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Properties

Label 20.0.11635480384...3641.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 17^{8}$
Root discriminant $14.23$
Ramified primes $3, 7, 17$
Class number $1$
Class group Trivial
Galois group $C_2^2\times D_5$ (as 20T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 3, -6, 8, -12, 24, -29, 36, -34, 16, 8, -14, 10, 1, -10, 9, -4, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 3 x^{18} - 4 x^{17} + 9 x^{16} - 10 x^{15} + x^{14} + 10 x^{13} - 14 x^{12} + 8 x^{11} + 16 x^{10} - 34 x^{9} + 36 x^{8} - 29 x^{7} + 24 x^{6} - 12 x^{5} + 8 x^{4} - 6 x^{3} + 3 x^{2} - x + 1 \)

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:  \(116354803848679984543641=3^{10}\cdot 7^{10}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.23$
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:  $3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{3}{13} a^{15} - \frac{2}{13} a^{14} - \frac{1}{13} a^{13} - \frac{3}{13} a^{12} - \frac{2}{13} a^{11} + \frac{1}{13} a^{10} - \frac{3}{13} a^{9} - \frac{3}{13} a^{8} - \frac{3}{13} a^{7} - \frac{4}{13} a^{6} - \frac{4}{13} a^{5} + \frac{6}{13} a^{4} + \frac{5}{13} a^{3} + \frac{1}{13} a^{2} + \frac{6}{13} a - \frac{2}{13}$, $\frac{1}{13} a^{17} + \frac{2}{13} a^{15} + \frac{6}{13} a^{14} - \frac{6}{13} a^{13} + \frac{2}{13} a^{12} - \frac{5}{13} a^{11} + \frac{1}{13} a^{9} + \frac{1}{13} a^{8} - \frac{3}{13} a^{6} - \frac{6}{13} a^{5} - \frac{3}{13} a^{4} + \frac{3}{13} a^{3} - \frac{4}{13} a^{2} + \frac{3}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{18} - \frac{1}{13} a^{15} - \frac{2}{13} a^{14} + \frac{4}{13} a^{13} + \frac{1}{13} a^{12} + \frac{4}{13} a^{11} - \frac{1}{13} a^{10} - \frac{6}{13} a^{9} + \frac{6}{13} a^{8} + \frac{3}{13} a^{7} + \frac{2}{13} a^{6} + \frac{5}{13} a^{5} + \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{1}{13} a^{2} - \frac{5}{13} a + \frac{4}{13}$, $\frac{1}{924443} a^{19} + \frac{25134}{924443} a^{18} - \frac{27874}{924443} a^{17} - \frac{12059}{924443} a^{16} + \frac{90230}{924443} a^{15} - \frac{187178}{924443} a^{14} - \frac{176292}{924443} a^{13} - \frac{2139}{71111} a^{12} + \frac{249890}{924443} a^{11} + \frac{23090}{54379} a^{10} - \frac{51290}{924443} a^{9} + \frac{2829}{71111} a^{8} + \frac{9571}{54379} a^{7} - \frac{125557}{924443} a^{6} + \frac{206561}{924443} a^{5} - \frac{130264}{924443} a^{4} + \frac{16853}{54379} a^{3} - \frac{346449}{924443} a^{2} + \frac{346771}{924443} a + \frac{280690}{924443}$

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:  \( -\frac{64590}{924443} a^{19} - \frac{367596}{924443} a^{18} + \frac{1415582}{924443} a^{17} - \frac{81126}{71111} a^{16} + \frac{1012970}{924443} a^{15} - \frac{3736306}{924443} a^{14} + \frac{343099}{71111} a^{13} + \frac{572491}{924443} a^{12} - \frac{5949199}{924443} a^{11} + \frac{375092}{54379} a^{10} - \frac{2302609}{924443} a^{9} - \frac{9563460}{924443} a^{8} + \frac{922391}{54379} a^{7} - \frac{12287346}{924443} a^{6} + \frac{6470820}{924443} a^{5} - \frac{6430639}{924443} a^{4} + \frac{178323}{54379} a^{3} - \frac{2699677}{924443} a^{2} + \frac{1670555}{924443} a - \frac{373205}{924443} \) (order $6$)
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:  \( 4706.41314047 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_5$ (as 20T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_2^2\times D_5$
Character table for $C_2^2\times D_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 5.1.14161.1, 10.0.48729742803.1, 10.2.341108199621.1, 10.0.1403737447.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
3Data not computed
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{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$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$