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Properties

Label 20.12.6125762557...0496.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{48}\cdot 31^{10}\cdot 227^{4}$
Root discriminant $86.97$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3747704, 7202352, -53866648, -102874864, 60019076, 41666720, -44511864, 13116408, 2157922, -4604316, 3158862, -1418768, 472819, -114296, 15368, -208, -623, 300, -44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 44*x^18 + 300*x^17 - 623*x^16 - 208*x^15 + 15368*x^14 - 114296*x^13 + 472819*x^12 - 1418768*x^11 + 3158862*x^10 - 4604316*x^9 + 2157922*x^8 + 13116408*x^7 - 44511864*x^6 + 41666720*x^5 + 60019076*x^4 - 102874864*x^3 - 53866648*x^2 + 7202352*x + 3747704)
 
gp: K = bnfinit(x^20 - 4*x^19 - 44*x^18 + 300*x^17 - 623*x^16 - 208*x^15 + 15368*x^14 - 114296*x^13 + 472819*x^12 - 1418768*x^11 + 3158862*x^10 - 4604316*x^9 + 2157922*x^8 + 13116408*x^7 - 44511864*x^6 + 41666720*x^5 + 60019076*x^4 - 102874864*x^3 - 53866648*x^2 + 7202352*x + 3747704, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 44 x^{18} + 300 x^{17} - 623 x^{16} - 208 x^{15} + 15368 x^{14} - 114296 x^{13} + 472819 x^{12} - 1418768 x^{11} + 3158862 x^{10} - 4604316 x^{9} + 2157922 x^{8} + 13116408 x^{7} - 44511864 x^{6} + 41666720 x^{5} + 60019076 x^{4} - 102874864 x^{3} - 53866648 x^{2} + 7202352 x + 3747704 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(612576255759573370734751264818904170496=2^{48}\cdot 31^{10}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{16} + \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{16} + \frac{1}{24} a^{15} - \frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{12} + \frac{1}{12} a^{10} + \frac{1}{6} a^{8} + \frac{5}{12} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{72} a^{18} - \frac{1}{24} a^{16} + \frac{1}{72} a^{14} - \frac{1}{12} a^{13} + \frac{1}{18} a^{12} + \frac{1}{9} a^{11} + \frac{1}{36} a^{10} + \frac{1}{18} a^{9} + \frac{7}{36} a^{8} + \frac{2}{9} a^{7} - \frac{1}{2} a^{6} + \frac{4}{9} a^{5} - \frac{1}{18} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{826090634717756481054360635016285951625526203545755179631836438335192} a^{19} - \frac{1269483540991860312961679629276393929700669862406880504948979552461}{413045317358878240527180317508142975812763101772877589815918219167596} a^{18} + \frac{4039516407837733079541176714126472028901746554763320610652066010215}{275363544905918827018120211672095317208508734515251726543945479445064} a^{17} - \frac{8616125588198378228872683200468051699022843086973566513958229873639}{275363544905918827018120211672095317208508734515251726543945479445064} a^{16} - \frac{24787093443539765115952474994304469841008318959878404031665241736781}{413045317358878240527180317508142975812763101772877589815918219167596} a^{15} - \frac{5470359987249254360335804909379078735693027415121961249246057310035}{206522658679439120263590158754071487906381550886438794907959109583798} a^{14} - \frac{91998655400104135519791337334073887666967630726511696946904003532445}{826090634717756481054360635016285951625526203545755179631836438335192} a^{13} + \frac{1736194653602736082320749863122417646937474935186199304519600893151}{15297974716995490389895567315116406511583818584180651474663637746948} a^{12} - \frac{55905888183389168101464371585986215962861168885289577148335962997517}{826090634717756481054360635016285951625526203545755179631836438335192} a^{11} - \frac{2906547679353400468634032528778902260774914514413186915349037260485}{30595949433990980779791134630232813023167637168361302949327275493896} a^{10} - \frac{7169163962617257965980010781509287917479880912524249429821363688195}{34420443113239853377265026459011914651063591814406465817993184930633} a^{9} - \frac{5171786252153684265951374704446528427012351208269411243224349830991}{68840886226479706754530052918023829302127183628812931635986369861266} a^{8} + \frac{42952375813038282270720247730306465972731993682404681813098823011055}{206522658679439120263590158754071487906381550886438794907959109583798} a^{7} + \frac{6718719048712272468689703784506252598126017257240689086004058134036}{103261329339719560131795079377035743953190775443219397453979554791899} a^{6} - \frac{32150835282341165480674217233887613586647641807116721791854105912921}{103261329339719560131795079377035743953190775443219397453979554791899} a^{5} - \frac{88272368344330017987807683154398681262130581449476599390041183468381}{206522658679439120263590158754071487906381550886438794907959109583798} a^{4} - \frac{37150208474048559726156115097848345908191656972011051340681726812917}{206522658679439120263590158754071487906381550886438794907959109583798} a^{3} + \frac{69353199052439372907679314585603899282768539296286753923242712190009}{206522658679439120263590158754071487906381550886438794907959109583798} a^{2} + \frac{43831894350847544861740365743900329169400172539490377858062099665778}{103261329339719560131795079377035743953190775443219397453979554791899} a + \frac{20684736105466417622401833644309932754503037558038083115779056941135}{103261329339719560131795079377035743953190775443219397453979554791899}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 3106142973120 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
31Data not computed
227Data not computed