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Properties

Label 20.8.77062297047...7401.1
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 109^{6}$
Root discriminant $27.82$
Ramified primes $11, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T254

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![197, -404, 823, 722, -1583, 1616, 1673, -1128, -3609, -2754, 2728, 2303, -744, -515, 94, 51, 11, -1, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 9*x^18 - x^17 + 11*x^16 + 51*x^15 + 94*x^14 - 515*x^13 - 744*x^12 + 2303*x^11 + 2728*x^10 - 2754*x^9 - 3609*x^8 - 1128*x^7 + 1673*x^6 + 1616*x^5 - 1583*x^4 + 722*x^3 + 823*x^2 - 404*x + 197)
 
gp: K = bnfinit(x^20 - x^19 - 9*x^18 - x^17 + 11*x^16 + 51*x^15 + 94*x^14 - 515*x^13 - 744*x^12 + 2303*x^11 + 2728*x^10 - 2754*x^9 - 3609*x^8 - 1128*x^7 + 1673*x^6 + 1616*x^5 - 1583*x^4 + 722*x^3 + 823*x^2 - 404*x + 197, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 9 x^{18} - x^{17} + 11 x^{16} + 51 x^{15} + 94 x^{14} - 515 x^{13} - 744 x^{12} + 2303 x^{11} + 2728 x^{10} - 2754 x^{9} - 3609 x^{8} - 1128 x^{7} + 1673 x^{6} + 1616 x^{5} - 1583 x^{4} + 722 x^{3} + 823 x^{2} - 404 x + 197 \)

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:  \(77062297047310879021301897401=11^{16}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.82$
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:  $11, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{3}{23} a^{16} + \frac{1}{23} a^{15} - \frac{4}{23} a^{14} - \frac{10}{23} a^{13} - \frac{5}{23} a^{12} + \frac{6}{23} a^{11} - \frac{11}{23} a^{10} + \frac{10}{23} a^{9} - \frac{10}{23} a^{8} + \frac{11}{23} a^{7} - \frac{9}{23} a^{6} + \frac{6}{23} a^{4} + \frac{4}{23} a^{3} + \frac{10}{23} a^{2} + \frac{7}{23} a - \frac{10}{23}$, $\frac{1}{23} a^{18} - \frac{8}{23} a^{16} - \frac{7}{23} a^{15} + \frac{2}{23} a^{14} + \frac{2}{23} a^{13} - \frac{2}{23} a^{12} - \frac{6}{23} a^{11} - \frac{3}{23} a^{10} + \frac{6}{23} a^{9} - \frac{5}{23} a^{8} + \frac{4}{23} a^{7} + \frac{4}{23} a^{6} + \frac{6}{23} a^{5} + \frac{9}{23} a^{4} - \frac{2}{23} a^{3} - \frac{8}{23} a + \frac{7}{23}$, $\frac{1}{10748225862010306261066739207153642389} a^{19} + \frac{69705349101333871507991047947112232}{10748225862010306261066739207153642389} a^{18} + \frac{189794027306952393939485300599407856}{10748225862010306261066739207153642389} a^{17} - \frac{3977213012820345966210430626859373611}{10748225862010306261066739207153642389} a^{16} + \frac{57810040347595327128849731451963119}{467314167913491576568119095963201843} a^{15} - \frac{1218860679407953264411846677738533937}{10748225862010306261066739207153642389} a^{14} - \frac{2269013613302770941568560969262212082}{10748225862010306261066739207153642389} a^{13} + \frac{2081397521179189845838949390824522240}{10748225862010306261066739207153642389} a^{12} - \frac{5338084660721027945165475688765451571}{10748225862010306261066739207153642389} a^{11} + \frac{4528449242507305684147083789655446738}{10748225862010306261066739207153642389} a^{10} - \frac{3427424907588023661478213585715142542}{10748225862010306261066739207153642389} a^{9} + \frac{1256918723466745402217037374798065481}{10748225862010306261066739207153642389} a^{8} - \frac{2962245546558416606127623125427928387}{10748225862010306261066739207153642389} a^{7} + \frac{2720071764199902315980073405326440789}{10748225862010306261066739207153642389} a^{6} + \frac{3954683329992385931033071441314371631}{10748225862010306261066739207153642389} a^{5} - \frac{1785029427015807551948560297922182827}{10748225862010306261066739207153642389} a^{4} - \frac{1172719465743604015860731963360233211}{10748225862010306261066739207153642389} a^{3} + \frac{4372875401959834709593235089549064933}{10748225862010306261066739207153642389} a^{2} + \frac{3392544105845118407162735237833005747}{10748225862010306261066739207153642389} a - \frac{3106915709438171706463206080946798350}{10748225862010306261066739207153642389}$

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:  \( 7510850.66348 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T254:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 28 conjugacy class representatives for t20n254
Character table for t20n254 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.23365118029.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
109Data not computed