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Properties

Label 20.0.32000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{25}$
Root discriminant $21.15$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $D_{20}$ (as 20T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, 50, 0, 35, 0, -420, 0, 865, 0, -798, 0, 485, 0, -200, 0, 55, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 55*x^16 - 200*x^14 + 485*x^12 - 798*x^10 + 865*x^8 - 420*x^6 + 35*x^4 + 50*x^2 + 5)
 
gp: K = bnfinit(x^20 - 10*x^18 + 55*x^16 - 200*x^14 + 485*x^12 - 798*x^10 + 865*x^8 - 420*x^6 + 35*x^4 + 50*x^2 + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 55 x^{16} - 200 x^{14} + 485 x^{12} - 798 x^{10} + 865 x^{8} - 420 x^{6} + 35 x^{4} + 50 x^{2} + 5 \)

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:  \(320000000000000000000000000=2^{30}\cdot 5^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.15$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{1}{8} a^{6} + \frac{5}{16} a^{4} - \frac{5}{16} a^{2} - \frac{5}{16}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{5}{16} a^{5} - \frac{5}{16} a^{3} - \frac{5}{16} a$, $\frac{1}{5554208} a^{18} - \frac{1}{32} a^{17} - \frac{51763}{2777104} a^{16} + \frac{1}{32} a^{15} - \frac{9}{682} a^{14} - \frac{3}{32} a^{13} + \frac{86839}{5554208} a^{12} + \frac{1}{16} a^{11} - \frac{151909}{1388552} a^{10} + \frac{3}{16} a^{9} + \frac{33492}{173569} a^{8} + \frac{3}{16} a^{7} - \frac{105887}{504928} a^{6} - \frac{5}{32} a^{5} - \frac{401743}{2777104} a^{4} + \frac{13}{32} a^{3} + \frac{16241}{347138} a^{2} - \frac{15}{32} a + \frac{296405}{5554208}$, $\frac{1}{5554208} a^{19} + \frac{70043}{5554208} a^{17} - \frac{1}{32} a^{16} - \frac{485}{10912} a^{15} + \frac{1}{32} a^{14} + \frac{303773}{2777104} a^{13} - \frac{3}{32} a^{12} + \frac{216889}{2777104} a^{11} + \frac{1}{16} a^{10} - \frac{679111}{2777104} a^{9} + \frac{3}{16} a^{8} - \frac{74329}{504928} a^{7} + \frac{3}{16} a^{6} + \frac{1452911}{5554208} a^{5} - \frac{5}{32} a^{4} + \frac{2169115}{5554208} a^{3} + \frac{13}{32} a^{2} - \frac{316429}{1388552} a - \frac{15}{32}$

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:  \( -\frac{575}{5456} a^{18} + \frac{1475}{1364} a^{16} - \frac{16545}{2728} a^{14} + \frac{122965}{5456} a^{12} - \frac{38441}{682} a^{10} + \frac{131885}{1364} a^{8} - \frac{55395}{496} a^{6} + \frac{90515}{1364} a^{4} - \frac{38125}{2728} a^{2} - \frac{20853}{5456} \) (order $4$)
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:  \( 665063.298986 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{20}$ (as 20T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $D_{20}$
Character table for $D_{20}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.320.1, 5.1.250000.1, 10.0.250000000000.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 $20$ R $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$5$5.10.13.8$x^{10} + 20 x^{4} + 10$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
5.10.12.7$x^{10} + 10 x^{8} + 5 x^{7} + 15 x^{6} + 5 x^{4} + 5 x^{2} - 20 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$