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Properties

Label 20.0.29305967939...1417.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{32}\cdot 7^{7}\cdot 79^{7}$
Root discriminant $52.89$
Ramified primes $3, 7, 79$
Class number $128$ (GRH)
Class group $[4, 32]$ (GRH)
Galois group $C_2\times S_5$ (as 20T65)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25767, 14364, -64161, -31560, 73287, -372, -31431, 20268, 2865, -13061, 5682, -196, -1841, 2028, -424, -229, 210, -119, 35, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 35*x^18 - 119*x^17 + 210*x^16 - 229*x^15 - 424*x^14 + 2028*x^13 - 1841*x^12 - 196*x^11 + 5682*x^10 - 13061*x^9 + 2865*x^8 + 20268*x^7 - 31431*x^6 - 372*x^5 + 73287*x^4 - 31560*x^3 - 64161*x^2 + 14364*x + 25767)
 
gp: K = bnfinit(x^20 - 6*x^19 + 35*x^18 - 119*x^17 + 210*x^16 - 229*x^15 - 424*x^14 + 2028*x^13 - 1841*x^12 - 196*x^11 + 5682*x^10 - 13061*x^9 + 2865*x^8 + 20268*x^7 - 31431*x^6 - 372*x^5 + 73287*x^4 - 31560*x^3 - 64161*x^2 + 14364*x + 25767, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 35 x^{18} - 119 x^{17} + 210 x^{16} - 229 x^{15} - 424 x^{14} + 2028 x^{13} - 1841 x^{12} - 196 x^{11} + 5682 x^{10} - 13061 x^{9} + 2865 x^{8} + 20268 x^{7} - 31431 x^{6} - 372 x^{5} + 73287 x^{4} - 31560 x^{3} - 64161 x^{2} + 14364 x + 25767 \)

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:  \(29305967939737460044866286228851417=3^{32}\cdot 7^{7}\cdot 79^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.89$
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:  $3, 7, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{12} + \frac{1}{3} a^{6}$, $\frac{1}{2280611401809202657599654515508375855074139373941} a^{19} - \frac{199561131281032922887805981623348578865313690363}{2280611401809202657599654515508375855074139373941} a^{18} + \frac{331386538927995177130480113336311625681906102839}{2280611401809202657599654515508375855074139373941} a^{17} - \frac{363001634546358189568496038896931043680658672054}{2280611401809202657599654515508375855074139373941} a^{16} + \frac{186114927191714038930705428380183598777843896989}{2280611401809202657599654515508375855074139373941} a^{15} - \frac{45248468309981445689197217241019937773293058798}{2280611401809202657599654515508375855074139373941} a^{14} + \frac{456970055268811371421419611318318079452960895527}{2280611401809202657599654515508375855074139373941} a^{13} - \frac{94945348772360946207568410874916757687301936789}{2280611401809202657599654515508375855074139373941} a^{12} - \frac{791961041071992806722174970473313752832556934019}{2280611401809202657599654515508375855074139373941} a^{11} + \frac{706843956185232421616729231035728788260964428298}{2280611401809202657599654515508375855074139373941} a^{10} + \frac{795429758122408240783915704799308208014228521957}{2280611401809202657599654515508375855074139373941} a^{9} + \frac{114004121910517732862115481045156407746512789465}{2280611401809202657599654515508375855074139373941} a^{8} + \frac{204528105509216530545241006113394099094429400592}{760203800603067552533218171836125285024713124647} a^{7} - \frac{63628306832973025413821466706460023087438386521}{760203800603067552533218171836125285024713124647} a^{6} + \frac{352017895132469793117558855127427046013385446112}{760203800603067552533218171836125285024713124647} a^{5} - \frac{43696247965181198766091556730960298514732369342}{760203800603067552533218171836125285024713124647} a^{4} + \frac{105544446986968280193128406174418617756186719839}{760203800603067552533218171836125285024713124647} a^{3} + \frac{112223203818014417783769842278217282477669731619}{760203800603067552533218171836125285024713124647} a^{2} + \frac{123113027572580562868903401646620022402147581733}{760203800603067552533218171836125285024713124647} a - \frac{8230452132800132745156518886674942257810834922}{760203800603067552533218171836125285024713124647}$

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

Class group and class number

$C_{4}\times C_{32}$, which has order $128$ (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:  \( -1 \) (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:  \( 60094247.8731 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T65):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

10.10.7279733310365817.1

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 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
79Data not computed