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Properties

Label 20.0.16706271954...6304.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}\cdot 23^{4}$
Root discriminant $45.83$
Ramified primes $2, 11, 23$
Class number $48$ (GRH)
Class group $[2, 2, 12]$ (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5039, 37372, 56910, -97790, -119520, -287708, 836746, -566126, 288907, -242876, 140920, -63948, 31178, -12254, 4372, -1328, 200, 0, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 + 200*x^16 - 1328*x^15 + 4372*x^14 - 12254*x^13 + 31178*x^12 - 63948*x^11 + 140920*x^10 - 242876*x^9 + 288907*x^8 - 566126*x^7 + 836746*x^6 - 287708*x^5 - 119520*x^4 - 97790*x^3 + 56910*x^2 + 37372*x + 5039)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 + 200*x^16 - 1328*x^15 + 4372*x^14 - 12254*x^13 + 31178*x^12 - 63948*x^11 + 140920*x^10 - 242876*x^9 + 288907*x^8 - 566126*x^7 + 836746*x^6 - 287708*x^5 - 119520*x^4 - 97790*x^3 + 56910*x^2 + 37372*x + 5039, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} + 200 x^{16} - 1328 x^{15} + 4372 x^{14} - 12254 x^{13} + 31178 x^{12} - 63948 x^{11} + 140920 x^{10} - 242876 x^{9} + 288907 x^{8} - 566126 x^{7} + 836746 x^{6} - 287708 x^{5} - 119520 x^{4} - 97790 x^{3} + 56910 x^{2} + 37372 x + 5039 \)

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:  \(1670627195488492909044967149666304=2^{30}\cdot 11^{18}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.83$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{989} a^{18} - \frac{357}{989} a^{17} - \frac{225}{989} a^{16} + \frac{297}{989} a^{15} + \frac{296}{989} a^{14} - \frac{485}{989} a^{13} - \frac{306}{989} a^{12} + \frac{75}{989} a^{11} + \frac{175}{989} a^{10} - \frac{484}{989} a^{9} + \frac{47}{989} a^{8} + \frac{233}{989} a^{7} + \frac{19}{989} a^{6} + \frac{421}{989} a^{5} + \frac{277}{989} a^{4} + \frac{103}{989} a^{3} + \frac{54}{989} a^{2} + \frac{49}{989} a - \frac{486}{989}$, $\frac{1}{334951962053401944008345505787909350881871393929488541109551} a^{19} - \frac{2164188461652669223960445486486360916958546625583628304}{4999283015722417074751425459521035087789125282529679718053} a^{18} + \frac{46163782783807997193362980629991397072858200615448234631374}{334951962053401944008345505787909350881871393929488541109551} a^{17} + \frac{3841683538167411544580972256603434771457285437412126109458}{7789580512869812651356872227625798857717939393709035839757} a^{16} + \frac{59967950460947093249196932958312295090344280477042435409678}{334951962053401944008345505787909350881871393929488541109551} a^{15} - \frac{45967547097668209241038123200605799644262935424852399104161}{334951962053401944008345505787909350881871393929488541109551} a^{14} - \frac{6577770640750408146027330637201180554303706240735968260821}{334951962053401944008345505787909350881871393929488541109551} a^{13} - \frac{128874123716253922574204215433569294188573487438878839930827}{334951962053401944008345505787909350881871393929488541109551} a^{12} + \frac{141250184245493059919020781975665193361269542667876998045554}{334951962053401944008345505787909350881871393929488541109551} a^{11} - \frac{35401327078906496244477352412348213704054686543256619037324}{334951962053401944008345505787909350881871393929488541109551} a^{10} - \frac{7730439364789811954901342215733531218408249519243152448356}{334951962053401944008345505787909350881871393929488541109551} a^{9} - \frac{141866547090551098546385148771971015211024927489858768832361}{334951962053401944008345505787909350881871393929488541109551} a^{8} - \frac{3084957300444886197674707038717126091509668313586534444215}{334951962053401944008345505787909350881871393929488541109551} a^{7} + \frac{47317662138815102355580194270138407554740920100322227512989}{334951962053401944008345505787909350881871393929488541109551} a^{6} - \frac{15720895611084039860009606610813183120318767551847808430204}{334951962053401944008345505787909350881871393929488541109551} a^{5} - \frac{42506646460629476962843426004752579811202539395854523950565}{334951962053401944008345505787909350881871393929488541109551} a^{4} + \frac{82963630092160551997812087210576940869066484911151724367522}{334951962053401944008345505787909350881871393929488541109551} a^{3} - \frac{140628528740924649976524863595033150291447195327855304378962}{334951962053401944008345505787909350881871393929488541109551} a^{2} - \frac{134780215577042586816104979946654134398291603904993031105917}{334951962053401944008345505787909350881871393929488541109551} a + \frac{97257810842691072608247234160229956180814561677840795326217}{334951962053401944008345505787909350881871393929488541109551}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}$, which has order $48$ (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:  \( 3932685.35305 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1277290832423936.1, 10.0.55534384018432.1, 10.0.5048580365312.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
11Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$