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Properties

Label 20.8.69576619586...5625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 97^{2}\cdot 27517559^{2}$
Root discriminant $19.59$
Ramified primes $5, 97, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1040

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 34, -82, 128, -267, 195, 531, -981, 130, 930, -840, 111, 275, -244, 96, 7, -37, 25, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 25*x^18 - 37*x^17 + 7*x^16 + 96*x^15 - 244*x^14 + 275*x^13 + 111*x^12 - 840*x^11 + 930*x^10 + 130*x^9 - 981*x^8 + 531*x^7 + 195*x^6 - 267*x^5 + 128*x^4 - 82*x^3 + 34*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 25*x^18 - 37*x^17 + 7*x^16 + 96*x^15 - 244*x^14 + 275*x^13 + 111*x^12 - 840*x^11 + 930*x^10 + 130*x^9 - 981*x^8 + 531*x^7 + 195*x^6 - 267*x^5 + 128*x^4 - 82*x^3 + 34*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 25 x^{18} - 37 x^{17} + 7 x^{16} + 96 x^{15} - 244 x^{14} + 275 x^{13} + 111 x^{12} - 840 x^{11} + 930 x^{10} + 130 x^{9} - 981 x^{8} + 531 x^{7} + 195 x^{6} - 267 x^{5} + 128 x^{4} - 82 x^{3} + 34 x^{2} - 2 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69576619586656130166015625=5^{10}\cdot 97^{2}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.59$
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:  $5, 97, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41} a^{18} + \frac{4}{41} a^{17} - \frac{17}{41} a^{16} + \frac{14}{41} a^{15} - \frac{17}{41} a^{14} - \frac{15}{41} a^{13} - \frac{1}{41} a^{12} + \frac{14}{41} a^{11} - \frac{9}{41} a^{9} + \frac{2}{41} a^{8} - \frac{20}{41} a^{7} - \frac{7}{41} a^{6} - \frac{8}{41} a^{5} - \frac{9}{41} a^{4} + \frac{17}{41} a^{3} - \frac{6}{41} a^{2} - \frac{3}{41} a + \frac{5}{41}$, $\frac{1}{114838279656104449} a^{19} - \frac{1073945644586949}{114838279656104449} a^{18} - \frac{19171618231749002}{114838279656104449} a^{17} - \frac{11543626086925869}{114838279656104449} a^{16} - \frac{521955926144332}{114838279656104449} a^{15} - \frac{155496881738160}{114838279656104449} a^{14} + \frac{13195605778994926}{114838279656104449} a^{13} - \frac{55402292324962451}{114838279656104449} a^{12} - \frac{20724591572374826}{114838279656104449} a^{11} + \frac{52806201805778013}{114838279656104449} a^{10} + \frac{37726883766943764}{114838279656104449} a^{9} - \frac{34967708587755539}{114838279656104449} a^{8} - \frac{32544942320507410}{114838279656104449} a^{7} + \frac{57226508432775455}{114838279656104449} a^{6} - \frac{40555954354043554}{114838279656104449} a^{5} + \frac{43902274887723513}{114838279656104449} a^{4} + \frac{55983076999381}{114838279656104449} a^{3} + \frac{20416265427565709}{114838279656104449} a^{2} + \frac{47672426744439794}{114838279656104449} a - \frac{51457726851422291}{114838279656104449}$

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:  \( 160028.668848 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1040:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1040 are not computed
Character table for t20n1040 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.85992371875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
97Data not computed
27517559Data not computed