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Properties

Label 20.8.96757855803...6009.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{2}\cdot 401^{10}$
Root discriminant $22.35$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1, -14, 92, -385, 1053, -1746, 1529, 158, -3109, 5091, -4149, 2022, -579, -29, 107, -76, 49, -27, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 14*x^18 - 27*x^17 + 49*x^16 - 76*x^15 + 107*x^14 - 29*x^13 - 579*x^12 + 2022*x^11 - 4149*x^10 + 5091*x^9 - 3109*x^8 + 158*x^7 + 1529*x^6 - 1746*x^5 + 1053*x^4 - 385*x^3 + 92*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 14*x^18 - 27*x^17 + 49*x^16 - 76*x^15 + 107*x^14 - 29*x^13 - 579*x^12 + 2022*x^11 - 4149*x^10 + 5091*x^9 - 3109*x^8 + 158*x^7 + 1529*x^6 - 1746*x^5 + 1053*x^4 - 385*x^3 + 92*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 14 x^{18} - 27 x^{17} + 49 x^{16} - 76 x^{15} + 107 x^{14} - 29 x^{13} - 579 x^{12} + 2022 x^{11} - 4149 x^{10} + 5091 x^{9} - 3109 x^{8} + 158 x^{7} + 1529 x^{6} - 1746 x^{5} + 1053 x^{4} - 385 x^{3} + 92 x^{2} - 14 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(967578558036712773184836009=3^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.35$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{39} a^{18} - \frac{1}{13} a^{17} - \frac{16}{39} a^{16} + \frac{4}{13} a^{15} - \frac{14}{39} a^{14} + \frac{2}{39} a^{13} + \frac{19}{39} a^{12} - \frac{4}{13} a^{11} + \frac{14}{39} a^{10} - \frac{2}{13} a^{9} - \frac{3}{13} a^{8} + \frac{5}{39} a^{7} - \frac{14}{39} a^{6} + \frac{3}{13} a^{5} - \frac{8}{39} a^{4} + \frac{5}{39} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{4}{13}$, $\frac{1}{31688810243375490611289} a^{19} - \frac{404888461968177128648}{31688810243375490611289} a^{18} - \frac{2112172255678913671058}{31688810243375490611289} a^{17} - \frac{11372651827721735174593}{31688810243375490611289} a^{16} + \frac{6808890511999808202842}{31688810243375490611289} a^{15} + \frac{4561301354494198394662}{10562936747791830203763} a^{14} + \frac{3483697226199536154434}{31688810243375490611289} a^{13} - \frac{2245392574659056750419}{31688810243375490611289} a^{12} + \frac{354192544120805397643}{31688810243375490611289} a^{11} + \frac{6926981867929555193507}{31688810243375490611289} a^{10} - \frac{2230640057215406958209}{31688810243375490611289} a^{9} + \frac{7888187650651841363399}{31688810243375490611289} a^{8} - \frac{4620839967948447769391}{10562936747791830203763} a^{7} + \frac{5790323527124573156159}{31688810243375490611289} a^{6} + \frac{280456716463196112037}{812533595983986938751} a^{5} + \frac{1650337285306525121924}{10562936747791830203763} a^{4} - \frac{1332699048396247306046}{10562936747791830203763} a^{3} - \frac{17177103017821083004}{2437600787951960816253} a^{2} - \frac{1736408979582367717637}{3520978915930610067921} a - \frac{10874010590240132436116}{31688810243375490611289}$

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:  $13$
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:  \( 938008.452217 \) (assuming GRH)
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, 10.4.31105924806003.1, 10.4.77570884803.1, 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed