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Properties

Label 20.12.6744255325...7456.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $31.01$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -20, -1694, 4278, -864, -10778, 16697, -5222, -11895, 16026, -7310, -1614, 4090, -2478, 640, 190, -268, 98, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 3*x^18 + 98*x^17 - 268*x^16 + 190*x^15 + 640*x^14 - 2478*x^13 + 4090*x^12 - 1614*x^11 - 7310*x^10 + 16026*x^9 - 11895*x^8 - 5222*x^7 + 16697*x^6 - 10778*x^5 - 864*x^4 + 4278*x^3 - 1694*x^2 - 20*x + 89)
 
gp: K = bnfinit(x^20 - 6*x^19 - 3*x^18 + 98*x^17 - 268*x^16 + 190*x^15 + 640*x^14 - 2478*x^13 + 4090*x^12 - 1614*x^11 - 7310*x^10 + 16026*x^9 - 11895*x^8 - 5222*x^7 + 16697*x^6 - 10778*x^5 - 864*x^4 + 4278*x^3 - 1694*x^2 - 20*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 3 x^{18} + 98 x^{17} - 268 x^{16} + 190 x^{15} + 640 x^{14} - 2478 x^{13} + 4090 x^{12} - 1614 x^{11} - 7310 x^{10} + 16026 x^{9} - 11895 x^{8} - 5222 x^{7} + 16697 x^{6} - 10778 x^{5} - 864 x^{4} + 4278 x^{3} - 1694 x^{2} - 20 x + 89 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(674425532537559799418460307456=2^{20}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.01$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{386140784589554707057132494839879} a^{19} - \frac{147382842349078453763346348013377}{386140784589554707057132494839879} a^{18} - \frac{56171273285658918918527256871402}{386140784589554707057132494839879} a^{17} - \frac{154696516364447632454127143203786}{386140784589554707057132494839879} a^{16} + \frac{116958697907834926145800128084228}{386140784589554707057132494839879} a^{15} + \frac{88385998795782380520717453517366}{386140784589554707057132494839879} a^{14} + \frac{106334639054442754224206460581378}{386140784589554707057132494839879} a^{13} + \frac{37235896082968375298751353350695}{386140784589554707057132494839879} a^{12} - \frac{93650848750010512617967719770856}{386140784589554707057132494839879} a^{11} - \frac{41177922799640336445912745562765}{386140784589554707057132494839879} a^{10} - \frac{179320140863355634133974929107248}{386140784589554707057132494839879} a^{9} + \frac{115742288337490976363351963312628}{386140784589554707057132494839879} a^{8} - \frac{6796997106660765784080348082623}{16788729764763248132918804123473} a^{7} - \frac{55717111977933506354642982108753}{386140784589554707057132494839879} a^{6} + \frac{118803244182962628293430670617056}{386140784589554707057132494839879} a^{5} - \frac{79120565837526862017241213343501}{386140784589554707057132494839879} a^{4} + \frac{33818933067376740407826926980070}{386140784589554707057132494839879} a^{3} + \frac{16628322076074699737368714442977}{386140784589554707057132494839879} a^{2} + \frac{52278824298352005886831918354512}{386140784589554707057132494839879} a - \frac{58727820347748341375537282230482}{386140784589554707057132494839879}$

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:  $15$
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:  \( 54067536.9661 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.52900342088704.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
241Data not computed