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Properties

Label 20.4.18660339843...5168.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{15}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $18.35$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4\times D_5$ (as 20T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -52, 151, -304, 634, -1104, 1413, -1585, 1523, -884, 34, 478, -614, 530, -348, 178, -74, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 74*x^17 + 178*x^16 - 348*x^15 + 530*x^14 - 614*x^13 + 478*x^12 + 34*x^11 - 884*x^10 + 1523*x^9 - 1585*x^8 + 1413*x^7 - 1104*x^6 + 634*x^5 - 304*x^4 + 151*x^3 - 52*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 74*x^17 + 178*x^16 - 348*x^15 + 530*x^14 - 614*x^13 + 478*x^12 + 34*x^11 - 884*x^10 + 1523*x^9 - 1585*x^8 + 1413*x^7 - 1104*x^6 + 634*x^5 - 304*x^4 + 151*x^3 - 52*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 26 x^{18} - 74 x^{17} + 178 x^{16} - 348 x^{15} + 530 x^{14} - 614 x^{13} + 478 x^{12} + 34 x^{11} - 884 x^{10} + 1523 x^{9} - 1585 x^{8} + 1413 x^{7} - 1104 x^{6} + 634 x^{5} - 304 x^{4} + 151 x^{3} - 52 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18660339843459071567495168=2^{15}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.35$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} + \frac{1}{17} a^{14} - \frac{6}{17} a^{13} - \frac{5}{17} a^{12} - \frac{7}{17} a^{11} + \frac{1}{17} a^{10} - \frac{7}{17} a^{9} - \frac{2}{17} a^{8} + \frac{5}{17} a^{7} + \frac{1}{17} a^{6} - \frac{4}{17} a^{5} - \frac{3}{17} a^{4} - \frac{2}{17} a^{3} + \frac{3}{17} a^{2} + \frac{6}{17} a - \frac{3}{17}$, $\frac{1}{17} a^{16} - \frac{7}{17} a^{14} + \frac{1}{17} a^{13} - \frac{2}{17} a^{12} + \frac{8}{17} a^{11} - \frac{8}{17} a^{10} + \frac{5}{17} a^{9} + \frac{7}{17} a^{8} - \frac{4}{17} a^{7} - \frac{5}{17} a^{6} + \frac{1}{17} a^{5} + \frac{1}{17} a^{4} + \frac{5}{17} a^{3} + \frac{3}{17} a^{2} + \frac{8}{17} a + \frac{3}{17}$, $\frac{1}{17} a^{17} + \frac{8}{17} a^{14} + \frac{7}{17} a^{13} + \frac{7}{17} a^{12} - \frac{6}{17} a^{11} - \frac{5}{17} a^{10} - \frac{8}{17} a^{9} - \frac{1}{17} a^{8} - \frac{4}{17} a^{7} + \frac{8}{17} a^{6} + \frac{7}{17} a^{5} + \frac{1}{17} a^{4} + \frac{6}{17} a^{3} - \frac{5}{17} a^{2} - \frac{6}{17} a - \frac{4}{17}$, $\frac{1}{221} a^{18} - \frac{2}{221} a^{17} + \frac{2}{221} a^{16} + \frac{3}{221} a^{15} + \frac{74}{221} a^{14} + \frac{110}{221} a^{13} + \frac{18}{221} a^{12} - \frac{10}{221} a^{11} - \frac{19}{221} a^{10} - \frac{25}{221} a^{9} - \frac{80}{221} a^{8} + \frac{1}{13} a^{7} + \frac{10}{221} a^{6} + \frac{94}{221} a^{5} - \frac{98}{221} a^{4} + \frac{37}{221} a^{3} - \frac{73}{221} a^{2} - \frac{108}{221} a - \frac{90}{221}$, $\frac{1}{782824606809011} a^{19} - \frac{1018476930953}{782824606809011} a^{18} - \frac{611394340001}{60217277446847} a^{17} + \frac{7107514275516}{782824606809011} a^{16} - \frac{8446981831549}{782824606809011} a^{15} - \frac{8980566849915}{782824606809011} a^{14} + \frac{344677473256283}{782824606809011} a^{13} + \frac{274196498828351}{782824606809011} a^{12} - \frac{161836264615106}{782824606809011} a^{11} - \frac{181546272156228}{782824606809011} a^{10} - \frac{324595565252921}{782824606809011} a^{9} + \frac{237451997031771}{782824606809011} a^{8} - \frac{5786064927165}{46048506282883} a^{7} - \frac{14743135937030}{60217277446847} a^{6} - \frac{340711880990602}{782824606809011} a^{5} + \frac{140489920747219}{782824606809011} a^{4} + \frac{60851640216591}{782824606809011} a^{3} - \frac{280153782178945}{782824606809011} a^{2} - \frac{244092559047262}{782824606809011} a + \frac{176717214229440}{782824606809011}$

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:  \( 49924.022929 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times D_5$ (as 20T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $D_4\times D_5$
Character table for $D_4\times D_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.113288.1, 5.1.14161.1, 10.2.3409076657.1

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 $20$ $20$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$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}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$