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Properties

Label 20.8.15418170861...0000.6
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 642, 426, -2588, -293, 4620, 2722, -11998, 1893, 16168, -16592, 2988, 5301, -3956, 528, 640, -370, 58, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 16*x^18 + 58*x^17 - 370*x^16 + 640*x^15 + 528*x^14 - 3956*x^13 + 5301*x^12 + 2988*x^11 - 16592*x^10 + 16168*x^9 + 1893*x^8 - 11998*x^7 + 2722*x^6 + 4620*x^5 - 293*x^4 - 2588*x^3 + 426*x^2 + 642*x - 199)
 
gp: K = bnfinit(x^20 - 8*x^19 + 16*x^18 + 58*x^17 - 370*x^16 + 640*x^15 + 528*x^14 - 3956*x^13 + 5301*x^12 + 2988*x^11 - 16592*x^10 + 16168*x^9 + 1893*x^8 - 11998*x^7 + 2722*x^6 + 4620*x^5 - 293*x^4 - 2588*x^3 + 426*x^2 + 642*x - 199, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 16 x^{18} + 58 x^{17} - 370 x^{16} + 640 x^{15} + 528 x^{14} - 3956 x^{13} + 5301 x^{12} + 2988 x^{11} - 16592 x^{10} + 16168 x^{9} + 1893 x^{8} - 11998 x^{7} + 2722 x^{6} + 4620 x^{5} - 293 x^{4} - 2588 x^{3} + 426 x^{2} + 642 x - 199 \)

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:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{7}{23} a^{17} - \frac{6}{23} a^{16} - \frac{1}{23} a^{15} + \frac{9}{23} a^{14} + \frac{5}{23} a^{13} - \frac{11}{23} a^{12} + \frac{5}{23} a^{11} + \frac{11}{23} a^{10} + \frac{11}{23} a^{9} + \frac{1}{23} a^{8} - \frac{3}{23} a^{7} - \frac{4}{23} a^{6} + \frac{7}{23} a^{5} + \frac{5}{23} a^{3} - \frac{11}{23} a^{2} - \frac{7}{23} a - \frac{7}{23}$, $\frac{1}{1826688465001470277237185517} a^{19} + \frac{15031842303914822635454093}{1826688465001470277237185517} a^{18} - \frac{529874182734968278040784088}{1826688465001470277237185517} a^{17} - \frac{15766143079670698971660194}{1826688465001470277237185517} a^{16} + \frac{387666720009290412305412361}{1826688465001470277237185517} a^{15} - \frac{21378128762595091141626710}{1826688465001470277237185517} a^{14} + \frac{880460250588157416549180519}{1826688465001470277237185517} a^{13} + \frac{810464047844055897954246316}{1826688465001470277237185517} a^{12} - \frac{749947199438677362975772854}{1826688465001470277237185517} a^{11} - \frac{281805489930359408437346645}{1826688465001470277237185517} a^{10} - \frac{669426402888784888842427162}{1826688465001470277237185517} a^{9} + \frac{781894603619417885621457706}{1826688465001470277237185517} a^{8} - \frac{25624745140602622475594948}{1826688465001470277237185517} a^{7} + \frac{321416221637910412274430959}{1826688465001470277237185517} a^{6} - \frac{193486855681543945054515619}{1826688465001470277237185517} a^{5} - \frac{179932312275919651045826039}{1826688465001470277237185517} a^{4} - \frac{349289919700337581434552986}{1826688465001470277237185517} a^{3} - \frac{453632097954283584776186182}{1826688465001470277237185517} a^{2} - \frac{36913990216105603095690908}{1826688465001470277237185517} a + \frac{535780600265426683070906356}{1826688465001470277237185517}$

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:  \( 11284822.2025 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.219503494144.1

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 $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$