↑

Properties

Label 20.0.22433904799...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{5}\cdot 401^{8}$
Root discriminant $46.51$
Ramified primes $2, 5, 401$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_4\times D_5$ (as 20T21)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -16, -136, 405, 478, -1796, -1252, 4444, 2708, -4608, -2670, 3039, 1552, -1048, -412, 247, 40, -20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 20*x^18 + 40*x^17 + 247*x^16 - 412*x^15 - 1048*x^14 + 1552*x^13 + 3039*x^12 - 2670*x^11 - 4608*x^10 + 2708*x^9 + 4444*x^8 - 1252*x^7 - 1796*x^6 + 478*x^5 + 405*x^4 - 136*x^3 - 16*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 20*x^18 + 40*x^17 + 247*x^16 - 412*x^15 - 1048*x^14 + 1552*x^13 + 3039*x^12 - 2670*x^11 - 4608*x^10 + 2708*x^9 + 4444*x^8 - 1252*x^7 - 1796*x^6 + 478*x^5 + 405*x^4 - 136*x^3 - 16*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 20 x^{18} + 40 x^{17} + 247 x^{16} - 412 x^{15} - 1048 x^{14} + 1552 x^{13} + 3039 x^{12} - 2670 x^{11} - 4608 x^{10} + 2708 x^{9} + 4444 x^{8} - 1252 x^{7} - 1796 x^{6} + 478 x^{5} + 405 x^{4} - 136 x^{3} - 16 x^{2} + 6 x + 1 \)

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:  \(2243390479903105373955109683200000=2^{30}\cdot 5^{5}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.51$
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:  $2, 5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{718118441} a^{18} - \frac{300283699}{718118441} a^{17} + \frac{144942748}{718118441} a^{16} - \frac{200195267}{718118441} a^{15} + \frac{248425198}{718118441} a^{14} - \frac{13507398}{718118441} a^{13} + \frac{267746316}{718118441} a^{12} - \frac{134665694}{718118441} a^{11} - \frac{292571811}{718118441} a^{10} - \frac{125664316}{718118441} a^{9} + \frac{111949152}{718118441} a^{8} - \frac{140894223}{718118441} a^{7} + \frac{315291970}{718118441} a^{6} + \frac{176751592}{718118441} a^{5} + \frac{114336927}{718118441} a^{4} + \frac{264959620}{718118441} a^{3} - \frac{194027970}{718118441} a^{2} + \frac{150315930}{718118441} a - \frac{298511441}{718118441}$, $\frac{1}{217149815169654844769} a^{19} - \frac{46150269302}{217149815169654844769} a^{18} - \frac{86821539462652254666}{217149815169654844769} a^{17} + \frac{59534038269066024595}{217149815169654844769} a^{16} - \frac{41310317007674331842}{217149815169654844769} a^{15} + \frac{15835002835756491889}{217149815169654844769} a^{14} + \frac{244966185457666317}{3680505341858556691} a^{13} - \frac{17405651776310195994}{217149815169654844769} a^{12} + \frac{49125672354581811610}{217149815169654844769} a^{11} + \frac{91363277043324401185}{217149815169654844769} a^{10} + \frac{80336313548446782556}{217149815169654844769} a^{9} - \frac{1426467870647714524}{12773518539391461457} a^{8} + \frac{68073236367966229379}{217149815169654844769} a^{7} - \frac{27501861943946235847}{217149815169654844769} a^{6} + \frac{73215397969135370418}{217149815169654844769} a^{5} - \frac{68289544919741351746}{217149815169654844769} a^{4} - \frac{63819302535866507670}{217149815169654844769} a^{3} + \frac{76873908335064564990}{217149815169654844769} a^{2} + \frac{53887273218792498827}{217149815169654844769} a + \frac{52286769722701206854}{217149815169654844769}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}$, which has order $5$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{218810443}{718118441} a^{19} - \frac{290127342}{718118441} a^{18} - \frac{4647839374}{718118441} a^{17} + \frac{5758658434}{718118441} a^{16} + \frac{59466283737}{718118441} a^{15} - \frac{52831506054}{718118441} a^{14} - \frac{284054379535}{718118441} a^{13} + \frac{175899403202}{718118441} a^{12} + \frac{866407340638}{718118441} a^{11} - \frac{100841695060}{718118441} a^{10} - \frac{1318291791828}{718118441} a^{9} - \frac{147612079192}{718118441} a^{8} + \frac{1232468388953}{718118441} a^{7} + \frac{438052862088}{718118441} a^{6} - \frac{433042528442}{718118441} a^{5} - \frac{178007687462}{718118441} a^{4} + \frac{86984200700}{718118441} a^{3} + \frac{28201762326}{718118441} a^{2} - \frac{8248433487}{718118441} a - \frac{1540727903}{718118441} \) (order $4$)
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:  \( 85215123.6135 \) (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

\(\Q(\sqrt{-1}) \), 4.0.320.1, 5.5.160801.1, 10.0.26477528679424.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$5$5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed