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Properties

Label 20.20.6234865940...1344.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{34}\cdot 881^{8}$
Root discriminant $48.95$
Ramified primes $2, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-673, -484, 14486, 19586, -62725, -73218, 127665, 102876, -149398, -63382, 102554, 12394, -39328, 3506, 7652, -1804, -616, 240, 5, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 616*x^16 - 1804*x^15 + 7652*x^14 + 3506*x^13 - 39328*x^12 + 12394*x^11 + 102554*x^10 - 63382*x^9 - 149398*x^8 + 102876*x^7 + 127665*x^6 - 73218*x^5 - 62725*x^4 + 19586*x^3 + 14486*x^2 - 484*x - 673)
 
gp: K = bnfinit(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 616*x^16 - 1804*x^15 + 7652*x^14 + 3506*x^13 - 39328*x^12 + 12394*x^11 + 102554*x^10 - 63382*x^9 - 149398*x^8 + 102876*x^7 + 127665*x^6 - 73218*x^5 - 62725*x^4 + 19586*x^3 + 14486*x^2 - 484*x - 673, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 5 x^{18} + 240 x^{17} - 616 x^{16} - 1804 x^{15} + 7652 x^{14} + 3506 x^{13} - 39328 x^{12} + 12394 x^{11} + 102554 x^{10} - 63382 x^{9} - 149398 x^{8} + 102876 x^{7} + 127665 x^{6} - 73218 x^{5} - 62725 x^{4} + 19586 x^{3} + 14486 x^{2} - 484 x - 673 \)

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:  \(6234865940162794507742264996921344=2^{34}\cdot 881^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.95$
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:  $2, 881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{11} - \frac{1}{9} a^{10} - \frac{1}{18} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{7}{18} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{2} a^{2} - \frac{5}{18} a - \frac{5}{18}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{6} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{9} a^{7} + \frac{5}{18} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{5}{18} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{1}{18}$, $\frac{1}{36} a^{16} - \frac{1}{36} a^{12} - \frac{1}{18} a^{11} - \frac{1}{12} a^{10} + \frac{1}{18} a^{9} + \frac{1}{9} a^{8} + \frac{7}{18} a^{7} - \frac{5}{12} a^{6} - \frac{7}{36} a^{4} + \frac{2}{9} a^{3} + \frac{11}{36} a^{2} - \frac{4}{9} a + \frac{7}{36}$, $\frac{1}{36} a^{17} - \frac{1}{36} a^{13} - \frac{1}{18} a^{12} - \frac{1}{12} a^{11} + \frac{1}{18} a^{10} + \frac{1}{9} a^{9} + \frac{7}{18} a^{8} - \frac{5}{12} a^{7} - \frac{7}{36} a^{5} + \frac{2}{9} a^{4} + \frac{11}{36} a^{3} - \frac{4}{9} a^{2} + \frac{7}{36} a$, $\frac{1}{8399010168} a^{18} - \frac{1}{933223352} a^{17} + \frac{73142125}{8399010168} a^{16} - \frac{14815640}{1049876271} a^{15} - \frac{133780469}{8399010168} a^{14} - \frac{162978467}{2799670056} a^{13} - \frac{290574211}{4199505084} a^{12} - \frac{305291279}{2799670056} a^{11} + \frac{93360679}{933223352} a^{10} + \frac{21743125}{349958757} a^{9} - \frac{3785059025}{8399010168} a^{8} + \frac{720326929}{8399010168} a^{7} + \frac{1252105693}{4199505084} a^{6} + \frac{2062388675}{8399010168} a^{5} - \frac{392798446}{1049876271} a^{4} + \frac{305207837}{8399010168} a^{3} - \frac{658395641}{4199505084} a^{2} - \frac{1677585367}{8399010168} a + \frac{2031297683}{8399010168}$, $\frac{1}{96580217921832} a^{19} + \frac{1435}{24145054480458} a^{18} + \frac{132643505201}{12072527240229} a^{17} - \frac{550876652089}{96580217921832} a^{16} - \frac{803443651643}{32193405973944} a^{15} + \frac{386745980425}{48290108960916} a^{14} + \frac{2381229779563}{32193405973944} a^{13} - \frac{3270779871325}{96580217921832} a^{12} - \frac{416874016255}{16096702986972} a^{11} - \frac{16001481465395}{96580217921832} a^{10} + \frac{12251953561111}{96580217921832} a^{9} - \frac{2651764369819}{12072527240229} a^{8} + \frac{23711024359267}{96580217921832} a^{7} - \frac{3823112218525}{96580217921832} a^{6} + \frac{20861935121863}{96580217921832} a^{5} - \frac{3086812815157}{96580217921832} a^{4} - \frac{619443182135}{3577045108216} a^{3} - \frac{10233230359651}{96580217921832} a^{2} + \frac{815078525012}{4024175746743} a - \frac{4509061966711}{96580217921832}$

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:  \( 41113221362.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.3104644.1, 10.10.1233768238942208.1, 10.10.78961167292301312.1, 10.10.616884119471104.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
881Data not computed