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Properties

Label 20.0.38082326211...6432.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 7^{8}\cdot 17^{10}$
Root discriminant $15.10$
Ramified primes $2, 7, 17$
Class number $1$
Class group Trivial
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, -12, 60, -164, 275, -318, 336, -454, 650, -748, 672, -506, 339, -194, 98, -58, 35, -16, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 35*x^16 - 58*x^15 + 98*x^14 - 194*x^13 + 339*x^12 - 506*x^11 + 672*x^10 - 748*x^9 + 650*x^8 - 454*x^7 + 336*x^6 - 318*x^5 + 275*x^4 - 164*x^3 + 60*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 35*x^16 - 58*x^15 + 98*x^14 - 194*x^13 + 339*x^12 - 506*x^11 + 672*x^10 - 748*x^9 + 650*x^8 - 454*x^7 + 336*x^6 - 318*x^5 + 275*x^4 - 164*x^3 + 60*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 16 x^{17} + 35 x^{16} - 58 x^{15} + 98 x^{14} - 194 x^{13} + 339 x^{12} - 506 x^{11} + 672 x^{10} - 748 x^{9} + 650 x^{8} - 454 x^{7} + 336 x^{6} - 318 x^{5} + 275 x^{4} - 164 x^{3} + 60 x^{2} - 12 x + 1 \)

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:  \(380823262111409623826432=2^{15}\cdot 7^{8}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{15} - \frac{3}{28} a^{14} - \frac{1}{14} a^{13} - \frac{3}{28} a^{12} + \frac{5}{28} a^{11} + \frac{1}{7} a^{10} - \frac{5}{14} a^{9} - \frac{3}{28} a^{8} - \frac{1}{2} a^{7} - \frac{3}{7} a^{6} - \frac{1}{14} a^{5} + \frac{5}{28} a^{4} + \frac{5}{14} a^{3} + \frac{3}{28} a^{2} + \frac{9}{28} a + \frac{13}{28}$, $\frac{1}{364} a^{16} + \frac{3}{182} a^{15} - \frac{85}{364} a^{14} + \frac{7}{52} a^{13} + \frac{17}{182} a^{12} - \frac{11}{52} a^{11} + \frac{17}{91} a^{10} - \frac{51}{364} a^{9} - \frac{139}{364} a^{8} - \frac{83}{182} a^{7} - \frac{31}{91} a^{6} - \frac{83}{364} a^{5} - \frac{1}{364} a^{4} - \frac{173}{364} a^{3} - \frac{87}{182} a^{2} + \frac{75}{182} a - \frac{149}{364}$, $\frac{1}{364} a^{17} - \frac{1}{91} a^{15} + \frac{1}{14} a^{14} + \frac{1}{7} a^{13} - \frac{43}{182} a^{12} + \frac{23}{364} a^{11} + \frac{9}{364} a^{10} + \frac{89}{364} a^{9} + \frac{135}{364} a^{8} + \frac{36}{91} a^{7} - \frac{15}{364} a^{6} + \frac{81}{364} a^{5} - \frac{32}{91} a^{4} - \frac{75}{182} a^{3} - \frac{93}{364} a^{2} - \frac{89}{182} a + \frac{7}{52}$, $\frac{1}{364} a^{18} - \frac{1}{182} a^{15} + \frac{25}{182} a^{14} + \frac{8}{91} a^{13} - \frac{7}{52} a^{12} - \frac{1}{28} a^{11} - \frac{29}{364} a^{10} - \frac{95}{364} a^{9} + \frac{27}{91} a^{8} + \frac{7}{52} a^{7} - \frac{155}{364} a^{6} - \frac{87}{182} a^{5} - \frac{25}{182} a^{4} - \frac{31}{364} a^{3} + \frac{31}{182} a^{2} + \frac{181}{364} a + \frac{1}{182}$, $\frac{1}{364} a^{19} - \frac{3}{364} a^{15} + \frac{57}{364} a^{14} - \frac{3}{364} a^{13} + \frac{17}{91} a^{12} + \frac{19}{182} a^{11} - \frac{37}{364} a^{10} + \frac{55}{182} a^{9} + \frac{37}{91} a^{8} + \frac{59}{364} a^{7} - \frac{3}{182} a^{6} + \frac{24}{91} a^{5} - \frac{44}{91} a^{4} - \frac{6}{91} a^{3} - \frac{45}{91} a^{2} + \frac{81}{364} a - \frac{51}{364}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2862.69597498 \)
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

\(\Q(\sqrt{17}) \), 4.0.2312.1, 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 ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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}$$[3]^{5}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_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$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$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$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$