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Properties

Label 20.0.11484071418...7904.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{14}\cdot 389^{4}$
Root discriminant $14.22$
Ramified primes $2, 3, 389$
Class number $1$
Class group Trivial
Galois group $D_5\wr C_2:C_2$ (as 20T96)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 56, -138, 309, -554, 842, -1094, 1267, -1264, 1112, -866, 622, -404, 243, -130, 66, -30, 13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 13*x^18 - 30*x^17 + 66*x^16 - 130*x^15 + 243*x^14 - 404*x^13 + 622*x^12 - 866*x^11 + 1112*x^10 - 1264*x^9 + 1267*x^8 - 1094*x^7 + 842*x^6 - 554*x^5 + 309*x^4 - 138*x^3 + 56*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 13*x^18 - 30*x^17 + 66*x^16 - 130*x^15 + 243*x^14 - 404*x^13 + 622*x^12 - 866*x^11 + 1112*x^10 - 1264*x^9 + 1267*x^8 - 1094*x^7 + 842*x^6 - 554*x^5 + 309*x^4 - 138*x^3 + 56*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 13 x^{18} - 30 x^{17} + 66 x^{16} - 130 x^{15} + 243 x^{14} - 404 x^{13} + 622 x^{12} - 866 x^{11} + 1112 x^{10} - 1264 x^{9} + 1267 x^{8} - 1094 x^{7} + 842 x^{6} - 554 x^{5} + 309 x^{4} - 138 x^{3} + 56 x^{2} - 12 x + 1 \)

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:  \(114840714180513475067904=2^{20}\cdot 3^{14}\cdot 389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.22$
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:  $2, 3, 389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{228} a^{18} - \frac{1}{19} a^{17} - \frac{2}{57} a^{16} - \frac{41}{114} a^{15} - \frac{7}{114} a^{14} + \frac{1}{3} a^{13} - \frac{1}{12} a^{12} - \frac{25}{57} a^{11} - \frac{1}{76} a^{10} - \frac{5}{114} a^{9} - \frac{53}{228} a^{8} - \frac{25}{114} a^{7} + \frac{10}{57} a^{6} + \frac{7}{57} a^{5} + \frac{7}{38} a^{4} - \frac{31}{114} a^{3} - \frac{5}{228} a^{2} + \frac{1}{19} a - \frac{21}{76}$, $\frac{1}{3701715432} a^{19} - \frac{561297}{1233905144} a^{18} - \frac{54718304}{462714429} a^{17} + \frac{45109247}{616952572} a^{16} + \frac{66966899}{925428858} a^{15} + \frac{699908975}{1850857716} a^{14} + \frac{34977623}{194827128} a^{13} - \frac{868493615}{3701715432} a^{12} - \frac{390200993}{1233905144} a^{11} - \frac{152975415}{1233905144} a^{10} - \frac{432350261}{1233905144} a^{9} - \frac{432661201}{1233905144} a^{8} - \frac{558583255}{1850857716} a^{7} + \frac{74171653}{154238143} a^{6} - \frac{486078935}{1850857716} a^{5} - \frac{39183423}{154238143} a^{4} - \frac{980728951}{3701715432} a^{3} - \frac{450161933}{3701715432} a^{2} + \frac{143114501}{3701715432} a + \frac{495251041}{3701715432}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1621212583}{3701715432} a^{19} + \frac{2140440355}{1233905144} a^{18} - \frac{878849936}{154238143} a^{17} + \frac{24374215633}{1850857716} a^{16} - \frac{8967204709}{308476286} a^{15} + \frac{105844956655}{1850857716} a^{14} - \frac{20825470313}{194827128} a^{13} + \frac{219595724935}{1233905144} a^{12} - \frac{338439884833}{1233905144} a^{11} + \frac{1413383214839}{3701715432} a^{10} - \frac{1811922885575}{3701715432} a^{9} + \frac{2056971412081}{3701715432} a^{8} - \frac{1027457362619}{1850857716} a^{7} + \frac{220113651248}{462714429} a^{6} - \frac{663045336275}{1850857716} a^{5} + \frac{105621675101}{462714429} a^{4} - \frac{151638431485}{1233905144} a^{3} + \frac{194426597383}{3701715432} a^{2} - \frac{67326387811}{3701715432} a + \frac{11322469765}{3701715432} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13043.728755 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2:C_2$ (as 20T96):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 16 conjugacy class representatives for $D_5\wr C_2:C_2$
Character table for $D_5\wr C_2:C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 10.0.112960521216.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
389Data not computed