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Properties

Label 20.10.4414566420...6624.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 401^{11}$
Root discriminant $38.22$
Ramified primes $2, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, -3557, -13515, -13433, 17493, 40199, 2410, -34067, -15437, 15480, 9673, -4392, -2789, 777, 423, -62, -27, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 27*x^17 - 62*x^16 + 423*x^15 + 777*x^14 - 2789*x^13 - 4392*x^12 + 9673*x^11 + 15480*x^10 - 15437*x^9 - 34067*x^8 + 2410*x^7 + 40199*x^6 + 17493*x^5 - 13433*x^4 - 13515*x^3 - 3557*x^2 + 81)
 
gp: K = bnfinit(x^20 - 5*x^18 - 27*x^17 - 62*x^16 + 423*x^15 + 777*x^14 - 2789*x^13 - 4392*x^12 + 9673*x^11 + 15480*x^10 - 15437*x^9 - 34067*x^8 + 2410*x^7 + 40199*x^6 + 17493*x^5 - 13433*x^4 - 13515*x^3 - 3557*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 27 x^{17} - 62 x^{16} + 423 x^{15} + 777 x^{14} - 2789 x^{13} - 4392 x^{12} + 9673 x^{11} + 15480 x^{10} - 15437 x^{9} - 34067 x^{8} + 2410 x^{7} + 40199 x^{6} + 17493 x^{5} - 13433 x^{4} - 13515 x^{3} - 3557 x^{2} + 81 \)

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:  \(-44145664201696349530694455706624=-\,2^{10}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $2, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{2} a^{13} + \frac{1}{3} a^{12} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{6} a^{12} - \frac{1}{2} a^{11} + \frac{1}{3} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{414} a^{18} + \frac{7}{414} a^{17} + \frac{16}{207} a^{16} + \frac{22}{207} a^{15} - \frac{1}{2} a^{14} + \frac{17}{69} a^{13} + \frac{10}{23} a^{12} - \frac{100}{207} a^{11} + \frac{121}{414} a^{10} - \frac{26}{207} a^{9} + \frac{1}{207} a^{8} - \frac{11}{138} a^{7} - \frac{167}{414} a^{6} - \frac{157}{414} a^{5} + \frac{59}{207} a^{4} + \frac{62}{207} a^{3} - \frac{181}{414} a^{2} + \frac{83}{414} a - \frac{5}{46}$, $\frac{1}{7893772399240439051017061007649974} a^{19} - \frac{9178417010602044259755113360483}{7893772399240439051017061007649974} a^{18} - \frac{19963527437893556781174555513047}{343207495619149523957263522071738} a^{17} + \frac{22859795609393472218902454190619}{303606630740016886577579269524999} a^{16} + \frac{42240132412931385888849026279456}{438542911068913280612058944869443} a^{15} + \frac{58165770808354429298013175950532}{1315628733206739841836176834608329} a^{14} - \frac{6364965100266579654133300811672}{146180970356304426870686314956481} a^{13} + \frac{2536537274818174871001914292819757}{7893772399240439051017061007649974} a^{12} - \frac{456317080398308368526661379140682}{3946886199620219525508530503824987} a^{11} + \frac{1436481417544894531193079095682904}{3946886199620219525508530503824987} a^{10} + \frac{1409911299916518069056597286301475}{7893772399240439051017061007649974} a^{9} + \frac{39882215047679062740205464215291}{877085822137826561224117889738886} a^{8} + \frac{174445541165078479222875386785901}{3946886199620219525508530503824987} a^{7} - \frac{1380744894095910155990021449890950}{3946886199620219525508530503824987} a^{6} + \frac{493105439585940939155910595613779}{7893772399240439051017061007649974} a^{5} - \frac{1734450592402713313841440560412696}{3946886199620219525508530503824987} a^{4} - \frac{1092186947593778015807542585565107}{7893772399240439051017061007649974} a^{3} + \frac{318773349001129291537490064772213}{3946886199620219525508530503824987} a^{2} + \frac{473921545660642510016704723973677}{2631257466413479683672353669216658} a + \frac{3950357032140618028164075855190}{146180970356304426870686314956481}$

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:  \( 410603324.945 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
401Data not computed