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Properties

Label 20.8.31427919143...8329.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{6}\cdot 401^{11}$
Root discriminant $37.57$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -90, -252, 39, 1562, 2630, -2788, -6094, 4717, 2741, 1183, 411, -1966, 561, 60, -90, 97, -72, 19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 19*x^18 - 72*x^17 + 97*x^16 - 90*x^15 + 60*x^14 + 561*x^13 - 1966*x^12 + 411*x^11 + 1183*x^10 + 2741*x^9 + 4717*x^8 - 6094*x^7 - 2788*x^6 + 2630*x^5 + 1562*x^4 + 39*x^3 - 252*x^2 - 90*x - 9)
 
gp: K = bnfinit(x^20 - 4*x^19 + 19*x^18 - 72*x^17 + 97*x^16 - 90*x^15 + 60*x^14 + 561*x^13 - 1966*x^12 + 411*x^11 + 1183*x^10 + 2741*x^9 + 4717*x^8 - 6094*x^7 - 2788*x^6 + 2630*x^5 + 1562*x^4 + 39*x^3 - 252*x^2 - 90*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 19 x^{18} - 72 x^{17} + 97 x^{16} - 90 x^{15} + 60 x^{14} + 561 x^{13} - 1966 x^{12} + 411 x^{11} + 1183 x^{10} + 2741 x^{9} + 4717 x^{8} - 6094 x^{7} - 2788 x^{6} + 2630 x^{5} + 1562 x^{4} + 39 x^{3} - 252 x^{2} - 90 x - 9 \)

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:  \(31427919143590467585816658408329=3^{6}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.57$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} + \frac{1}{3} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} + \frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{6} a^{13} + \frac{1}{12} a^{11} - \frac{1}{2} a^{10} + \frac{5}{12} a^{9} + \frac{1}{12} a^{8} + \frac{5}{12} a^{7} - \frac{1}{6} a^{6} + \frac{5}{12} a^{5} - \frac{1}{12} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4201128} a^{18} + \frac{7069}{175047} a^{17} + \frac{1747}{175047} a^{16} + \frac{9457}{175047} a^{15} + \frac{338113}{4201128} a^{14} - \frac{170023}{2100564} a^{13} - \frac{18855}{466792} a^{12} - \frac{1888057}{4201128} a^{11} + \frac{2021515}{4201128} a^{10} + \frac{437593}{2100564} a^{9} + \frac{833803}{2100564} a^{8} - \frac{1089577}{4201128} a^{7} - \frac{58235}{221112} a^{6} - \frac{2006207}{4201128} a^{5} - \frac{336781}{4201128} a^{4} + \frac{52685}{1400376} a^{3} - \frac{290585}{1400376} a^{2} - \frac{62857}{233396} a - \frac{134851}{466792}$, $\frac{1}{37398337199953819846655247504} a^{19} - \frac{614724550690854498659}{37398337199953819846655247504} a^{18} - \frac{99331840544728400651958439}{3116528099996151653887937292} a^{17} + \frac{5294695549746569271029311}{84230489189085179834809116} a^{16} + \frac{2333259865635148280343317701}{37398337199953819846655247504} a^{15} + \frac{207658106697262227866357627}{37398337199953819846655247504} a^{14} - \frac{2156767936153235854245181325}{37398337199953819846655247504} a^{13} - \frac{582846128276369044288499329}{9349584299988454961663811876} a^{12} + \frac{1721116061105167444595196439}{6233056199992303307775874584} a^{11} + \frac{1777096331215394547559557121}{4155370799994868871850583056} a^{10} - \frac{4517830263194665269223630805}{9349584299988454961663811876} a^{9} - \frac{8109006021199511003940257063}{37398337199953819846655247504} a^{8} + \frac{152005969912795027233543573}{692561799999144811975097176} a^{7} + \frac{2012506249351031913454537781}{9349584299988454961663811876} a^{6} - \frac{1397119333646769140137241977}{3116528099996151653887937292} a^{5} + \frac{8490156502964203532539717459}{18699168599976909923327623752} a^{4} + \frac{1512836316503025011902889263}{3116528099996151653887937292} a^{3} - \frac{1835480680795083275942369945}{4155370799994868871850583056} a^{2} - \frac{490403400348375220503586739}{1385123599998289623950194352} a + \frac{1069916919427520660318330741}{4155370799994868871850583056}$

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:  \( 254264402.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

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 $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

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

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.2.0.1$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}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed