15*x^18 + 82*x^17 + 939*x^16 - 3668*x^15 + 15256*x^14 - 37616*x^13 + 120164*x^12 + 133364*x^11 + 6370066*x^10 - 23278548*x^9 + 266289852*x^8 - 545513248*x^7 + 3789497873*x^6 - 5663152336*x^5 + 25947045609*x^4 - 21357492094*x^3 + 113274078335*x^2 - 6996693352*x + 216476277061, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 15 x^{18} + 82 x^{17} + 939 x^{16} - 3668 x^{15} + 15256 x^{14} - 37616 x^{13} + 120164 x^{12} + 133364 x^{11} + 6370066 x^{10} - 23278548 x^{9} + 266289852 x^{8} - 545513248 x^{7} + 3789497873 x^{6} - 5663152336 x^{5} + 25947045609 x^{4} - 21357492094 x^{3} + 113274078335 x^{2} - 6996693352 x + 216476277061 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69444824657298740769595326110530848000000000000000=2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 61^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $310.51$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3660=2^{2}\cdot 3\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{3660}(1,·)$, $\chi_{3660}(1667,·)$, $\chi_{3660}(203,·)$, $\chi_{3660}(3023,·)$, $\chi_{3660}(2449,·)$, $\chi_{3660}(2327,·)$, $\chi_{3660}(2521,·)$, $\chi_{3660}(2929,·)$, $\chi_{3660}(863,·)$, $\chi_{3660}(3169,·)$, $\chi_{3660}(3047,·)$, $\chi_{3660}(3181,·)$, $\chi_{3660}(1583,·)$, $\chi_{3660}(241,·)$, $\chi_{3660}(949,·)$, $\chi_{3660}(2807,·)$, $\chi_{3660}(1681,·)$, $\chi_{3660}(827,·)$, $\chi_{3660}(1789,·)$, $\chi_{3660}(1343,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{33} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{4}{11} a + \frac{1}{3}$, $\frac{1}{957} a^{12} + \frac{1}{319} a^{11} - \frac{5}{87} a^{10} - \frac{23}{87} a^{9} + \frac{35}{87} a^{8} - \frac{10}{29} a^{7} + \frac{14}{87} a^{6} - \frac{8}{87} a^{5} + \frac{20}{87} a^{4} + \frac{11}{29} a^{3} - \frac{386}{957} a^{2} - \frac{14}{957} a - \frac{1}{3}$, $\frac{1}{957} a^{13} - \frac{2}{319} a^{11} - \frac{8}{87} a^{10} + \frac{17}{87} a^{9} + \frac{13}{29} a^{8} - \frac{4}{29} a^{7} - \frac{7}{29} a^{6} - \frac{14}{87} a^{5} - \frac{9}{29} a^{4} + \frac{439}{957} a^{3} - \frac{4}{29} a^{2} - \frac{16}{957} a - \frac{1}{3}$, $\frac{1}{957} a^{14} - \frac{4}{319} a^{11} - \frac{13}{87} a^{10} - \frac{4}{29} a^{9} + \frac{8}{29} a^{8} + \frac{31}{87} a^{7} + \frac{4}{29} a^{6} + \frac{41}{87} a^{5} - \frac{155}{957} a^{4} + \frac{4}{29} a^{3} + \frac{20}{87} a^{2} - \frac{142}{957} a - \frac{1}{3}$, $\frac{1}{957} a^{15} + \frac{3}{319} a^{11} - \frac{14}{87} a^{10} - \frac{20}{87} a^{9} - \frac{13}{87} a^{8} + \frac{2}{29} a^{6} + \frac{65}{957} a^{5} + \frac{20}{87} a^{4} + \frac{10}{87} a^{3} - \frac{28}{87} a^{2} - \frac{284}{957} a$, $\frac{1}{957} a^{16} - \frac{7}{957} a^{11} - \frac{4}{87} a^{10} - \frac{3}{29} a^{9} + \frac{4}{87} a^{8} - \frac{14}{87} a^{7} - \frac{364}{957} a^{6} - \frac{8}{29} a^{5} + \frac{11}{29} a^{4} - \frac{35}{87} a^{3} - \frac{1}{3} a^{2} - \frac{367}{957} a - \frac{1}{3}$, $\frac{1}{305283} a^{17} + \frac{18}{101761} a^{16} + \frac{74}{305283} a^{15} + \frac{113}{305283} a^{14} - \frac{142}{305283} a^{13} - \frac{34}{305283} a^{12} + \frac{125}{9251} a^{11} - \frac{815}{9251} a^{10} + \frac{12869}{27753} a^{9} - \frac{1636}{9251} a^{8} - \frac{670}{3509} a^{7} - \frac{43570}{305283} a^{6} + \frac{77927}{305283} a^{5} - \frac{2885}{305283} a^{4} - \frac{28395}{101761} a^{3} - \frac{47926}{305283} a^{2} - \frac{208}{957} a - \frac{1}{3}$, $\frac{1}{784162480751767852512179457} a^{18} + \frac{690674332456821787796}{784162480751767852512179457} a^{17} - \frac{2382224820518614375142}{6480681659105519442249417} a^{16} + \frac{35938947719575022587810}{261387493583922617504059819} a^{15} - \frac{64563285764995830968779}{784162480751767852512179457} a^{14} + \frac{106003403055173331882896}{784162480751767852512179457} a^{13} + \frac{34485564339059305441542}{261387493583922617504059819} a^{12} + \frac{1041084088535715658036571}{71287498250160713864743587} a^{11} - \frac{3279984816938953150725899}{71287498250160713864743587} a^{10} + \frac{6172697511132035616780502}{23762499416720237954914529} a^{9} + \frac{250864265750851129688646532}{784162480751767852512179457} a^{8} + \frac{280592826932305541262926684}{784162480751767852512179457} a^{7} - \frac{1797455936884334466935798}{23762499416720237954914529} a^{6} - \frac{171817805977896883963192640}{784162480751767852512179457} a^{5} - \frac{81846552697320057309329131}{784162480751767852512179457} a^{4} - \frac{114687539155168053947513417}{784162480751767852512179457} a^{3} - \frac{159722856262609726679971799}{784162480751767852512179457} a^{2} - \frac{5703396277640485033716}{28255052814173885796569} a - \frac{1189148418216363510845}{2568641164924898708779}$, $\frac{1}{56022758438497633359847328649923376319846445011795713} a^{19} + \frac{33207966494024778863035465}{56022758438497633359847328649923376319846445011795713} a^{18} + \frac{41571021814382333854823922882421414557313643240}{56022758438497633359847328649923376319846445011795713} a^{17} + \frac{8277435414031952417086823786988280651225849653961}{18674252812832544453282442883307792106615481670598571} a^{16} - \frac{5749798628681938544118889257021160473883039703936}{56022758438497633359847328649923376319846445011795713} a^{15} - \frac{2870068625908873649297701271286161321121713028459}{18674252812832544453282442883307792106615481670598571} a^{14} - \frac{16972728190647038236906799436783183838735201682750}{56022758438497633359847328649923376319846445011795713} a^{13} - \frac{17033730981345092634722565625524625278417808874362}{56022758438497633359847328649923376319846445011795713} a^{12} - \frac{10286510006555078336012503555500615340108013639611}{5092978039863421214531575331811216029076949546526883} a^{11} + \frac{124635869332273825053726210367170560510696863939001}{5092978039863421214531575331811216029076949546526883} a^{10} - \frac{1285651987126731425748961981526097753717949431566975}{18674252812832544453282442883307792106615481670598571} a^{9} + \frac{26457500939361093961469155065288412590493122480047147}{56022758438497633359847328649923376319846445011795713} a^{8} + \frac{19198945226346916723426167674427683703602209557892678}{56022758438497633359847328649923376319846445011795713} a^{7} - \frac{26614955944101294345398432519003020681954899370582630}{56022758438497633359847328649923376319846445011795713} a^{6} - \frac{12932010107623830608000546294195153032098015569748428}{56022758438497633359847328649923376319846445011795713} a^{5} + \frac{4478254958232747113379519581674283541422885736337628}{18674252812832544453282442883307792106615481670598571} a^{4} - \frac{4947827983283953270306586598079220254657981193747903}{56022758438497633359847328649923376319846445011795713} a^{3} - \frac{27165149085810289378293154564443530023344430748230601}{56022758438497633359847328649923376319846445011795713} a^{2} + \frac{80128966003755143047978657829427178995116427620092}{175619932409083490156261218338317794106101708500927} a + \frac{91413933261723398819268204603234962022248109173}{550532703476750752840944258113848884345146421633}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{10}\times C_{64857610}$, which has order $648576100$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 366097410.81371444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.18000.1, 5.5.13845841.1, 10.10.599085353116503125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R $20$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
5Data not computed
$61$61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$