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Properties

Label 20.0.16863534401...000.11
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $204.81$
Ramified primes $2, 5, 7, 11$
Class number $32862976$ (GRH)
Class group $[2, 8, 2053936]$ (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![51445143565729, 0, -2497505487390, 0, 399450190019, 0, -2034260966, 0, 1604479779, 0, 38824576, 0, 3954160, 0, 98526, 0, 3961, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 54*x^18 + 3961*x^16 + 98526*x^14 + 3954160*x^12 + 38824576*x^10 + 1604479779*x^8 - 2034260966*x^6 + 399450190019*x^4 - 2497505487390*x^2 + 51445143565729)
 
gp: K = bnfinit(x^20 + 54*x^18 + 3961*x^16 + 98526*x^14 + 3954160*x^12 + 38824576*x^10 + 1604479779*x^8 - 2034260966*x^6 + 399450190019*x^4 - 2497505487390*x^2 + 51445143565729, 1)
 

Normalized defining polynomial

\( x^{20} + 54 x^{18} + 3961 x^{16} + 98526 x^{14} + 3954160 x^{12} + 38824576 x^{10} + 1604479779 x^{8} - 2034260966 x^{6} + 399450190019 x^{4} - 2497505487390 x^{2} + 51445143565729 \)

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:  \(16863534401027144382603387201975746560000000000=2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.81$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1539,·)$, $\chi_{3080}(1989,·)$, $\chi_{3080}(391,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(139,·)$, $\chi_{3080}(1231,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(699,·)$, $\chi_{3080}(2269,·)$, $\chi_{3080}(2911,·)$, $\chi_{3080}(2659,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(2631,·)$, $\chi_{3080}(1709,·)$, $\chi_{3080}(2351,·)$, $\chi_{3080}(309,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2939,·)$, $\chi_{3080}(1149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} + \frac{1}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{7} a^{6} + \frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{49} a^{8} + \frac{2}{49} a^{6} - \frac{2}{49} a^{4} - \frac{3}{49} a^{2} - \frac{10}{49}$, $\frac{1}{49} a^{9} + \frac{2}{49} a^{7} - \frac{2}{49} a^{5} - \frac{3}{49} a^{3} - \frac{10}{49} a$, $\frac{1}{98} a^{10} - \frac{1}{98} a^{8} + \frac{3}{49} a^{6} + \frac{3}{98} a^{4} + \frac{13}{98} a^{2} - \frac{47}{98}$, $\frac{1}{98} a^{11} - \frac{1}{98} a^{9} + \frac{3}{49} a^{7} + \frac{3}{98} a^{5} + \frac{13}{98} a^{3} - \frac{47}{98} a$, $\frac{1}{686} a^{12} + \frac{3}{686} a^{10} + \frac{1}{343} a^{8} - \frac{1}{686} a^{6} - \frac{17}{686} a^{4} + \frac{327}{686} a^{2} - \frac{10}{343}$, $\frac{1}{686} a^{13} + \frac{3}{686} a^{11} + \frac{1}{343} a^{9} - \frac{1}{686} a^{7} - \frac{17}{686} a^{5} + \frac{327}{686} a^{3} - \frac{10}{343} a$, $\frac{1}{686} a^{14} - \frac{3}{49} a^{6} - \frac{3}{98} a^{4} - \frac{5}{49} a^{2} - \frac{311}{686}$, $\frac{1}{1372} a^{15} - \frac{1}{1372} a^{14} - \frac{1}{1372} a^{13} - \frac{3}{1372} a^{11} - \frac{1}{686} a^{9} - \frac{1}{98} a^{8} - \frac{41}{1372} a^{7} + \frac{1}{98} a^{6} + \frac{47}{686} a^{5} - \frac{1}{28} a^{4} + \frac{387}{1372} a^{3} + \frac{1}{98} a^{2} + \frac{591}{1372} a - \frac{431}{1372}$, $\frac{1}{9604} a^{16} + \frac{1}{2401} a^{14} - \frac{1}{1372} a^{13} - \frac{1}{1372} a^{12} - \frac{3}{1372} a^{11} + \frac{1}{686} a^{10} + \frac{3}{343} a^{9} - \frac{13}{1372} a^{8} + \frac{29}{1372} a^{7} + \frac{22}{343} a^{6} - \frac{11}{1372} a^{5} - \frac{1}{343} a^{4} + \frac{317}{1372} a^{3} + \frac{809}{9604} a^{2} - \frac{30}{343} a + \frac{1367}{9604}$, $\frac{1}{220892} a^{17} - \frac{73}{220892} a^{15} - \frac{11}{15778} a^{13} - \frac{1}{1372} a^{12} + \frac{135}{31556} a^{11} + \frac{1}{343} a^{10} + \frac{197}{31556} a^{9} + \frac{5}{1372} a^{8} + \frac{291}{31556} a^{7} - \frac{27}{1372} a^{6} - \frac{597}{15778} a^{5} + \frac{5}{686} a^{4} - \frac{51707}{110446} a^{3} + \frac{155}{686} a^{2} + \frac{13801}{55223} a + \frac{433}{1372}$, $\frac{1}{15289862426346454063112930884148466082564} a^{18} + \frac{55579637453740477578733415228132189}{2184266060906636294730418697735495154652} a^{16} - \frac{31805053319297209682491928736830397}{58358253535673488790507369786826206422} a^{14} - \frac{1}{1372} a^{13} + \frac{503324917713788145146263027774713269}{2184266060906636294730418697735495154652} a^{12} + \frac{1}{343} a^{11} - \frac{9454788061942079579541476007586465973}{2184266060906636294730418697735495154652} a^{10} + \frac{5}{1372} a^{9} - \frac{17818101804946003297901224565658715449}{2184266060906636294730418697735495154652} a^{8} - \frac{27}{1372} a^{7} - \frac{19626202237680363115588647955558765668}{546066515226659073682604674433873788663} a^{6} + \frac{5}{686} a^{5} - \frac{62100827313840993223127614909996287650}{3822465606586613515778232721037116520641} a^{4} + \frac{155}{686} a^{3} + \frac{2864000381054047711980451456025587407}{22288429192924860150310394874851991374} a^{2} + \frac{433}{1372} a - \frac{11285335608373249092199720472060873802}{166194156808113631120792727001613761767}$, $\frac{1}{4768128307793715353127704383290814999381500836} a^{19} - \frac{5026413131508597710557077737526636980151}{4768128307793715353127704383290814999381500836} a^{17} + \frac{3772209438851263188501291400591715773303}{36397926013692483611661865521303931292988556} a^{15} - \frac{118791023995925080676576455996885597238798}{170290296706918405468846585117529107120767887} a^{13} - \frac{236651320633727017859483919776444483497823}{48654370487690972991099024319294030605933682} a^{11} - \frac{1}{196} a^{10} + \frac{3343391089533057079984236013557576808123647}{681161186827673621875386340470116428483071548} a^{9} + \frac{1}{196} a^{8} - \frac{381430920785785530215884999441194965050569}{29615703775116244429364623498700714281872676} a^{7} + \frac{2}{49} a^{6} - \frac{63556306276502484923678630526569741316421813}{1192032076948428838281926095822703749845375209} a^{5} - \frac{3}{196} a^{4} + \frac{2030594550267007578357964308151304799333450155}{4768128307793715353127704383290814999381500836} a^{3} + \frac{1}{196} a^{2} + \frac{2044199109697410787681745115793008275458661063}{4768128307793715353127704383290814999381500836} a - \frac{79}{196}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{2053936}$, which has order $32862976$ (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:  \( 14452469.589232503 \) (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

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-770}) \), \(\Q(\sqrt{10}, \sqrt{-77})\), \(\Q(\zeta_{11})^+\), 10.0.40581147486860288.1, 10.10.21950349414400000.1, 10.0.4058114748686028800000.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
7Data not computed
11Data not computed