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Properties

Label 20.8.16075758655...0000.1
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 $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-500, 500, 2200, -5000, 4320, -1530, -220, -110, -414, 2358, -2437, 1253, -1057, 1096, -599, 181, -57, 8, 13, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 13*x^18 + 8*x^17 - 57*x^16 + 181*x^15 - 599*x^14 + 1096*x^13 - 1057*x^12 + 1253*x^11 - 2437*x^10 + 2358*x^9 - 414*x^8 - 110*x^7 - 220*x^6 - 1530*x^5 + 4320*x^4 - 5000*x^3 + 2200*x^2 + 500*x - 500)
 
gp: K = bnfinit(x^20 - 7*x^19 + 13*x^18 + 8*x^17 - 57*x^16 + 181*x^15 - 599*x^14 + 1096*x^13 - 1057*x^12 + 1253*x^11 - 2437*x^10 + 2358*x^9 - 414*x^8 - 110*x^7 - 220*x^6 - 1530*x^5 + 4320*x^4 - 5000*x^3 + 2200*x^2 + 500*x - 500, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 13 x^{18} + 8 x^{17} - 57 x^{16} + 181 x^{15} - 599 x^{14} + 1096 x^{13} - 1057 x^{12} + 1253 x^{11} - 2437 x^{10} + 2358 x^{9} - 414 x^{8} - 110 x^{7} - 220 x^{6} - 1530 x^{5} + 4320 x^{4} - 5000 x^{3} + 2200 x^{2} + 500 x - 500 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{5} a^{12} + \frac{3}{10} a^{11} + \frac{3}{10} a^{10} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} - \frac{3}{10} a^{7} - \frac{1}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{10} a^{16} + \frac{3}{10} a^{13} + \frac{3}{10} a^{12} - \frac{2}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{2}{5} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{14} + \frac{1}{10} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{3}{10} a^{10} + \frac{3}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{3}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{50} a^{18} - \frac{1}{25} a^{17} - \frac{1}{25} a^{16} - \frac{1}{25} a^{15} - \frac{1}{25} a^{14} + \frac{1}{50} a^{13} + \frac{3}{25} a^{12} + \frac{3}{25} a^{11} + \frac{4}{25} a^{10} - \frac{7}{50} a^{9} + \frac{4}{25} a^{8} - \frac{11}{25} a^{7} + \frac{21}{50} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{786366367600754028049394150} a^{19} - \frac{494320496794060392840203}{157273273520150805609878830} a^{18} + \frac{14392612869835706516134489}{786366367600754028049394150} a^{17} - \frac{3540760449016297050506163}{393183183800377014024697075} a^{16} + \frac{14367364618559293644806362}{393183183800377014024697075} a^{15} - \frac{18451540673929036391499924}{393183183800377014024697075} a^{14} + \frac{124027610257566179051522489}{393183183800377014024697075} a^{13} - \frac{173659114670187805078404746}{393183183800377014024697075} a^{12} - \frac{2752449342577918129081657}{78636636760075402804939415} a^{11} - \frac{144284377920391982655385933}{393183183800377014024697075} a^{10} + \frac{48828578059824308933717987}{393183183800377014024697075} a^{9} - \frac{190219227395113403267702478}{393183183800377014024697075} a^{8} - \frac{97810414696905996389941943}{786366367600754028049394150} a^{7} - \frac{102313617673596419752286713}{786366367600754028049394150} a^{6} + \frac{61892834400273696394063013}{157273273520150805609878830} a^{5} + \frac{6499642745211809846635612}{78636636760075402804939415} a^{4} - \frac{15754172644136231874830764}{78636636760075402804939415} a^{3} - \frac{22962390514670142282797436}{78636636760075402804939415} a^{2} + \frac{4480314310959441586990172}{15727327352015080560987883} a - \frac{7335679058001481592866189}{15727327352015080560987883}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 30582771.3695 \) (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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$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}$
3469Data not computed