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Properties

Label 20.0.26745536440...8049.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 79^{10}$
Root discriminant $23.52$
Ramified primes $7, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6399, -25596, 32929, -2776, -23849, 11228, 9220, -12332, 5829, -746, 591, -2840, 3808, -2570, 917, -100, -45, 8, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 + 8*x^17 - 45*x^16 - 100*x^15 + 917*x^14 - 2570*x^13 + 3808*x^12 - 2840*x^11 + 591*x^10 - 746*x^9 + 5829*x^8 - 12332*x^7 + 9220*x^6 + 11228*x^5 - 23849*x^4 - 2776*x^3 + 32929*x^2 - 25596*x + 6399)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 + 8*x^17 - 45*x^16 - 100*x^15 + 917*x^14 - 2570*x^13 + 3808*x^12 - 2840*x^11 + 591*x^10 - 746*x^9 + 5829*x^8 - 12332*x^7 + 9220*x^6 + 11228*x^5 - 23849*x^4 - 2776*x^3 + 32929*x^2 - 25596*x + 6399, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} + 8 x^{17} - 45 x^{16} - 100 x^{15} + 917 x^{14} - 2570 x^{13} + 3808 x^{12} - 2840 x^{11} + 591 x^{10} - 746 x^{9} + 5829 x^{8} - 12332 x^{7} + 9220 x^{6} + 11228 x^{5} - 23849 x^{4} - 2776 x^{3} + 32929 x^{2} - 25596 x + 6399 \)

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:  \(2674553644040763237187428049=7^{10}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.52$
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:  $7, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} + \frac{4}{9} a^{6} - \frac{1}{6} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{18} a - \frac{1}{2}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{4}{9} a^{7} - \frac{1}{6} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{18} a^{2} + \frac{1}{6} a$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{13} - \frac{1}{18} a^{11} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} + \frac{4}{9} a^{8} + \frac{1}{6} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{1}{2}$, $\frac{1}{18} a^{17} + \frac{1}{18} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{9} a - \frac{1}{2}$, $\frac{1}{9666} a^{18} + \frac{59}{3222} a^{17} + \frac{7}{3222} a^{16} + \frac{28}{4833} a^{15} - \frac{53}{3222} a^{14} - \frac{43}{4833} a^{13} + \frac{575}{9666} a^{12} + \frac{509}{9666} a^{11} - \frac{151}{3222} a^{10} - \frac{227}{3222} a^{9} + \frac{151}{3222} a^{8} - \frac{1837}{4833} a^{7} + \frac{10}{179} a^{6} - \frac{4525}{9666} a^{5} - \frac{1487}{9666} a^{4} - \frac{1169}{9666} a^{3} - \frac{4813}{9666} a^{2} + \frac{51}{179} a + \frac{68}{179}$, $\frac{1}{127358559100519806572094091084794} a^{19} - \frac{274653970278454825226835233}{63679279550259903286047045542397} a^{18} + \frac{483993171998928578420856828911}{42452853033506602190698030361598} a^{17} + \frac{681302408209197018367194056887}{63679279550259903286047045542397} a^{16} - \frac{756324361347671466785796609661}{63679279550259903286047045542397} a^{15} + \frac{1037213455187128108420021215265}{127358559100519806572094091084794} a^{14} + \frac{5850016112149044755428445905771}{127358559100519806572094091084794} a^{13} + \frac{11399483896402649305802213135}{237166776723500570897754359562} a^{12} - \frac{335898739302581020757647957567}{63679279550259903286047045542397} a^{11} + \frac{287456657623649557335141483940}{21226426516753301095349015180799} a^{10} - \frac{5586581857783277744659184609891}{42452853033506602190698030361598} a^{9} - \frac{12649405287158417567786343607415}{127358559100519806572094091084794} a^{8} - \frac{17297321806623373840490484881}{1480913477913021006652256873079} a^{7} + \frac{19779173739901026567552472801979}{127358559100519806572094091084794} a^{6} + \frac{55395571731965834957843625637589}{127358559100519806572094091084794} a^{5} - \frac{542808143713536112967262837332}{21226426516753301095349015180799} a^{4} - \frac{21873548935795472010991633048858}{63679279550259903286047045542397} a^{3} + \frac{25596156794637752065983000850067}{63679279550259903286047045542397} a^{2} + \frac{975539786740563407234198820731}{42452853033506602190698030361598} a - \frac{1105397033344570818080069295288}{2358491835194811232816557242311}$

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:  $9$
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:  \( 561201.28151 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-79}) \), \(\Q(\sqrt{553}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{-79})\), 5.1.6241.1 x5, 10.0.3077056399.1, 10.2.51716086897993.1 x5, 10.0.654634011367.1 x5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$