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Properties

Label 20.0.48181783941...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 11^{16}$
Root discriminant $43.07$
Ramified primes $2, 5, 11$
Class number $775$ (GRH)
Class group $[5, 155]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![767371, 24970, 1041689, 78782, 714780, 38138, 311223, -1416, 93762, -5580, 19969, -3100, 3374, -536, 322, -66, 79, 6, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 6*x^17 + 79*x^16 - 66*x^15 + 322*x^14 - 536*x^13 + 3374*x^12 - 3100*x^11 + 19969*x^10 - 5580*x^9 + 93762*x^8 - 1416*x^7 + 311223*x^6 + 38138*x^5 + 714780*x^4 + 78782*x^3 + 1041689*x^2 + 24970*x + 767371)
 
gp: K = bnfinit(x^20 - 2*x^19 - 5*x^18 + 6*x^17 + 79*x^16 - 66*x^15 + 322*x^14 - 536*x^13 + 3374*x^12 - 3100*x^11 + 19969*x^10 - 5580*x^9 + 93762*x^8 - 1416*x^7 + 311223*x^6 + 38138*x^5 + 714780*x^4 + 78782*x^3 + 1041689*x^2 + 24970*x + 767371, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 5 x^{18} + 6 x^{17} + 79 x^{16} - 66 x^{15} + 322 x^{14} - 536 x^{13} + 3374 x^{12} - 3100 x^{11} + 19969 x^{10} - 5580 x^{9} + 93762 x^{8} - 1416 x^{7} + 311223 x^{6} + 38138 x^{5} + 714780 x^{4} + 78782 x^{3} + 1041689 x^{2} + 24970 x + 767371 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(481817839414250422927360000000000=2^{30}\cdot 5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.07$
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 Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(9,·)$, $\chi_{440}(331,·)$, $\chi_{440}(81,·)$, $\chi_{440}(339,·)$, $\chi_{440}(89,·)$, $\chi_{440}(201,·)$, $\chi_{440}(91,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(291,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(419,·)$, $\chi_{440}(49,·)$, $\chi_{440}(179,·)$, $\chi_{440}(59,·)$, $\chi_{440}(251,·)$, $\chi_{440}(379,·)$, $\chi_{440}(411,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43361} a^{18} + \frac{7482}{43361} a^{17} + \frac{10332}{43361} a^{16} + \frac{9729}{43361} a^{15} + \frac{15562}{43361} a^{14} + \frac{7160}{43361} a^{13} + \frac{14364}{43361} a^{12} - \frac{2665}{43361} a^{11} - \frac{15632}{43361} a^{10} - \frac{21553}{43361} a^{9} + \frac{20190}{43361} a^{8} + \frac{11543}{43361} a^{7} + \frac{11407}{43361} a^{6} + \frac{7194}{43361} a^{5} - \frac{11798}{43361} a^{4} - \frac{12640}{43361} a^{3} + \frac{7380}{43361} a^{2} + \frac{20507}{43361} a + \frac{13278}{43361}$, $\frac{1}{118423719626501019541955343687288493729454032137293} a^{19} + \frac{59448363528291919990537769284204803715535379}{118423719626501019541955343687288493729454032137293} a^{18} + \frac{8426745177794735997483119345663284920504375051887}{118423719626501019541955343687288493729454032137293} a^{17} - \frac{22097867806214740114443814997945843613211033070}{282634175719572839002280056532908099592969050447} a^{16} + \frac{39880604771278528752047721471692342555400007728412}{118423719626501019541955343687288493729454032137293} a^{15} + \frac{45811769977849404640252186408187846919884480492468}{118423719626501019541955343687288493729454032137293} a^{14} + \frac{52691518922449196555273573099499216853177552590168}{118423719626501019541955343687288493729454032137293} a^{13} - \frac{34612719316516362355881855242609531152960569056488}{118423719626501019541955343687288493729454032137293} a^{12} - \frac{19176119341538220875435172057707173808415435804292}{118423719626501019541955343687288493729454032137293} a^{11} - \frac{56186132104873574134245806039642472515443163734482}{118423719626501019541955343687288493729454032137293} a^{10} - \frac{33723610050575970938876532291736806644367948282745}{118423719626501019541955343687288493729454032137293} a^{9} - \frac{27005611904510293628284921734335646591521484433659}{118423719626501019541955343687288493729454032137293} a^{8} - \frac{24683967535824881085101312757868198805063801880031}{118423719626501019541955343687288493729454032137293} a^{7} - \frac{21961147173486309756368189899350977256059195778}{1330603591309000219572531951542567345274764406037} a^{6} - \frac{15245527884860725630624140157670200742832630775309}{118423719626501019541955343687288493729454032137293} a^{5} - \frac{27219053436320729117228447107426385323222599380068}{118423719626501019541955343687288493729454032137293} a^{4} - \frac{53400964005058629063691262902343010909016601882552}{118423719626501019541955343687288493729454032137293} a^{3} + \frac{31727130864551176909129348539122131054860396389971}{118423719626501019541955343687288493729454032137293} a^{2} + \frac{5582892269010578534295596320904133446568346198539}{118423719626501019541955343687288493729454032137293} a - \frac{51103056768034399879889172759212834612310783364168}{118423719626501019541955343687288493729454032137293}$

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

Class group and class number

$C_{5}\times C_{155}$, which has order $775$ (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:  $9$
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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.7024111812608.1, 10.0.21950349414400000.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$