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Properties

Label 20.12.1802032470...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{31}$
Root discriminant $36.54$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:C_5:C_4$ (as 20T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -3840, 1280, 38400, 44160, -18432, -43200, -2880, 16320, 1440, -6176, -720, 2320, 480, -560, -48, 140, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 140*x^16 - 48*x^15 - 560*x^14 + 480*x^13 + 2320*x^12 - 720*x^11 - 6176*x^10 + 1440*x^9 + 16320*x^8 - 2880*x^7 - 43200*x^6 - 18432*x^5 + 44160*x^4 + 38400*x^3 + 1280*x^2 - 3840*x + 256)
 
gp: K = bnfinit(x^20 - 20*x^18 + 140*x^16 - 48*x^15 - 560*x^14 + 480*x^13 + 2320*x^12 - 720*x^11 - 6176*x^10 + 1440*x^9 + 16320*x^8 - 2880*x^7 - 43200*x^6 - 18432*x^5 + 44160*x^4 + 38400*x^3 + 1280*x^2 - 3840*x + 256, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} + 140 x^{16} - 48 x^{15} - 560 x^{14} + 480 x^{13} + 2320 x^{12} - 720 x^{11} - 6176 x^{10} + 1440 x^{9} + 16320 x^{8} - 2880 x^{7} - 43200 x^{6} - 18432 x^{5} + 44160 x^{4} + 38400 x^{3} + 1280 x^{2} - 3840 x + 256 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18020324707031250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.54$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{8} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{16} a^{10}$, $\frac{1}{16} a^{11}$, $\frac{1}{32} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{13}$, $\frac{1}{64} a^{14} - \frac{1}{4} a^{4}$, $\frac{1}{64} a^{15}$, $\frac{1}{128} a^{16} - \frac{1}{32} a^{11} - \frac{1}{8} a^{6} - \frac{1}{2} a$, $\frac{1}{128} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{18}$, $\frac{1}{324990005968846144256} a^{19} + \frac{9616785444701327}{20311875373052884016} a^{18} - \frac{264941252059345261}{162495002984423072128} a^{17} - \frac{2615678449206971}{20311875373052884016} a^{16} - \frac{699537243158731}{1846534124822989456} a^{15} + \frac{458705946485933453}{81247501492211536064} a^{14} - \frac{219502882414767819}{20311875373052884016} a^{13} - \frac{410997600895085839}{40623750746105768032} a^{12} - \frac{94271533411640479}{5077968843263221004} a^{11} - \frac{124793192001078039}{5077968843263221004} a^{10} - \frac{1144356063933232683}{20311875373052884016} a^{9} - \frac{137026061557324061}{5077968843263221004} a^{8} + \frac{558153458121354849}{10155937686526442008} a^{7} - \frac{193246019975728295}{5077968843263221004} a^{6} + \frac{12756903056264863}{230816765602873682} a^{5} - \frac{1211905193823683867}{5077968843263221004} a^{4} - \frac{320853319162872029}{2538984421631610502} a^{3} + \frac{299728489551298333}{2538984421631610502} a^{2} - \frac{12057241059092418}{66815379516621329} a - \frac{96186829822320940}{1269492210815805251}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 247750749.896 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5:C_4$ (as 20T88):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 11 conjugacy class representatives for $C_2^4:C_5:C_4$
Character table for $C_2^4:C_5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.11250000.1, 10.10.632812500000000.1

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed