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Properties

Label 20.8.79692860778...7424.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{18}\cdot 241^{3}$
Root discriminant $27.87$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 208, 248, -1672, -2126, 1213, 4195, -11, -3885, -2011, 1660, 1607, 49, -374, -129, 41, 29, 11, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 9*x^18 + 11*x^17 + 29*x^16 + 41*x^15 - 129*x^14 - 374*x^13 + 49*x^12 + 1607*x^11 + 1660*x^10 - 2011*x^9 - 3885*x^8 - 11*x^7 + 4195*x^6 + 1213*x^5 - 2126*x^4 - 1672*x^3 + 248*x^2 + 208*x - 32)
 
gp: K = bnfinit(x^20 - 2*x^19 - 9*x^18 + 11*x^17 + 29*x^16 + 41*x^15 - 129*x^14 - 374*x^13 + 49*x^12 + 1607*x^11 + 1660*x^10 - 2011*x^9 - 3885*x^8 - 11*x^7 + 4195*x^6 + 1213*x^5 - 2126*x^4 - 1672*x^3 + 248*x^2 + 208*x - 32, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 9 x^{18} + 11 x^{17} + 29 x^{16} + 41 x^{15} - 129 x^{14} - 374 x^{13} + 49 x^{12} + 1607 x^{11} + 1660 x^{10} - 2011 x^{9} - 3885 x^{8} - 11 x^{7} + 4195 x^{6} + 1213 x^{5} - 2126 x^{4} - 1672 x^{3} + 248 x^{2} + 208 x - 32 \)

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:  \(79692860778363999735970407424=2^{10}\cdot 11^{18}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.87$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{6152} a^{18} - \frac{31}{1538} a^{17} - \frac{137}{6152} a^{16} + \frac{1349}{6152} a^{15} - \frac{53}{6152} a^{14} - \frac{1549}{6152} a^{13} + \frac{793}{6152} a^{12} + \frac{378}{769} a^{11} - \frac{2935}{6152} a^{10} + \frac{2565}{6152} a^{9} - \frac{783}{3076} a^{8} - \frac{439}{6152} a^{7} - \frac{3015}{6152} a^{6} + \frac{2707}{6152} a^{5} - \frac{1667}{6152} a^{4} + \frac{1955}{6152} a^{3} - \frac{331}{1538} a^{2} - \frac{85}{769} a - \frac{124}{769}$, $\frac{1}{13719138160089775867159746223376} a^{19} - \frac{374955315736783334473939595}{6859569080044887933579873111688} a^{18} + \frac{1027652446467253848400219992483}{13719138160089775867159746223376} a^{17} + \frac{1695954064806378521529979397639}{13719138160089775867159746223376} a^{16} - \frac{5735190173805715095488914036547}{13719138160089775867159746223376} a^{15} - \frac{611589997905326841463190184207}{13719138160089775867159746223376} a^{14} + \frac{2570399471715228210538009279471}{13719138160089775867159746223376} a^{13} + \frac{1333341114165308623399757317841}{6859569080044887933579873111688} a^{12} - \frac{764500495506501181680193859211}{13719138160089775867159746223376} a^{11} + \frac{2240211145354162437223401571931}{13719138160089775867159746223376} a^{10} + \frac{595682681684008029001589103017}{3429784540022443966789936555844} a^{9} + \frac{3796287976248320260211846582753}{13719138160089775867159746223376} a^{8} + \frac{6179215059583272582156361329903}{13719138160089775867159746223376} a^{7} + \frac{5650918302226277298951720672269}{13719138160089775867159746223376} a^{6} - \frac{2636759500559635347150180288293}{13719138160089775867159746223376} a^{5} + \frac{5417436250080927278503092016837}{13719138160089775867159746223376} a^{4} + \frac{365184917090729362259581860149}{6859569080044887933579873111688} a^{3} - \frac{818647840854628738784995744371}{3429784540022443966789936555844} a^{2} - \frac{464788189895097860530375920}{37280266739374390943368875607} a + \frac{123134505401579532794775550955}{857446135005610991697484138961}$

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:  \( 7666043.93763 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
241Data not computed