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Properties

Label 20.4.31070022585...6289.1
Degree $20$
Signature $[4, 8]$
Discriminant $17^{2}\cdot 401^{10}$
Root discriminant $26.58$
Ramified primes $17, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 0, -208, 236, -444, 585, 355, -852, 1974, -2089, 366, -769, 1164, -256, 34, -114, 73, -13, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 - 13*x^17 + 73*x^16 - 114*x^15 + 34*x^14 - 256*x^13 + 1164*x^12 - 769*x^11 + 366*x^10 - 2089*x^9 + 1974*x^8 - 852*x^7 + 355*x^6 + 585*x^5 - 444*x^4 + 236*x^3 - 208*x^2 - 32)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 - 13*x^17 + 73*x^16 - 114*x^15 + 34*x^14 - 256*x^13 + 1164*x^12 - 769*x^11 + 366*x^10 - 2089*x^9 + 1974*x^8 - 852*x^7 + 355*x^6 + 585*x^5 - 444*x^4 + 236*x^3 - 208*x^2 - 32, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} - 13 x^{17} + 73 x^{16} - 114 x^{15} + 34 x^{14} - 256 x^{13} + 1164 x^{12} - 769 x^{11} + 366 x^{10} - 2089 x^{9} + 1974 x^{8} - 852 x^{7} + 355 x^{6} + 585 x^{5} - 444 x^{4} + 236 x^{3} - 208 x^{2} - 32 \)

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:  \(31070022585845554605601956289=17^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.58$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} - \frac{1}{18} a^{12} - \frac{1}{6} a^{10} + \frac{4}{9} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{5}{18} a^{2} + \frac{1}{6} a + \frac{2}{9}$, $\frac{1}{54} a^{15} + \frac{1}{54} a^{13} - \frac{1}{54} a^{12} - \frac{7}{54} a^{10} - \frac{7}{54} a^{9} + \frac{4}{9} a^{8} - \frac{7}{18} a^{7} - \frac{7}{54} a^{6} + \frac{17}{54} a^{5} - \frac{1}{3} a^{4} - \frac{8}{27} a^{3} + \frac{5}{54} a^{2} + \frac{7}{54} a - \frac{7}{27}$, $\frac{1}{162} a^{16} + \frac{1}{162} a^{15} + \frac{1}{162} a^{14} + \frac{1}{18} a^{13} + \frac{4}{81} a^{12} + \frac{11}{162} a^{11} - \frac{7}{81} a^{10} - \frac{55}{162} a^{9} + \frac{25}{54} a^{8} - \frac{1}{162} a^{7} - \frac{35}{162} a^{6} + \frac{53}{162} a^{5} + \frac{1}{81} a^{4} - \frac{28}{81} a^{3} - \frac{4}{27} a^{2} - \frac{25}{162} a + \frac{2}{81}$, $\frac{1}{972} a^{17} + \frac{1}{486} a^{16} + \frac{1}{486} a^{15} - \frac{17}{972} a^{14} - \frac{37}{972} a^{13} + \frac{23}{486} a^{12} - \frac{5}{162} a^{11} - \frac{25}{162} a^{10} + \frac{86}{243} a^{9} - \frac{385}{972} a^{8} - \frac{4}{27} a^{7} + \frac{41}{108} a^{6} - \frac{101}{243} a^{5} + \frac{7}{18} a^{4} + \frac{379}{972} a^{3} + \frac{437}{972} a^{2} - \frac{40}{81} a - \frac{80}{243}$, $\frac{1}{5832} a^{18} - \frac{1}{2916} a^{16} - \frac{7}{1944} a^{15} - \frac{1}{1944} a^{14} - \frac{7}{972} a^{13} - \frac{61}{2916} a^{12} + \frac{11}{162} a^{11} + \frac{40}{729} a^{10} - \frac{1073}{5832} a^{9} + \frac{1285}{2916} a^{8} + \frac{91}{648} a^{7} + \frac{401}{2916} a^{6} - \frac{473}{1458} a^{5} - \frac{1673}{5832} a^{4} - \frac{161}{1944} a^{3} + \frac{107}{1458} a^{2} - \frac{245}{1458} a - \frac{244}{729}$, $\frac{1}{12181979495289706416} a^{19} - \frac{19873622522255}{761373718455606651} a^{18} - \frac{267715047720421}{6090989747644853208} a^{17} + \frac{31844704154158003}{12181979495289706416} a^{16} - \frac{970388910835265}{4060659831763235472} a^{15} + \frac{14550752098615591}{676776638627205912} a^{14} - \frac{236755989844223293}{6090989747644853208} a^{13} - \frac{22617078797897291}{761373718455606651} a^{12} - \frac{176181129599015719}{3045494873822426604} a^{11} + \frac{583528846104401389}{4060659831763235472} a^{10} - \frac{191167768057313545}{2030329915881617736} a^{9} + \frac{148429346562291515}{12181979495289706416} a^{8} + \frac{2946889263121748825}{6090989747644853208} a^{7} + \frac{139977671705693173}{1522747436911213302} a^{6} - \frac{20058736493706157}{12181979495289706416} a^{5} - \frac{186409587543939347}{12181979495289706416} a^{4} + \frac{620137060439854349}{3045494873822426604} a^{3} - \frac{81702062091144391}{253791239485202217} a^{2} + \frac{9350691897055841}{761373718455606651} a + \frac{192050876685495109}{761373718455606651}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2562961.55608 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.2.176266907234017.1, 10.2.439568347217.1, 10.10.10368641602001.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed