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Properties

Label 20.8.67442553253...7456.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $31.01$
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![-23, -112, -138, -206, -314, -94, 1191, 498, -601, 450, 156, 104, 330, -12, 72, 28, -32, 6, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 6*x^17 - 32*x^16 + 28*x^15 + 72*x^14 - 12*x^13 + 330*x^12 + 104*x^11 + 156*x^10 + 450*x^9 - 601*x^8 + 498*x^7 + 1191*x^6 - 94*x^5 - 314*x^4 - 206*x^3 - 138*x^2 - 112*x - 23)
 
gp: K = bnfinit(x^20 - 2*x^19 - 5*x^18 + 6*x^17 - 32*x^16 + 28*x^15 + 72*x^14 - 12*x^13 + 330*x^12 + 104*x^11 + 156*x^10 + 450*x^9 - 601*x^8 + 498*x^7 + 1191*x^6 - 94*x^5 - 314*x^4 - 206*x^3 - 138*x^2 - 112*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 5 x^{18} + 6 x^{17} - 32 x^{16} + 28 x^{15} + 72 x^{14} - 12 x^{13} + 330 x^{12} + 104 x^{11} + 156 x^{10} + 450 x^{9} - 601 x^{8} + 498 x^{7} + 1191 x^{6} - 94 x^{5} - 314 x^{4} - 206 x^{3} - 138 x^{2} - 112 x - 23 \)

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:  \(674425532537559799418460307456=2^{20}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.01$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{15150422158757397739606029678061} a^{19} - \frac{221958272874322963180934580067}{15150422158757397739606029678061} a^{18} - \frac{2734318697565660231581918179012}{15150422158757397739606029678061} a^{17} + \frac{4993447939740193746883974122855}{15150422158757397739606029678061} a^{16} - \frac{3122903047871855609155132703229}{15150422158757397739606029678061} a^{15} + \frac{1330060758280965210097710036231}{15150422158757397739606029678061} a^{14} + \frac{1509389339119912140349639298563}{15150422158757397739606029678061} a^{13} + \frac{1022036900603477256711977809781}{15150422158757397739606029678061} a^{12} + \frac{327694112416903897558177251677}{15150422158757397739606029678061} a^{11} + \frac{3580328841675759718611549992750}{15150422158757397739606029678061} a^{10} - \frac{1261668643654453405420820135573}{15150422158757397739606029678061} a^{9} + \frac{5535622359960166625774870513511}{15150422158757397739606029678061} a^{8} - \frac{1300903795788835879478187119745}{15150422158757397739606029678061} a^{7} - \frac{7017130985503566442203759468236}{15150422158757397739606029678061} a^{6} + \frac{7343205670047346204333318031926}{15150422158757397739606029678061} a^{5} + \frac{4728487519289023799072821420530}{15150422158757397739606029678061} a^{4} - \frac{4557899254302075425131277251810}{15150422158757397739606029678061} a^{3} + \frac{4714384287168288345353756884520}{15150422158757397739606029678061} a^{2} - \frac{4599583601934256995428451704154}{15150422158757397739606029678061} a - \frac{5002409405921679535639806867524}{15150422158757397739606029678061}$

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:  \( 22430428.5916 \) (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

\(\Q(\zeta_{11})^+\), 10.10.52900342088704.1

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 $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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