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Properties

Label 20.10.3009103398...3776.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 11^{18}\cdot 727^{2}$
Root discriminant $23.65$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -306, -176, 828, 28, -1155, 416, 931, 121, -314, -210, 218, 66, -47, 5, -44, -9, 26, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 26*x^17 - 9*x^16 - 44*x^15 + 5*x^14 - 47*x^13 + 66*x^12 + 218*x^11 - 210*x^10 - 314*x^9 + 121*x^8 + 931*x^7 + 416*x^6 - 1155*x^5 + 28*x^4 + 828*x^3 - 176*x^2 - 306*x - 43)
 
gp: K = bnfinit(x^20 - 5*x^19 + 26*x^17 - 9*x^16 - 44*x^15 + 5*x^14 - 47*x^13 + 66*x^12 + 218*x^11 - 210*x^10 - 314*x^9 + 121*x^8 + 931*x^7 + 416*x^6 - 1155*x^5 + 28*x^4 + 828*x^3 - 176*x^2 - 306*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 26 x^{17} - 9 x^{16} - 44 x^{15} + 5 x^{14} - 47 x^{13} + 66 x^{12} + 218 x^{11} - 210 x^{10} - 314 x^{9} + 121 x^{8} + 931 x^{7} + 416 x^{6} - 1155 x^{5} + 28 x^{4} + 828 x^{3} - 176 x^{2} - 306 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3009103398689521267119563776=-\,2^{10}\cdot 11^{18}\cdot 727^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{19883656226990381654793538199} a^{19} + \frac{1648022082951243150442210893}{19883656226990381654793538199} a^{18} + \frac{2958517707011655474853610752}{19883656226990381654793538199} a^{17} + \frac{3218933340522492993302087728}{19883656226990381654793538199} a^{16} + \frac{3468689529448897553278266}{13684553494143414765859283} a^{15} - \frac{4155976215276032107425867930}{19883656226990381654793538199} a^{14} + \frac{6022764571743662278259323341}{19883656226990381654793538199} a^{13} + \frac{2715815380457376256873758212}{19883656226990381654793538199} a^{12} - \frac{2818420277168787213674853830}{19883656226990381654793538199} a^{11} + \frac{3507963378952899641869382989}{19883656226990381654793538199} a^{10} - \frac{4258924487164078920660767154}{19883656226990381654793538199} a^{9} - \frac{3310552737155237914678173592}{19883656226990381654793538199} a^{8} + \frac{6083044934739591163599156711}{19883656226990381654793538199} a^{7} + \frac{3323249811376148322661125306}{19883656226990381654793538199} a^{6} - \frac{1615818108028193593386442022}{19883656226990381654793538199} a^{5} - \frac{219956978470934381288446223}{864506792477842680643197313} a^{4} + \frac{4315986599227545370015881540}{19883656226990381654793538199} a^{3} - \frac{5289306910744260375445165640}{19883656226990381654793538199} a^{2} - \frac{128741509914325636732297358}{864506792477842680643197313} a - \frac{4611355832361859049284925409}{19883656226990381654793538199}$

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:  $14$
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:  \( 1944038.88538 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
727Data not computed