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Properties

Label 20.0.46098141783...3549.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $24.16$
Ramified primes $3, 61, 397$
Class number $1$
Class group Trivial
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 35, -54, 75, -123, 14, 166, -62, 477, -216, 38, -66, -248, 268, -211, 162, -74, 29, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 29*x^18 - 74*x^17 + 162*x^16 - 211*x^15 + 268*x^14 - 248*x^13 - 66*x^12 + 38*x^11 - 216*x^10 + 477*x^9 - 62*x^8 + 166*x^7 + 14*x^6 - 123*x^5 + 75*x^4 - 54*x^3 + 35*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 29*x^18 - 74*x^17 + 162*x^16 - 211*x^15 + 268*x^14 - 248*x^13 - 66*x^12 + 38*x^11 - 216*x^10 + 477*x^9 - 62*x^8 + 166*x^7 + 14*x^6 - 123*x^5 + 75*x^4 - 54*x^3 + 35*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 29 x^{18} - 74 x^{17} + 162 x^{16} - 211 x^{15} + 268 x^{14} - 248 x^{13} - 66 x^{12} + 38 x^{11} - 216 x^{10} + 477 x^{9} - 62 x^{8} + 166 x^{7} + 14 x^{6} - 123 x^{5} + 75 x^{4} - 54 x^{3} + 35 x^{2} - 5 x + 1 \)

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:  \(4609814178311274644985033549=3^{10}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.16$
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:  $3, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{544364391359875372262509} a^{19} + \frac{81521633813561959436245}{544364391359875372262509} a^{18} + \frac{78646525732615514573615}{544364391359875372262509} a^{17} + \frac{85250134461082951717455}{544364391359875372262509} a^{16} - \frac{225187415063332318425113}{544364391359875372262509} a^{15} + \frac{15923660631916102937155}{41874183950759644020193} a^{14} - \frac{261131489285514460060752}{544364391359875372262509} a^{13} - \frac{59453681595430607189890}{544364391359875372262509} a^{12} + \frac{116210929573842724690345}{544364391359875372262509} a^{11} - \frac{184916868654636457030021}{544364391359875372262509} a^{10} - \frac{251638494485469311419859}{544364391359875372262509} a^{9} + \frac{22927920005323292330740}{544364391359875372262509} a^{8} + \frac{29627206397249711435294}{544364391359875372262509} a^{7} + \frac{67421331342344256232435}{544364391359875372262509} a^{6} - \frac{47285986960020336015061}{544364391359875372262509} a^{5} - \frac{143751247034046120471950}{544364391359875372262509} a^{4} + \frac{13401104351022184108501}{41874183950759644020193} a^{3} + \frac{80494551292274121377789}{544364391359875372262509} a^{2} + \frac{160105350835215309705795}{544364391359875372262509} a + \frac{29143534409606058771502}{544364391359875372262509}$

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:  \( -\frac{14730284694584}{323387475417031} a^{19} + \frac{112421076038934}{323387475417031} a^{18} - \frac{399961828702114}{323387475417031} a^{17} + \frac{1072219316949963}{323387475417031} a^{16} - \frac{2522015813003376}{323387475417031} a^{15} + \frac{3610773478163458}{323387475417031} a^{14} - \frac{5753995167439339}{323387475417031} a^{13} + \frac{6159160950924071}{323387475417031} a^{12} - \frac{1763619871935758}{323387475417031} a^{11} + \frac{3223862078233971}{323387475417031} a^{10} + \frac{5375148533488920}{323387475417031} a^{9} - \frac{10608433072122575}{323387475417031} a^{8} + \frac{1410871577934436}{323387475417031} a^{7} - \frac{9817338600889712}{323387475417031} a^{6} + \frac{2003612791308047}{323387475417031} a^{5} + \frac{4100260810048387}{323387475417031} a^{4} - \frac{1250797725403599}{323387475417031} a^{3} + \frac{2679639770228372}{323387475417031} a^{2} - \frac{1889372947102143}{323387475417031} a + \frac{347686673457693}{323387475417031} \) (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:  \( 756145.319367 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1, 5.5.24217.1, 10.0.142510530627.1

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 24 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 $20$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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}$
61Data not computed
397Data not computed