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Properties

Label 20.8.48739661569...3529.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{6}\cdot 401^{8}$
Root discriminant $15.29$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -3, -1, 28, -3, -102, 56, 145, -48, -143, -48, 145, 56, -102, -3, 28, -1, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 - x^17 + 28*x^16 - 3*x^15 - 102*x^14 + 56*x^13 + 145*x^12 - 48*x^11 - 143*x^10 - 48*x^9 + 145*x^8 + 56*x^7 - 102*x^6 - 3*x^5 + 28*x^4 - x^3 - 3*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 3*x^18 - x^17 + 28*x^16 - 3*x^15 - 102*x^14 + 56*x^13 + 145*x^12 - 48*x^11 - 143*x^10 - 48*x^9 + 145*x^8 + 56*x^7 - 102*x^6 - 3*x^5 + 28*x^4 - x^3 - 3*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 3 x^{18} - x^{17} + 28 x^{16} - 3 x^{15} - 102 x^{14} + 56 x^{13} + 145 x^{12} - 48 x^{11} - 143 x^{10} - 48 x^{9} + 145 x^{8} + 56 x^{7} - 102 x^{6} - 3 x^{5} + 28 x^{4} - x^{3} - 3 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(487396615698744004253529=3^{6}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.29$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{3}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a$, $\frac{1}{795504} a^{18} - \frac{12943}{795504} a^{17} + \frac{41767}{795504} a^{16} + \frac{51163}{795504} a^{15} - \frac{41959}{397752} a^{14} + \frac{2863}{33146} a^{13} - \frac{17807}{99438} a^{12} - \frac{81239}{198876} a^{11} - \frac{77525}{265168} a^{10} + \frac{288599}{795504} a^{9} - \frac{77525}{265168} a^{8} - \frac{7880}{49719} a^{7} - \frac{85333}{198876} a^{6} + \frac{2863}{33146} a^{5} + \frac{156917}{397752} a^{4} - \frac{346589}{795504} a^{3} - \frac{355985}{795504} a^{2} - \frac{211819}{795504} a - \frac{198875}{795504}$, $\frac{1}{795504} a^{19} - \frac{4315}{132584} a^{17} - \frac{1481}{397752} a^{16} + \frac{60587}{795504} a^{15} - \frac{10063}{397752} a^{14} - \frac{38537}{397752} a^{13} - \frac{9209}{132584} a^{12} - \frac{109559}{795504} a^{11} - \frac{14980}{49719} a^{10} - \frac{5332}{49719} a^{9} - \frac{256607}{795504} a^{8} - \frac{19813}{132584} a^{7} + \frac{169207}{397752} a^{6} - \frac{58987}{397752} a^{5} - \frac{335989}{795504} a^{4} - \frac{56021}{397752} a^{3} + \frac{38541}{132584} a^{2} - \frac{2819}{33146} a + \frac{311257}{795504}$

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:  $13$
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:  \( 12914.3678688 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.4.77570884803.1, 10.8.698137963227.1, 10.6.232712654409.1

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 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed