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Properties

Label 20.0.46897939275...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{14}\cdot 17^{15}$
Root discriminant $68.16$
Ramified primes $2, 5, 17$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2049074, 7547296, 780298, -8034072, 9613058, -7916862, 7917568, -5325116, 3045364, -1791172, 895115, -363102, 160191, -57078, 17758, -5768, 1352, -296, 61, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 61*x^18 - 296*x^17 + 1352*x^16 - 5768*x^15 + 17758*x^14 - 57078*x^13 + 160191*x^12 - 363102*x^11 + 895115*x^10 - 1791172*x^9 + 3045364*x^8 - 5325116*x^7 + 7917568*x^6 - 7916862*x^5 + 9613058*x^4 - 8034072*x^3 + 780298*x^2 + 7547296*x + 2049074)
 
gp: K = bnfinit(x^20 - 6*x^19 + 61*x^18 - 296*x^17 + 1352*x^16 - 5768*x^15 + 17758*x^14 - 57078*x^13 + 160191*x^12 - 363102*x^11 + 895115*x^10 - 1791172*x^9 + 3045364*x^8 - 5325116*x^7 + 7917568*x^6 - 7916862*x^5 + 9613058*x^4 - 8034072*x^3 + 780298*x^2 + 7547296*x + 2049074, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 61 x^{18} - 296 x^{17} + 1352 x^{16} - 5768 x^{15} + 17758 x^{14} - 57078 x^{13} + 160191 x^{12} - 363102 x^{11} + 895115 x^{10} - 1791172 x^{9} + 3045364 x^{8} - 5325116 x^{7} + 7917568 x^{6} - 7916862 x^{5} + 9613058 x^{4} - 8034072 x^{3} + 780298 x^{2} + 7547296 x + 2049074 \)

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:  \(4689793927593682195251200000000000000=2^{28}\cdot 5^{14}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.16$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11}$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{24} a^{16} - \frac{1}{12} a^{15} - \frac{1}{24} a^{14} - \frac{1}{8} a^{12} - \frac{5}{12} a^{11} - \frac{7}{24} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a + \frac{5}{12}$, $\frac{1}{48} a^{17} - \frac{1}{48} a^{16} - \frac{1}{16} a^{15} - \frac{1}{48} a^{14} - \frac{1}{16} a^{13} + \frac{11}{48} a^{12} + \frac{7}{48} a^{11} + \frac{19}{48} a^{10} - \frac{1}{24} a^{9} + \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{5}{12} a^{6} + \frac{1}{8} a^{5} - \frac{1}{12} a^{4} - \frac{1}{24} a^{3} - \frac{1}{24} a^{2} + \frac{3}{8} a - \frac{7}{24}$, $\frac{1}{10896} a^{18} + \frac{113}{10896} a^{17} + \frac{47}{10896} a^{16} + \frac{337}{10896} a^{15} - \frac{785}{10896} a^{14} - \frac{427}{10896} a^{13} - \frac{2375}{10896} a^{12} + \frac{4409}{10896} a^{11} - \frac{611}{1362} a^{10} + \frac{46}{681} a^{9} - \frac{1099}{5448} a^{8} - \frac{379}{1362} a^{7} + \frac{1211}{5448} a^{6} + \frac{302}{681} a^{5} - \frac{829}{5448} a^{4} - \frac{1667}{5448} a^{3} - \frac{2365}{5448} a^{2} + \frac{1219}{5448} a + \frac{251}{2724}$, $\frac{1}{8483004743563911681840078619106235610143661254799073143616544} a^{19} + \frac{84637373450907578775984802758459578862868995479355955411}{8483004743563911681840078619106235610143661254799073143616544} a^{18} - \frac{1519886597507641725832639436342333612651701184349338919475}{1413834123927318613640013103184372601690610209133178857269424} a^{17} - \frac{14384952144400446135960425287009164990806502479072430208739}{4241502371781955840920039309553117805071830627399536571808272} a^{16} - \frac{79266697520968375142599596815935681727157449806329579033393}{1413834123927318613640013103184372601690610209133178857269424} a^{15} + \frac{326019860596977758958237068879036085344953386582036810301955}{4241502371781955840920039309553117805071830627399536571808272} a^{14} + \frac{502384962968170887913377090638151977534506214829541382680769}{2120751185890977920460019654776558902535915313699768285904136} a^{13} - \frac{119510639589961498376655505640227853773718196498856540317151}{4241502371781955840920039309553117805071830627399536571808272} a^{12} - \frac{429567131496140469657613762150681275971618686491800123314353}{2827668247854637227280026206368745203381220418266357714538848} a^{11} - \frac{3970102963521050169429958934947503687399254846473652776992745}{8483004743563911681840078619106235610143661254799073143616544} a^{10} + \frac{85528330039434900568326676360183688222622863936669465126771}{176729265490914826705001637898046575211326276141647357158678} a^{9} + \frac{126320198240446162517487455373962534784708347672442646837499}{353458530981829653410003275796093150422652552283294714317356} a^{8} - \frac{86001393366112632638739979559938661819247611965256367247301}{176729265490914826705001637898046575211326276141647357158678} a^{7} - \frac{68504781016124425870950250253957267811201154392353710650307}{265093898236372240057502456847069862816989414212471035738017} a^{6} + \frac{566310277638842560774521467376067878787075544986260211631261}{2120751185890977920460019654776558902535915313699768285904136} a^{5} + \frac{1683962604738446953253835434279385596895548688537049942483025}{4241502371781955840920039309553117805071830627399536571808272} a^{4} + \frac{219459469433630480313931127056305340815473345508808163998553}{530187796472744480115004913694139725633978828424942071476034} a^{3} + \frac{110747656329284276260379223667579031073837489718487024653583}{1060375592945488960230009827388279451267957656849884142952068} a^{2} + \frac{414229466108696846588525716768366773032309807782176592880019}{1413834123927318613640013103184372601690610209133178857269424} a + \frac{1935973832681762296318880564268947488527950175923957186278529}{4241502371781955840920039309553117805071830627399536571808272}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 2581061275.0473027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.1965200.1, 5.1.578000.2, 10.2.5679428000000.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
5Data not computed
17Data not computed