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Properties

Label 20.8.16075758655...0000.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $32.38$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-80, 720, -2240, 2400, 2376, -10924, 16092, -12272, 2194, 6010, -6943, 2779, 857, -1494, 585, 57, -135, 46, 1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + x^18 + 46*x^17 - 135*x^16 + 57*x^15 + 585*x^14 - 1494*x^13 + 857*x^12 + 2779*x^11 - 6943*x^10 + 6010*x^9 + 2194*x^8 - 12272*x^7 + 16092*x^6 - 10924*x^5 + 2376*x^4 + 2400*x^3 - 2240*x^2 + 720*x - 80)
 
gp: K = bnfinit(x^20 - 5*x^19 + x^18 + 46*x^17 - 135*x^16 + 57*x^15 + 585*x^14 - 1494*x^13 + 857*x^12 + 2779*x^11 - 6943*x^10 + 6010*x^9 + 2194*x^8 - 12272*x^7 + 16092*x^6 - 10924*x^5 + 2376*x^4 + 2400*x^3 - 2240*x^2 + 720*x - 80, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + x^{18} + 46 x^{17} - 135 x^{16} + 57 x^{15} + 585 x^{14} - 1494 x^{13} + 857 x^{12} + 2779 x^{11} - 6943 x^{10} + 6010 x^{9} + 2194 x^{8} - 12272 x^{7} + 16092 x^{6} - 10924 x^{5} + 2376 x^{4} + 2400 x^{3} - 2240 x^{2} + 720 x - 80 \)

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:  \(1607575865553562716800000000000=2^{16}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.38$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{3}{10} a^{11} - \frac{1}{2} a^{9} - \frac{1}{10} a^{8} - \frac{3}{10} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{14} + \frac{3}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{3}{10} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{15} - \frac{1}{20} a^{14} - \frac{1}{20} a^{13} + \frac{3}{20} a^{11} - \frac{3}{20} a^{10} - \frac{1}{4} a^{9} - \frac{1}{5} a^{8} + \frac{7}{20} a^{7} + \frac{7}{20} a^{6} - \frac{9}{20} a^{5}$, $\frac{1}{20} a^{16} - \frac{1}{20} a^{13} - \frac{1}{20} a^{12} + \frac{3}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{4} a^{9} - \frac{9}{20} a^{8} - \frac{9}{20} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{17} - \frac{1}{20} a^{14} - \frac{1}{20} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{9}{20} a^{10} + \frac{7}{20} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{20} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4}$, $\frac{1}{2600} a^{18} - \frac{61}{2600} a^{17} + \frac{19}{2600} a^{16} - \frac{1}{52} a^{15} + \frac{113}{2600} a^{14} - \frac{41}{2600} a^{13} - \frac{193}{2600} a^{12} + \frac{411}{1300} a^{11} - \frac{1011}{2600} a^{10} + \frac{529}{2600} a^{9} + \frac{27}{200} a^{8} + \frac{441}{1300} a^{7} - \frac{303}{1300} a^{6} + \frac{99}{1300} a^{5} - \frac{19}{50} a^{4} - \frac{19}{65} a^{3} + \frac{8}{65} a^{2} + \frac{19}{65} a - \frac{29}{65}$, $\frac{1}{12394306137908398600} a^{19} + \frac{260122497942082}{1549288267238549825} a^{18} + \frac{36182585995416899}{1549288267238549825} a^{17} - \frac{60136588379095637}{12394306137908398600} a^{16} + \frac{164791210143673193}{12394306137908398600} a^{15} - \frac{21207434544283041}{619715306895419930} a^{14} + \frac{625384365166813}{619715306895419930} a^{13} - \frac{1150878618079881409}{12394306137908398600} a^{12} + \frac{65048507770974233}{12394306137908398600} a^{11} + \frac{2512118154023674881}{6197153068954199300} a^{10} - \frac{1363169410095859073}{6197153068954199300} a^{9} + \frac{6152899723591175319}{12394306137908398600} a^{8} - \frac{1265653599382691071}{6197153068954199300} a^{7} - \frac{17053017324456283}{1549288267238549825} a^{6} + \frac{669222631974986529}{6197153068954199300} a^{5} + \frac{400895390748608713}{1549288267238549825} a^{4} + \frac{58395424056771889}{123943061379083986} a^{3} - \frac{17707156394941735}{61971530689541993} a^{2} - \frac{102756034305841811}{309857653447709965} a - \frac{41385626331502488}{309857653447709965}$

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:  \( 50967066.8569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 R $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3469Data not computed