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Properties

Label 20.0.13932535636...1609.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 1583^{2}$
Root discriminant $18.08$
Ramified primes $11, 1583$
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![1024, -1536, -1024, 2304, 576, -1664, -432, 688, 416, -132, -263, -66, 104, 86, -27, -52, 9, 18, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 4*x^18 + 18*x^17 + 9*x^16 - 52*x^15 - 27*x^14 + 86*x^13 + 104*x^12 - 66*x^11 - 263*x^10 - 132*x^9 + 416*x^8 + 688*x^7 - 432*x^6 - 1664*x^5 + 576*x^4 + 2304*x^3 - 1024*x^2 - 1536*x + 1024)
 
gp: K = bnfinit(x^20 - 3*x^19 - 4*x^18 + 18*x^17 + 9*x^16 - 52*x^15 - 27*x^14 + 86*x^13 + 104*x^12 - 66*x^11 - 263*x^10 - 132*x^9 + 416*x^8 + 688*x^7 - 432*x^6 - 1664*x^5 + 576*x^4 + 2304*x^3 - 1024*x^2 - 1536*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 4 x^{18} + 18 x^{17} + 9 x^{16} - 52 x^{15} - 27 x^{14} + 86 x^{13} + 104 x^{12} - 66 x^{11} - 263 x^{10} - 132 x^{9} + 416 x^{8} + 688 x^{7} - 432 x^{6} - 1664 x^{5} + 576 x^{4} + 2304 x^{3} - 1024 x^{2} - 1536 x + 1024 \)

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:  \(13932535636789734453691609=11^{18}\cdot 1583^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.08$
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:  $11, 1583$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{3}{8} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{3}{16} a^{10} - \frac{1}{8} a^{9} + \frac{5}{16} a^{8} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{5}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{3}{32} a^{11} + \frac{7}{16} a^{10} + \frac{5}{32} a^{9} - \frac{1}{2} a^{8} - \frac{1}{8} a^{7} + \frac{7}{16} a^{6} - \frac{11}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} - \frac{3}{64} a^{12} + \frac{7}{32} a^{11} - \frac{27}{64} a^{10} + \frac{1}{4} a^{9} + \frac{7}{16} a^{8} + \frac{7}{32} a^{7} - \frac{11}{64} a^{6} + \frac{15}{32} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{16} - \frac{1}{64} a^{15} + \frac{1}{64} a^{14} - \frac{3}{128} a^{13} + \frac{7}{64} a^{12} - \frac{27}{128} a^{11} + \frac{1}{8} a^{10} + \frac{7}{32} a^{9} - \frac{25}{64} a^{8} - \frac{11}{128} a^{7} - \frac{17}{64} a^{6} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{11008} a^{18} + \frac{3}{11008} a^{17} - \frac{37}{5504} a^{16} - \frac{1}{5504} a^{15} + \frac{317}{11008} a^{14} + \frac{325}{5504} a^{13} - \frac{631}{11008} a^{12} - \frac{605}{2752} a^{11} + \frac{151}{1376} a^{10} - \frac{1041}{5504} a^{9} + \frac{2029}{11008} a^{8} - \frac{67}{5504} a^{7} + \frac{329}{2752} a^{6} + \frac{225}{1376} a^{5} - \frac{35}{172} a^{4} - \frac{45}{172} a^{3} - \frac{19}{172} a^{2} - \frac{37}{86} a + \frac{2}{43}$, $\frac{1}{22016} a^{19} - \frac{1}{22016} a^{18} - \frac{1}{256} a^{17} - \frac{25}{11008} a^{16} - \frac{19}{22016} a^{15} - \frac{309}{11008} a^{14} - \frac{1167}{22016} a^{13} + \frac{157}{1376} a^{12} + \frac{673}{2752} a^{11} - \frac{1565}{11008} a^{10} - \frac{6843}{22016} a^{9} + \frac{4131}{11008} a^{8} + \frac{2699}{5504} a^{7} - \frac{119}{688} a^{6} + \frac{49}{688} a^{5} + \frac{9}{344} a^{4} + \frac{4}{43} a^{3} + \frac{11}{43} a^{2} + \frac{33}{86} a - \frac{4}{43}$

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:  \( -\frac{1925}{22016} a^{19} + \frac{6271}{22016} a^{18} + \frac{2851}{5504} a^{17} - \frac{20141}{11008} a^{16} - \frac{45045}{22016} a^{15} + \frac{29169}{5504} a^{14} + \frac{155615}{22016} a^{13} - \frac{76515}{11008} a^{12} - \frac{13503}{688} a^{11} - \frac{48023}{11008} a^{10} + \frac{768339}{22016} a^{9} + \frac{241509}{5504} a^{8} - \frac{75705}{2752} a^{7} - \frac{316855}{2752} a^{6} - \frac{40543}{1376} a^{5} + \frac{134431}{688} a^{4} + \frac{31833}{344} a^{3} - \frac{9384}{43} a^{2} - \frac{6455}{86} a + \frac{5553}{43} \) (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:  \( 179575.849258 \)
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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\), 10.10.3732631194853.1, 10.0.339330108623.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$
1583Data not computed