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Properties

Label 20.0.44040105040...1001.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 89^{2}$
Root discriminant $13.56$
Ramified primes $11, 89$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 30, -67, 99, -98, 58, -3, -34, 44, -43, 44, -34, -3, 58, -98, 99, -67, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 30*x^18 - 67*x^17 + 99*x^16 - 98*x^15 + 58*x^14 - 3*x^13 - 34*x^12 + 44*x^11 - 43*x^10 + 44*x^9 - 34*x^8 - 3*x^7 + 58*x^6 - 98*x^5 + 99*x^4 - 67*x^3 + 30*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 30*x^18 - 67*x^17 + 99*x^16 - 98*x^15 + 58*x^14 - 3*x^13 - 34*x^12 + 44*x^11 - 43*x^10 + 44*x^9 - 34*x^8 - 3*x^7 + 58*x^6 - 98*x^5 + 99*x^4 - 67*x^3 + 30*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 30 x^{18} - 67 x^{17} + 99 x^{16} - 98 x^{15} + 58 x^{14} - 3 x^{13} - 34 x^{12} + 44 x^{11} - 43 x^{10} + 44 x^{9} - 34 x^{8} - 3 x^{7} + 58 x^{6} - 98 x^{5} + 99 x^{4} - 67 x^{3} + 30 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44040105040171965561001=11^{18}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.56$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{23} a^{19} - \frac{9}{23} a^{18} - \frac{7}{23} a^{17} + \frac{9}{23} a^{16} - \frac{2}{23} a^{15} - \frac{4}{23} a^{14} - \frac{7}{23} a^{13} + \frac{4}{23} a^{12} + \frac{8}{23} a^{11} - \frac{10}{23} a^{10} - \frac{10}{23} a^{9} + \frac{8}{23} a^{8} + \frac{4}{23} a^{7} - \frac{7}{23} a^{6} - \frac{4}{23} a^{5} - \frac{2}{23} a^{4} + \frac{9}{23} a^{3} - \frac{7}{23} a^{2} - \frac{9}{23} a + \frac{1}{23}$

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:  \( -\frac{351}{23} a^{19} + \frac{2837}{23} a^{18} - \frac{10354}{23} a^{17} + \frac{21750}{23} a^{16} - \frac{28899}{23} a^{15} + \frac{24427}{23} a^{14} - \frac{9871}{23} a^{13} - \frac{4969}{23} a^{12} + \frac{11820}{23} a^{11} - \frac{11325}{23} a^{10} + \frac{10962}{23} a^{9} - \frac{11939}{23} a^{8} + \frac{7750}{23} a^{7} + \frac{5447}{23} a^{6} - \frac{21113}{23} a^{5} + \frac{29038}{23} a^{4} - \frac{24549}{23} a^{3} + \frac{13313}{23} a^{2} - \frac{4224}{23} a + \frac{615}{23} \) (order $22$)
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:  \( 10681.2000756 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\), 10.2.19077940409.1, 10.8.209857344499.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$