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Properties

Label 20.4.38591891691...2096.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{65}\cdot 3^{21}$
Root discriminant $30.15$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, -120, 672, -2064, 3810, -4416, 2952, -384, -1494, 1632, -384, -864, 982, -272, -236, 208, -23, -44, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 28*x^18 - 44*x^17 - 23*x^16 + 208*x^15 - 236*x^14 - 272*x^13 + 982*x^12 - 864*x^11 - 384*x^10 + 1632*x^9 - 1494*x^8 - 384*x^7 + 2952*x^6 - 4416*x^5 + 3810*x^4 - 2064*x^3 + 672*x^2 - 120*x + 6)
 
gp: K = bnfinit(x^20 - 8*x^19 + 28*x^18 - 44*x^17 - 23*x^16 + 208*x^15 - 236*x^14 - 272*x^13 + 982*x^12 - 864*x^11 - 384*x^10 + 1632*x^9 - 1494*x^8 - 384*x^7 + 2952*x^6 - 4416*x^5 + 3810*x^4 - 2064*x^3 + 672*x^2 - 120*x + 6, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 28 x^{18} - 44 x^{17} - 23 x^{16} + 208 x^{15} - 236 x^{14} - 272 x^{13} + 982 x^{12} - 864 x^{11} - 384 x^{10} + 1632 x^{9} - 1494 x^{8} - 384 x^{7} + 2952 x^{6} - 4416 x^{5} + 3810 x^{4} - 2064 x^{3} + 672 x^{2} - 120 x + 6 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(385918916912697952676238852096=2^{65}\cdot 3^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.15$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12}$, $\frac{1}{15} a^{18} + \frac{1}{15} a^{17} + \frac{1}{15} a^{16} - \frac{1}{15} a^{15} + \frac{2}{15} a^{14} + \frac{2}{15} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{3311192432118285} a^{19} + \frac{43241447050337}{3311192432118285} a^{18} - \frac{140780243202206}{1103730810706095} a^{17} - \frac{99447214374055}{662238486423657} a^{16} + \frac{740957198319101}{3311192432118285} a^{15} + \frac{168558373166006}{367910270235365} a^{14} + \frac{1360746473578387}{3311192432118285} a^{13} + \frac{295517215233191}{3311192432118285} a^{12} + \frac{998436847969}{100339164609645} a^{11} - \frac{63846863712155}{220746162141219} a^{10} + \frac{31832693933022}{73582054047073} a^{9} - \frac{416389492457414}{1103730810706095} a^{8} - \frac{69148306688326}{220746162141219} a^{7} + \frac{521892570415493}{1103730810706095} a^{6} + \frac{195493091192348}{1103730810706095} a^{5} - \frac{15322495900553}{33446388203215} a^{4} - \frac{376464103488116}{1103730810706095} a^{3} - \frac{125550362608991}{367910270235365} a^{2} + \frac{1634814423428}{20067832921929} a + \frac{145202894903587}{1103730810706095}$

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:  $11$
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:  \( 39401688.4754 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.13824.1, 10.2.5283615080448.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.15.3$x^{12} - 3 x^{11} - 3 x^{9} - 3 x^{8} + 3 x^{7} - 3 x^{6} - 3 x^{4} - 3$$12$$1$$15$12T42$[3/2]_{4}^{6}$