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Properties

Label 20.2.27487790694...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{52}\cdot 5^{14}$
Root discriminant $18.70$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 8, -39, 90, -185, 272, -244, 32, 371, -784, 1045, -1038, 819, -552, 296, -152, 65, -24, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 11*x^18 - 24*x^17 + 65*x^16 - 152*x^15 + 296*x^14 - 552*x^13 + 819*x^12 - 1038*x^11 + 1045*x^10 - 784*x^9 + 371*x^8 + 32*x^7 - 244*x^6 + 272*x^5 - 185*x^4 + 90*x^3 - 39*x^2 + 8*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 11*x^18 - 24*x^17 + 65*x^16 - 152*x^15 + 296*x^14 - 552*x^13 + 819*x^12 - 1038*x^11 + 1045*x^10 - 784*x^9 + 371*x^8 + 32*x^7 - 244*x^6 + 272*x^5 - 185*x^4 + 90*x^3 - 39*x^2 + 8*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 11 x^{18} - 24 x^{17} + 65 x^{16} - 152 x^{15} + 296 x^{14} - 552 x^{13} + 819 x^{12} - 1038 x^{11} + 1045 x^{10} - 784 x^{9} + 371 x^{8} + 32 x^{7} - 244 x^{6} + 272 x^{5} - 185 x^{4} + 90 x^{3} - 39 x^{2} + 8 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-27487790694400000000000000=-\,2^{52}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.70$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{82} a^{18} - \frac{15}{82} a^{17} + \frac{8}{41} a^{16} - \frac{3}{41} a^{15} + \frac{7}{41} a^{14} - \frac{7}{41} a^{13} + \frac{16}{41} a^{12} - \frac{15}{41} a^{11} - \frac{33}{82} a^{10} + \frac{7}{82} a^{9} + \frac{4}{41} a^{8} - \frac{2}{41} a^{7} + \frac{5}{41} a^{6} + \frac{3}{41} a^{5} - \frac{4}{41} a^{4} - \frac{13}{41} a^{3} + \frac{33}{82} a^{2} + \frac{9}{82} a - \frac{15}{41}$, $\frac{1}{1230311393030606} a^{19} - \frac{767125378257}{1230311393030606} a^{18} + \frac{61825100165130}{615155696515303} a^{17} - \frac{10161575433523}{615155696515303} a^{16} - \frac{152504507610137}{615155696515303} a^{15} + \frac{248215124237643}{615155696515303} a^{14} - \frac{59464378783726}{615155696515303} a^{13} - \frac{202239961353698}{615155696515303} a^{12} + \frac{358854125644235}{1230311393030606} a^{11} + \frac{472999323640299}{1230311393030606} a^{10} - \frac{140838985209129}{615155696515303} a^{9} - \frac{142222888282862}{615155696515303} a^{8} + \frac{294455679607824}{615155696515303} a^{7} - \frac{263305738953602}{615155696515303} a^{6} + \frac{113132433950722}{615155696515303} a^{5} - \frac{230074580336930}{615155696515303} a^{4} + \frac{33533405958475}{1230311393030606} a^{3} + \frac{118434264804295}{1230311393030606} a^{2} + \frac{227894851289288}{615155696515303} a - \frac{306633427614925}{615155696515303}$

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:  $10$
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:  \( 67620.9124596 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 5.1.256000.1, 10.2.327680000000.1

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
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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$