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Properties

Label 20.0.41931483557...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{15}\cdot 13^{15}$
Root discriminant $60.41$
Ramified primes $2, 5, 13$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![426546, 316440, 570906, 825312, 1079534, 1102378, 781396, 149956, -88708, -51952, 19883, 19694, 14759, 1652, 1188, -528, 30, -70, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 11*x^18 - 70*x^17 + 30*x^16 - 528*x^15 + 1188*x^14 + 1652*x^13 + 14759*x^12 + 19694*x^11 + 19883*x^10 - 51952*x^9 - 88708*x^8 + 149956*x^7 + 781396*x^6 + 1102378*x^5 + 1079534*x^4 + 825312*x^3 + 570906*x^2 + 316440*x + 426546)
 
gp: K = bnfinit(x^20 - 2*x^19 + 11*x^18 - 70*x^17 + 30*x^16 - 528*x^15 + 1188*x^14 + 1652*x^13 + 14759*x^12 + 19694*x^11 + 19883*x^10 - 51952*x^9 - 88708*x^8 + 149956*x^7 + 781396*x^6 + 1102378*x^5 + 1079534*x^4 + 825312*x^3 + 570906*x^2 + 316440*x + 426546, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 11 x^{18} - 70 x^{17} + 30 x^{16} - 528 x^{15} + 1188 x^{14} + 1652 x^{13} + 14759 x^{12} + 19694 x^{11} + 19883 x^{10} - 51952 x^{9} - 88708 x^{8} + 149956 x^{7} + 781396 x^{6} + 1102378 x^{5} + 1079534 x^{4} + 825312 x^{3} + 570906 x^{2} + 316440 x + 426546 \)

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:  \(419314835571431481344000000000000000=2^{28}\cdot 5^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.41$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} - \frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{4}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{5577392767139525175369045068629705783020206030292490119341883} a^{19} + \frac{89751054727077820060771788204120106158511061094813831923698}{5577392767139525175369045068629705783020206030292490119341883} a^{18} - \frac{32217998171972017099490285240939207589370506905112022630678}{796770395305646453624149295518529397574315147184641445620269} a^{17} + \frac{23417997198502152996087928792451751658068336889358364925223}{796770395305646453624149295518529397574315147184641445620269} a^{16} + \frac{221131095171248253718574295356619113765214974297852918546650}{1859130922379841725123015022876568594340068676764163373113961} a^{15} - \frac{54998277421990602431951304832857693451093242142942364504177}{1859130922379841725123015022876568594340068676764163373113961} a^{14} + \frac{77917974168880007512608720703758135961206818484649776758042}{619710307459947241707671674292189531446689558921387791037987} a^{13} - \frac{1665773782809041574834150703661177871730047609519328837810846}{5577392767139525175369045068629705783020206030292490119341883} a^{12} - \frac{469261255693373272774849063632244075878154717078483550927857}{5577392767139525175369045068629705783020206030292490119341883} a^{11} - \frac{1046927022945673630210547109585551559376944841745756161019404}{5577392767139525175369045068629705783020206030292490119341883} a^{10} - \frac{2735821374935857660634418007610521880825440177304240296613017}{5577392767139525175369045068629705783020206030292490119341883} a^{9} - \frac{812558222942954515620584611705790062915195082755484169327223}{5577392767139525175369045068629705783020206030292490119341883} a^{8} + \frac{191551184060774094392572768006875289096248201696316872753245}{796770395305646453624149295518529397574315147184641445620269} a^{7} - \frac{1370521943710212068515254078764723181796143415336597013313968}{5577392767139525175369045068629705783020206030292490119341883} a^{6} + \frac{1475731237578980144073663317523721396056567357668351957221441}{5577392767139525175369045068629705783020206030292490119341883} a^{5} + \frac{399165406134131828653135304289697327632831544206465560592094}{5577392767139525175369045068629705783020206030292490119341883} a^{4} - \frac{1641287055285566460625746146526834318526266446959891262723011}{5577392767139525175369045068629705783020206030292490119341883} a^{3} - \frac{559085020006374806920458762576929216718581755174227581278595}{1859130922379841725123015022876568594340068676764163373113961} a^{2} + \frac{33854922842066794304821141508436626612161759924327594000118}{206570102486649080569223891430729843815563186307129263679329} a - \frac{89817610290822740174150715416106210043898192138854914714622}{206570102486649080569223891430729843815563186307129263679329}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  \( 679419183.0669895 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{65}) \), 4.0.4394000.2, 5.1.338000.1, 10.2.7425860000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
5Data not computed
13Data not computed