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Properties

Label 20.8.32809060263...8125.3
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 401^{10}$
Root discriminant $66.96$
Ramified primes $5, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![559701, 4193337, 2800900, -5082669, -2474812, 3098069, -11999, -967984, 571581, 54973, -170685, 31738, 13339, -7737, 2166, 251, -339, 103, -22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 22*x^18 + 103*x^17 - 339*x^16 + 251*x^15 + 2166*x^14 - 7737*x^13 + 13339*x^12 + 31738*x^11 - 170685*x^10 + 54973*x^9 + 571581*x^8 - 967984*x^7 - 11999*x^6 + 3098069*x^5 - 2474812*x^4 - 5082669*x^3 + 2800900*x^2 + 4193337*x + 559701)
 
gp: K = bnfinit(x^20 - x^19 - 22*x^18 + 103*x^17 - 339*x^16 + 251*x^15 + 2166*x^14 - 7737*x^13 + 13339*x^12 + 31738*x^11 - 170685*x^10 + 54973*x^9 + 571581*x^8 - 967984*x^7 - 11999*x^6 + 3098069*x^5 - 2474812*x^4 - 5082669*x^3 + 2800900*x^2 + 4193337*x + 559701, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 22 x^{18} + 103 x^{17} - 339 x^{16} + 251 x^{15} + 2166 x^{14} - 7737 x^{13} + 13339 x^{12} + 31738 x^{11} - 170685 x^{10} + 54973 x^{9} + 571581 x^{8} - 967984 x^{7} - 11999 x^{6} + 3098069 x^{5} - 2474812 x^{4} - 5082669 x^{3} + 2800900 x^{2} + 4193337 x + 559701 \)

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:  \(3280906026328914297094848663330078125=5^{15}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.96$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{51} a^{18} + \frac{13}{51} a^{17} + \frac{13}{51} a^{16} + \frac{2}{17} a^{15} - \frac{8}{17} a^{14} + \frac{2}{51} a^{13} + \frac{10}{51} a^{12} + \frac{14}{51} a^{11} - \frac{22}{51} a^{10} - \frac{4}{51} a^{9} - \frac{23}{51} a^{8} + \frac{2}{17} a^{7} + \frac{7}{17} a^{6} + \frac{20}{51} a^{5} - \frac{16}{51} a^{4} + \frac{7}{17} a^{3} + \frac{8}{51} a^{2} + \frac{25}{51} a + \frac{7}{17}$, $\frac{1}{149394249630850065423633892592430045528012442199716348443777321} a^{19} + \frac{513573285104720311239532757366817221871881850624427914711724}{149394249630850065423633892592430045528012442199716348443777321} a^{18} - \frac{24211284370518095228056437861186456881511812642775424865907176}{149394249630850065423633892592430045528012442199716348443777321} a^{17} - \frac{2973769528695653000074335651115942716302743231493690781269759}{49798083210283355141211297530810015176004147399905449481259107} a^{16} - \frac{13181148101811956533194505508675880723071015705663832596388657}{49798083210283355141211297530810015176004147399905449481259107} a^{15} - \frac{32390165540956392213924945667062275617897638575387155280402287}{149394249630850065423633892592430045528012442199716348443777321} a^{14} + \frac{11728326014055727131168870925379858935874402416691509090889812}{149394249630850065423633892592430045528012442199716348443777321} a^{13} + \frac{31155752313167259240440027053711768791030234706716259763479847}{149394249630850065423633892592430045528012442199716348443777321} a^{12} - \frac{60553321446349783449570542507785763008742749455587383629935257}{149394249630850065423633892592430045528012442199716348443777321} a^{11} + \frac{50915552792015092935505692378390727886438546878350767487657998}{149394249630850065423633892592430045528012442199716348443777321} a^{10} + \frac{33535186151769668620549126520298197515668750790703591038598293}{149394249630850065423633892592430045528012442199716348443777321} a^{9} + \frac{825587707069787088528424143641177379391149730953924000523235}{2929299012369609125953605737106471480941420435288555851838771} a^{8} - \frac{21212967295141491101760726550311177323559684829566900193913568}{49798083210283355141211297530810015176004147399905449481259107} a^{7} + \frac{53241196102470912305994358618770241968865803693028188545951249}{149394249630850065423633892592430045528012442199716348443777321} a^{6} + \frac{64845458744658638156984144227667652479830860780589422253510616}{149394249630850065423633892592430045528012442199716348443777321} a^{5} - \frac{5734014428610806953564164714336866749392412625917460110755063}{49798083210283355141211297530810015176004147399905449481259107} a^{4} + \frac{60577152459360125456905646202246938842502762023389299684759846}{149394249630850065423633892592430045528012442199716348443777321} a^{3} - \frac{70667573938825083474151586196013227908449858891775710061185112}{149394249630850065423633892592430045528012442199716348443777321} a^{2} - \frac{17017446955214683182758163398786854978793547347881791854677735}{49798083210283355141211297530810015176004147399905449481259107} a + \frac{1675564519004144256018953280395626964953986037648213824533126}{49798083210283355141211297530810015176004147399905449481259107}$

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:  \( 57933797723.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
401Data not computed