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Properties

Label 20.8.68022091096...6129.1
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 23^{6}$
Root discriminant $17.44$
Ramified primes $11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -20, 77, 189, -214, -108, 71, -124, 260, -206, 81, -6, -60, 79, -74, 46, -23, 8, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 8*x^17 - 23*x^16 + 46*x^15 - 74*x^14 + 79*x^13 - 60*x^12 - 6*x^11 + 81*x^10 - 206*x^9 + 260*x^8 - 124*x^7 + 71*x^6 - 108*x^5 - 214*x^4 + 189*x^3 + 77*x^2 - 20*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 8*x^17 - 23*x^16 + 46*x^15 - 74*x^14 + 79*x^13 - 60*x^12 - 6*x^11 + 81*x^10 - 206*x^9 + 260*x^8 - 124*x^7 + 71*x^6 - 108*x^5 - 214*x^4 + 189*x^3 + 77*x^2 - 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 8 x^{17} - 23 x^{16} + 46 x^{15} - 74 x^{14} + 79 x^{13} - 60 x^{12} - 6 x^{11} + 81 x^{10} - 206 x^{9} + 260 x^{8} - 124 x^{7} + 71 x^{6} - 108 x^{5} - 214 x^{4} + 189 x^{3} + 77 x^{2} - 20 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6802209109663753569286129=11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.44$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{398} a^{18} + \frac{61}{398} a^{17} - \frac{21}{398} a^{16} - \frac{49}{398} a^{15} - \frac{6}{199} a^{14} - \frac{9}{199} a^{13} - \frac{65}{199} a^{12} + \frac{88}{199} a^{11} + \frac{90}{199} a^{10} + \frac{59}{199} a^{9} - \frac{145}{398} a^{8} + \frac{74}{199} a^{7} + \frac{89}{398} a^{6} + \frac{74}{199} a^{5} - \frac{37}{199} a^{4} - \frac{191}{398} a^{3} + \frac{119}{398} a^{2} + \frac{14}{199} a + \frac{95}{199}$, $\frac{1}{254861758323380393726} a^{19} + \frac{104027912088833706}{127430879161690196863} a^{18} - \frac{24161878636073674498}{127430879161690196863} a^{17} - \frac{19087928922717923095}{127430879161690196863} a^{16} - \frac{12525068269418628470}{127430879161690196863} a^{15} + \frac{26196064180106245112}{127430879161690196863} a^{14} - \frac{88774190311957817647}{254861758323380393726} a^{13} + \frac{45946314365625116523}{127430879161690196863} a^{12} + \frac{57697516601619557682}{127430879161690196863} a^{11} + \frac{13324433051831289308}{127430879161690196863} a^{10} - \frac{37523485893636636273}{254861758323380393726} a^{9} - \frac{123097433427376414607}{254861758323380393726} a^{8} - \frac{44794005080652496623}{254861758323380393726} a^{7} + \frac{57548845197407708055}{254861758323380393726} a^{6} - \frac{104107008847439956439}{254861758323380393726} a^{5} - \frac{19011846163223262163}{127430879161690196863} a^{4} - \frac{59844005456402019022}{127430879161690196863} a^{3} - \frac{72081351201063207043}{254861758323380393726} a^{2} - \frac{35638339166152422608}{127430879161690196863} a + \frac{68955107744735803057}{254861758323380393726}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 52598.7023471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.2608104505127.1, 10.4.4930254263.1, 10.6.113395848049.1

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
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$