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Properties

Label 20.0.40787561206...2517.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{12}\cdot 37^{9}$
Root discriminant $21.41$
Ramified primes $11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_5$ (as 20T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, 74, -252, 587, -1072, 1813, -2665, 3582, -4222, 4260, -3681, 2822, -1922, 1207, -674, 333, -131, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 40*x^18 - 131*x^17 + 333*x^16 - 674*x^15 + 1207*x^14 - 1922*x^13 + 2822*x^12 - 3681*x^11 + 4260*x^10 - 4222*x^9 + 3582*x^8 - 2665*x^7 + 1813*x^6 - 1072*x^5 + 587*x^4 - 252*x^3 + 74*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 40*x^18 - 131*x^17 + 333*x^16 - 674*x^15 + 1207*x^14 - 1922*x^13 + 2822*x^12 - 3681*x^11 + 4260*x^10 - 4222*x^9 + 3582*x^8 - 2665*x^7 + 1813*x^6 - 1072*x^5 + 587*x^4 - 252*x^3 + 74*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 40 x^{18} - 131 x^{17} + 333 x^{16} - 674 x^{15} + 1207 x^{14} - 1922 x^{13} + 2822 x^{12} - 3681 x^{11} + 4260 x^{10} - 4222 x^{9} + 3582 x^{8} - 2665 x^{7} + 1813 x^{6} - 1072 x^{5} + 587 x^{4} - 252 x^{3} + 74 x^{2} - 13 x + 1 \)

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:  \(407875612060900496297202517=11^{12}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.41$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{228} a^{18} + \frac{4}{57} a^{17} + \frac{17}{228} a^{16} - \frac{1}{76} a^{15} - \frac{2}{57} a^{14} - \frac{25}{228} a^{13} + \frac{25}{228} a^{12} - \frac{1}{228} a^{11} + \frac{71}{228} a^{10} + \frac{16}{57} a^{9} - \frac{73}{228} a^{8} + \frac{22}{57} a^{7} + \frac{9}{76} a^{6} + \frac{5}{76} a^{5} + \frac{5}{228} a^{3} + \frac{23}{228} a^{2} + \frac{13}{228} a - \frac{7}{228}$, $\frac{1}{1460803602129672} a^{19} - \frac{462948389907}{486934534043224} a^{18} + \frac{10234426147503}{486934534043224} a^{17} - \frac{1164242075252}{182600450266209} a^{16} + \frac{20997908582693}{1460803602129672} a^{15} + \frac{113127090639}{486934534043224} a^{14} - \frac{131978228707883}{730401801064836} a^{13} + \frac{18973065033757}{121733633510806} a^{12} - \frac{7637488091375}{730401801064836} a^{11} + \frac{210410539458839}{486934534043224} a^{10} + \frac{701089989102319}{1460803602129672} a^{9} + \frac{617696772548491}{1460803602129672} a^{8} + \frac{706285031571979}{1460803602129672} a^{7} - \frac{20991348760889}{121733633510806} a^{6} - \frac{32011371879917}{486934534043224} a^{5} - \frac{54551120325683}{486934534043224} a^{4} + \frac{18567463222207}{365200900532418} a^{3} - \frac{48617207717159}{365200900532418} a^{2} + \frac{328642563625411}{730401801064836} a + \frac{243449232877231}{486934534043224}$

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:  $9$
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:  \( 172377.141044 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

10.2.89734879333.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$