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Properties

Label 20.0.19940812824...2849.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 179^{10}$
Root discriminant $23.17$
Ramified primes $3, 179$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 128, 0, 1536, 0, 5547, 0, 2806, 0, -1103, 0, 823, 0, -355, 0, 99, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 + 99*x^16 - 355*x^14 + 823*x^12 - 1103*x^10 + 2806*x^8 + 5547*x^6 + 1536*x^4 + 128*x^2 + 64)
 
gp: K = bnfinit(x^20 - 14*x^18 + 99*x^16 - 355*x^14 + 823*x^12 - 1103*x^10 + 2806*x^8 + 5547*x^6 + 1536*x^4 + 128*x^2 + 64, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} + 99 x^{16} - 355 x^{14} + 823 x^{12} - 1103 x^{10} + 2806 x^{8} + 5547 x^{6} + 1536 x^{4} + 128 x^{2} + 64 \)

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:  \(1994081282499911259959052849=3^{10}\cdot 179^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.17$
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:  $3, 179$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{3}{16} a^{7} - \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{12} + \frac{1}{32} a^{10} + \frac{3}{32} a^{8} - \frac{1}{4} a^{7} - \frac{5}{32} a^{6} - \frac{5}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{9} + \frac{5}{32} a^{7} + \frac{5}{32} a^{5} - \frac{7}{16} a^{3} - \frac{1}{2}$, $\frac{1}{224} a^{16} + \frac{3}{224} a^{14} - \frac{5}{224} a^{12} + \frac{5}{224} a^{10} + \frac{17}{224} a^{8} - \frac{1}{32} a^{6} - \frac{9}{112} a^{4} + \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{448} a^{17} - \frac{1}{448} a^{16} - \frac{1}{112} a^{15} - \frac{3}{448} a^{14} - \frac{3}{112} a^{13} + \frac{5}{448} a^{12} - \frac{1}{224} a^{11} - \frac{5}{448} a^{10} - \frac{1}{112} a^{9} + \frac{39}{448} a^{8} - \frac{3}{16} a^{7} + \frac{1}{64} a^{6} + \frac{17}{448} a^{5} + \frac{9}{224} a^{4} - \frac{9}{56} a^{3} + \frac{9}{56} a^{2} - \frac{3}{28} a - \frac{1}{7}$, $\frac{1}{296380041664} a^{18} + \frac{549546605}{296380041664} a^{16} - \frac{1}{64} a^{15} + \frac{154250471}{296380041664} a^{14} - \frac{1}{64} a^{13} - \frac{6213200591}{296380041664} a^{12} - \frac{1}{64} a^{11} + \frac{993056311}{12886088768} a^{10} - \frac{3}{64} a^{9} - \frac{23578878811}{296380041664} a^{8} - \frac{11}{64} a^{7} - \frac{426327375}{3614390752} a^{6} + \frac{5}{64} a^{5} + \frac{16124227}{298169056} a^{4} - \frac{3}{8} a^{3} + \frac{6499677655}{37047505208} a^{2} + \frac{1}{4} a - \frac{1800149376}{4630938151}$, $\frac{1}{296380041664} a^{19} - \frac{4000571}{10585001488} a^{17} + \frac{400071549}{42340005952} a^{15} + \frac{1725550525}{296380041664} a^{13} - \frac{560177603}{12886088768} a^{11} - \frac{1}{8} a^{10} - \frac{20932628439}{296380041664} a^{9} - \frac{1}{8} a^{8} - \frac{652226797}{3614390752} a^{7} + \frac{589132715}{4174366784} a^{5} + \frac{1}{8} a^{4} - \frac{216786129}{1323125186} a^{3} + \frac{1}{8} a^{2} - \frac{292485787}{9261876302} a$

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:  \( -\frac{16109}{12593696} a^{19} + \frac{251853}{12593696} a^{17} - \frac{1970409}{12593696} a^{15} + \frac{8430907}{12593696} a^{13} - \frac{1017015}{547552} a^{11} + \frac{42507395}{12593696} a^{9} - \frac{40950193}{6296848} a^{7} - \frac{3053}{11086} a^{5} + \frac{47457361}{6296848} a^{3} + \frac{426354}{393553} a + \frac{1}{2} \) (order $6$)
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:  \( 6730201.64731 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-179}) \), \(\Q(\sqrt{537}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-179})\), 5.1.32041.1 x5, 10.0.183765996899.1, 10.2.44655137246457.1 x5, 10.0.249470040483.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$179$179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$