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Properties

Label 20.6.62230646487...3200.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 3^{16}\cdot 5^{2}\cdot 223^{5}$
Root discriminant $21.86$
Ramified primes $2, 3, 5, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T989

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -102, 44, -82, 159, 116, 85, 20, -216, 162, -321, 76, -18, 20, -61, 32, -19, 0, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 19*x^16 + 32*x^15 - 61*x^14 + 20*x^13 - 18*x^12 + 76*x^11 - 321*x^10 + 162*x^9 - 216*x^8 + 20*x^7 + 85*x^6 + 116*x^5 + 159*x^4 - 82*x^3 + 44*x^2 - 102*x + 13)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 - 19*x^16 + 32*x^15 - 61*x^14 + 20*x^13 - 18*x^12 + 76*x^11 - 321*x^10 + 162*x^9 - 216*x^8 + 20*x^7 + 85*x^6 + 116*x^5 + 159*x^4 - 82*x^3 + 44*x^2 - 102*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} - 19 x^{16} + 32 x^{15} - 61 x^{14} + 20 x^{13} - 18 x^{12} + 76 x^{11} - 321 x^{10} + 162 x^{9} - 216 x^{8} + 20 x^{7} + 85 x^{6} + 116 x^{5} + 159 x^{4} - 82 x^{3} + 44 x^{2} - 102 x + 13 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-622306464875034504147763200=-\,2^{20}\cdot 3^{16}\cdot 5^{2}\cdot 223^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.86$
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:  $2, 3, 5, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} - \frac{2}{5} a^{15} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{10} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{51075953029588169453880246605} a^{19} - \frac{217098559513357557673940711}{51075953029588169453880246605} a^{18} - \frac{2795066515171109997295717628}{10215190605917633890776049321} a^{17} - \frac{10374184583644622371398441372}{51075953029588169453880246605} a^{16} + \frac{4018961983386964553869265981}{10215190605917633890776049321} a^{15} - \frac{9080813218654586912389643532}{51075953029588169453880246605} a^{14} + \frac{12455768975850863395898356538}{51075953029588169453880246605} a^{13} + \frac{599443225958872227687501100}{10215190605917633890776049321} a^{12} + \frac{11698572969152126044869787208}{51075953029588169453880246605} a^{11} - \frac{3824109316853559267867542939}{10215190605917633890776049321} a^{10} - \frac{1088146912991514797957371865}{10215190605917633890776049321} a^{9} + \frac{13112377100615058261256554472}{51075953029588169453880246605} a^{8} + \frac{669045949318756536193161284}{10215190605917633890776049321} a^{7} + \frac{2318498735668060977138439304}{51075953029588169453880246605} a^{6} + \frac{15001993822429373083209145687}{51075953029588169453880246605} a^{5} + \frac{2848716311449341048841857639}{10215190605917633890776049321} a^{4} - \frac{3358477113558380660304464152}{51075953029588169453880246605} a^{3} + \frac{23789124772293500969523687242}{51075953029588169453880246605} a^{2} + \frac{12890955106555496577775683906}{51075953029588169453880246605} a + \frac{202370995951339362801090486}{785783892762894914675080717}$

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:  $12$
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:  \( 644232.370107 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T989:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1966080
The 265 conjugacy class representatives for t20n989 are not computed
Character table for t20n989 is not computed

Intermediate fields

5.3.18063.1, 10.6.1670512481280.1

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 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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
223Data not computed