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Properties

Label 20.20.1694434321...3609.1
Degree $20$
Signature $[20, 0]$
Discriminant $397^{2}\cdot 401^{10}$
Root discriminant $36.43$
Ramified primes $397, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -13, 277, -895, -1348, 7243, 1135, -20001, 3507, 25301, -7177, -16433, 5124, 5752, -1758, -1091, 313, 105, -28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 28*x^18 + 105*x^17 + 313*x^16 - 1091*x^15 - 1758*x^14 + 5752*x^13 + 5124*x^12 - 16433*x^11 - 7177*x^10 + 25301*x^9 + 3507*x^8 - 20001*x^7 + 1135*x^6 + 7243*x^5 - 1348*x^4 - 895*x^3 + 277*x^2 - 13*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 28*x^18 + 105*x^17 + 313*x^16 - 1091*x^15 - 1758*x^14 + 5752*x^13 + 5124*x^12 - 16433*x^11 - 7177*x^10 + 25301*x^9 + 3507*x^8 - 20001*x^7 + 1135*x^6 + 7243*x^5 - 1348*x^4 - 895*x^3 + 277*x^2 - 13*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 28 x^{18} + 105 x^{17} + 313 x^{16} - 1091 x^{15} - 1758 x^{14} + 5752 x^{13} + 5124 x^{12} - 16433 x^{11} - 7177 x^{10} + 25301 x^{9} + 3507 x^{8} - 20001 x^{7} + 1135 x^{6} + 7243 x^{5} - 1348 x^{4} - 895 x^{3} + 277 x^{2} - 13 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16944343217067584829876535393609=397^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.43$
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:  $397, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{2}{9} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{2}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{85428568580499125252163} a^{19} - \frac{2669849579754239879783}{85428568580499125252163} a^{18} - \frac{3577114193185701043970}{85428568580499125252163} a^{17} + \frac{13264160874105063950717}{85428568580499125252163} a^{16} - \frac{9304494009683623814368}{85428568580499125252163} a^{15} - \frac{13266510899630763428383}{85428568580499125252163} a^{14} - \frac{34347072569930327672435}{85428568580499125252163} a^{13} - \frac{4589793794165336577826}{28476189526833041750721} a^{12} - \frac{3470059327300041612220}{28476189526833041750721} a^{11} + \frac{35261119499536793295400}{85428568580499125252163} a^{10} + \frac{28930051967161419245623}{85428568580499125252163} a^{9} - \frac{30027476573603148166739}{85428568580499125252163} a^{8} + \frac{34302096124726825018394}{85428568580499125252163} a^{7} - \frac{41887779440833357801241}{85428568580499125252163} a^{6} + \frac{980598198511488417524}{9492063175611013916907} a^{5} - \frac{5354557501646879773994}{85428568580499125252163} a^{4} + \frac{27199883989343610568897}{85428568580499125252163} a^{3} + \frac{33991319527003433393335}{85428568580499125252163} a^{2} - \frac{330141098053497890665}{3164021058537004638969} a - \frac{8258671471082753818678}{85428568580499125252163}$

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:  $19$
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:  \( 1861846863.26 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

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

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.4116350715994397.1, 10.10.10265213755597.1, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
397Data not computed
401Data not computed