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Properties

Label 20.0.73404803840...4928.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{8}\cdot 17^{9}$
Root discriminant $22.04$
Ramified primes $2, 7, 17$
Class number $1$
Class group Trivial
Galois group $D_4\times D_5$ (as 20T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 35, 128, 332, 644, 816, 668, 322, 160, 220, 10, 135, -42, 27, 12, 8, -14, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 14*x^18 - 14*x^17 + 8*x^16 + 12*x^15 + 27*x^14 - 42*x^13 + 135*x^12 + 10*x^11 + 220*x^10 + 160*x^9 + 322*x^8 + 668*x^7 + 816*x^6 + 644*x^5 + 332*x^4 + 128*x^3 + 35*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 14*x^18 - 14*x^17 + 8*x^16 + 12*x^15 + 27*x^14 - 42*x^13 + 135*x^12 + 10*x^11 + 220*x^10 + 160*x^9 + 322*x^8 + 668*x^7 + 816*x^6 + 644*x^5 + 332*x^4 + 128*x^3 + 35*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 14 x^{18} - 14 x^{17} + 8 x^{16} + 12 x^{15} + 27 x^{14} - 42 x^{13} + 135 x^{12} + 10 x^{11} + 220 x^{10} + 160 x^{9} + 322 x^{8} + 668 x^{7} + 816 x^{6} + 644 x^{5} + 332 x^{4} + 128 x^{3} + 35 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(734048038403921797267324928=2^{30}\cdot 7^{8}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.04$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{17} - \frac{3}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{637} a^{18} + \frac{1}{637} a^{17} + \frac{4}{637} a^{16} + \frac{264}{637} a^{15} - \frac{97}{637} a^{14} - \frac{19}{49} a^{13} + \frac{5}{49} a^{12} + \frac{257}{637} a^{11} - \frac{10}{637} a^{10} + \frac{18}{637} a^{9} + \frac{5}{637} a^{8} - \frac{85}{637} a^{7} + \frac{277}{637} a^{6} + \frac{4}{49} a^{5} - \frac{258}{637} a^{4} - \frac{61}{637} a^{3} - \frac{289}{637} a^{2} + \frac{144}{637} a - \frac{188}{637}$, $\frac{1}{9842551036094260939} a^{19} + \frac{721573174622397}{1406078719442037277} a^{18} + \frac{495085801206277053}{9842551036094260939} a^{17} - \frac{348401628919474339}{9842551036094260939} a^{16} + \frac{5984320842161436}{427937001569315693} a^{15} - \frac{61902887612498854}{124589253621446341} a^{14} + \frac{36233897644251086}{757119310468789303} a^{13} + \frac{3317830694034346864}{9842551036094260939} a^{12} + \frac{4698352466747609004}{9842551036094260939} a^{11} + \frac{553766148069113027}{1406078719442037277} a^{10} + \frac{805831165840614656}{9842551036094260939} a^{9} + \frac{3279752696354748552}{9842551036094260939} a^{8} - \frac{1287481833127054133}{9842551036094260939} a^{7} - \frac{3820350653568038647}{9842551036094260939} a^{6} - \frac{984579622237835843}{9842551036094260939} a^{5} + \frac{2107663157409422894}{9842551036094260939} a^{4} + \frac{3847245624492955989}{9842551036094260939} a^{3} - \frac{1890471464239356783}{9842551036094260939} a^{2} - \frac{2723618587509024859}{9842551036094260939} a + \frac{957673717271562136}{9842551036094260939}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 311018.518626 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times D_5$ (as 20T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $D_4\times D_5$
Character table for $D_4\times D_5$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.1088.2, 5.1.14161.1, 10.2.6571095523328.1

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 $20$ $20$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$