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Properties

Label 20.0.20278473101...7376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{10}\cdot 401^{8}$
Root discriminant $82.29$
Ramified primes $2, 7, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\times A_4$ (as 20T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17123, -13398, 35838, -20038, 18589, -16984, 21364, -13704, 8903, -4132, 4838, -3450, -353, 806, 478, -436, -14, 64, -2, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 2*x^18 + 64*x^17 - 14*x^16 - 436*x^15 + 478*x^14 + 806*x^13 - 353*x^12 - 3450*x^11 + 4838*x^10 - 4132*x^9 + 8903*x^8 - 13704*x^7 + 21364*x^6 - 16984*x^5 + 18589*x^4 - 20038*x^3 + 35838*x^2 - 13398*x + 17123)
 
gp: K = bnfinit(x^20 - 6*x^19 - 2*x^18 + 64*x^17 - 14*x^16 - 436*x^15 + 478*x^14 + 806*x^13 - 353*x^12 - 3450*x^11 + 4838*x^10 - 4132*x^9 + 8903*x^8 - 13704*x^7 + 21364*x^6 - 16984*x^5 + 18589*x^4 - 20038*x^3 + 35838*x^2 - 13398*x + 17123, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 2 x^{18} + 64 x^{17} - 14 x^{16} - 436 x^{15} + 478 x^{14} + 806 x^{13} - 353 x^{12} - 3450 x^{11} + 4838 x^{10} - 4132 x^{9} + 8903 x^{8} - 13704 x^{7} + 21364 x^{6} - 16984 x^{5} + 18589 x^{4} - 20038 x^{3} + 35838 x^{2} - 13398 x + 17123 \)

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:  \(202784731012754939641986471226953957376=2^{30}\cdot 7^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.29$
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:  $2, 7, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{4}{13} a^{16} - \frac{1}{13} a^{15} - \frac{1}{13} a^{14} + \frac{3}{13} a^{13} + \frac{4}{13} a^{10} - \frac{1}{13} a^{9} - \frac{2}{13} a^{8} - \frac{4}{13} a^{7} + \frac{4}{13} a^{6} + \frac{5}{13} a^{5} - \frac{5}{13} a^{4} - \frac{5}{13} a^{3} - \frac{6}{13} a^{2} + \frac{6}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{18} - \frac{4}{13} a^{16} + \frac{3}{13} a^{15} - \frac{6}{13} a^{14} + \frac{1}{13} a^{13} + \frac{4}{13} a^{11} - \frac{4}{13} a^{10} + \frac{2}{13} a^{9} + \frac{4}{13} a^{8} - \frac{6}{13} a^{7} + \frac{2}{13} a^{6} + \frac{1}{13} a^{5} + \frac{2}{13} a^{4} + \frac{1}{13} a^{3} + \frac{4}{13} a^{2} - \frac{6}{13} a + \frac{6}{13}$, $\frac{1}{37645568501116350216497243075029069615006241} a^{19} + \frac{7125729384846329896168994983030892687633}{2895812961624334632038249467309928431923557} a^{18} + \frac{182764806493982988867392460640524405324088}{37645568501116350216497243075029069615006241} a^{17} + \frac{16324437831509296829970927571013737787257547}{37645568501116350216497243075029069615006241} a^{16} + \frac{17592207235144369971638266181937772464058308}{37645568501116350216497243075029069615006241} a^{15} - \frac{10998042047911819142908032834985674048768894}{37645568501116350216497243075029069615006241} a^{14} - \frac{13020112875650641262271751118411444486741654}{37645568501116350216497243075029069615006241} a^{13} - \frac{14402659641346401419303359580435431355994914}{37645568501116350216497243075029069615006241} a^{12} - \frac{2822428401609716678788793598363019022308189}{37645568501116350216497243075029069615006241} a^{11} + \frac{1339495483227671289940126615512522952020456}{2895812961624334632038249467309928431923557} a^{10} + \frac{8632702480879627254298483240495949373402042}{37645568501116350216497243075029069615006241} a^{9} + \frac{399291561871845082102463814682023358851518}{37645568501116350216497243075029069615006241} a^{8} - \frac{11344274497795235730247890019056904029969218}{37645568501116350216497243075029069615006241} a^{7} - \frac{16033353937538149289802816441724551349496871}{37645568501116350216497243075029069615006241} a^{6} + \frac{11285820924913652277861294097562573260513930}{37645568501116350216497243075029069615006241} a^{5} - \frac{153112966342230665544745099123512112020103}{37645568501116350216497243075029069615006241} a^{4} + \frac{1440312891258408463104264722185152572613349}{2895812961624334632038249467309928431923557} a^{3} - \frac{18455329955790570940708678777829744186755950}{37645568501116350216497243075029069615006241} a^{2} + \frac{10162489952790614490360882588059224613859416}{37645568501116350216497243075029069615006241} a + \frac{14609638266929485216365194483625120253307472}{37645568501116350216497243075029069615006241}$

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:  $9$
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:  \( 99133769895.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.0.3136.1, 5.5.160801.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
7Data not computed
401Data not computed