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Properties

Label 20.0.12569066605...8944.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}\cdot 1451^{2}$
Root discriminant $50.70$
Ramified primes $2, 11, 1451$
Class number $1756$ (GRH)
Class group $[2, 878]$ (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1233541, -3721474, 6937796, -9762542, 11152518, -10659646, 8726846, -6131704, 3717519, -1926386, 842392, -301356, 84550, -16140, 1638, 184, 56, -54, 28, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 28*x^18 - 54*x^17 + 56*x^16 + 184*x^15 + 1638*x^14 - 16140*x^13 + 84550*x^12 - 301356*x^11 + 842392*x^10 - 1926386*x^9 + 3717519*x^8 - 6131704*x^7 + 8726846*x^6 - 10659646*x^5 + 11152518*x^4 - 9762542*x^3 + 6937796*x^2 - 3721474*x + 1233541)
 
gp: K = bnfinit(x^20 - 6*x^19 + 28*x^18 - 54*x^17 + 56*x^16 + 184*x^15 + 1638*x^14 - 16140*x^13 + 84550*x^12 - 301356*x^11 + 842392*x^10 - 1926386*x^9 + 3717519*x^8 - 6131704*x^7 + 8726846*x^6 - 10659646*x^5 + 11152518*x^4 - 9762542*x^3 + 6937796*x^2 - 3721474*x + 1233541, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 28 x^{18} - 54 x^{17} + 56 x^{16} + 184 x^{15} + 1638 x^{14} - 16140 x^{13} + 84550 x^{12} - 301356 x^{11} + 842392 x^{10} - 1926386 x^{9} + 3717519 x^{8} - 6131704 x^{7} + 8726846 x^{6} - 10659646 x^{5} + 11152518 x^{4} - 9762542 x^{3} + 6937796 x^{2} - 3721474 x + 1233541 \)

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:  \(12569066605710630176407970532818944=2^{30}\cdot 11^{18}\cdot 1451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
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:  $2, 11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{92311724094158979711149106269708729906316293518254134566591} a^{19} + \frac{25804702688606378118532774669790100625449870593022765156032}{92311724094158979711149106269708729906316293518254134566591} a^{18} - \frac{193631782671083300002167133612295222141195286176712365576}{92311724094158979711149106269708729906316293518254134566591} a^{17} - \frac{39620575110344484669015265921535416112558211624989165261793}{92311724094158979711149106269708729906316293518254134566591} a^{16} - \frac{19362379168750093178607202653036508014059754202471139641864}{92311724094158979711149106269708729906316293518254134566591} a^{15} + \frac{13006567712183861447723867477127722824987539510432149509531}{92311724094158979711149106269708729906316293518254134566591} a^{14} + \frac{9627220658052071376418177636753248515898205274750226527199}{92311724094158979711149106269708729906316293518254134566591} a^{13} + \frac{18718978028896175970922083445456594240183608047951967424623}{92311724094158979711149106269708729906316293518254134566591} a^{12} + \frac{16168877453439796444193690828522921920891107905536623026135}{92311724094158979711149106269708729906316293518254134566591} a^{11} - \frac{19779226401461038784079917628129141479406990398704970484087}{92311724094158979711149106269708729906316293518254134566591} a^{10} + \frac{33267931683385511913472723478133246958655452377144732964202}{92311724094158979711149106269708729906316293518254134566591} a^{9} - \frac{17042537655821918713731001209594682232102051821715791413549}{92311724094158979711149106269708729906316293518254134566591} a^{8} + \frac{34333279191668959235937513456155567811177672986357529855485}{92311724094158979711149106269708729906316293518254134566591} a^{7} + \frac{21350606699714095372863027459537878642824279086573479143489}{92311724094158979711149106269708729906316293518254134566591} a^{6} + \frac{31565282433241902939693797600061432250775087257828835646518}{92311724094158979711149106269708729906316293518254134566591} a^{5} + \frac{35931448362037599687440948272335459025771524027513674542323}{92311724094158979711149106269708729906316293518254134566591} a^{4} - \frac{41677772209322040921857334665780836530462537216431887434912}{92311724094158979711149106269708729906316293518254134566591} a^{3} - \frac{11514310973279953112134762461624130351111505422350470309070}{92311724094158979711149106269708729906316293518254134566591} a^{2} - \frac{22321588181515781684098981338708007146862388441850326526060}{92311724094158979711149106269708729906316293518254134566591} a - \frac{978637390964339981842130965315539607540917345745886448799}{2146784281259511156073235029528109997821309151587305455037}$

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

Class group and class number

$C_{2}\times C_{878}$, which has order $1756$ (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:  $9$
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:  \( 281202.490766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1451Data not computed