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Properties

Label 20.20.1114719476...6001.1
Degree $20$
Signature $[20, 0]$
Discriminant $401^{15}$
Root discriminant $89.61$
Ramified prime $401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:C_4$ (as 20T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6072119, 21505161, 106222904, 4965810, -318011368, -314665312, 52059719, 187263905, 42546574, -38883595, -15663089, 3491841, 2164979, -117981, -152812, -1381, 5889, 177, -119, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 119*x^18 + 177*x^17 + 5889*x^16 - 1381*x^15 - 152812*x^14 - 117981*x^13 + 2164979*x^12 + 3491841*x^11 - 15663089*x^10 - 38883595*x^9 + 42546574*x^8 + 187263905*x^7 + 52059719*x^6 - 314665312*x^5 - 318011368*x^4 + 4965810*x^3 + 106222904*x^2 + 21505161*x - 6072119)
 
gp: K = bnfinit(x^20 - 3*x^19 - 119*x^18 + 177*x^17 + 5889*x^16 - 1381*x^15 - 152812*x^14 - 117981*x^13 + 2164979*x^12 + 3491841*x^11 - 15663089*x^10 - 38883595*x^9 + 42546574*x^8 + 187263905*x^7 + 52059719*x^6 - 314665312*x^5 - 318011368*x^4 + 4965810*x^3 + 106222904*x^2 + 21505161*x - 6072119, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 119 x^{18} + 177 x^{17} + 5889 x^{16} - 1381 x^{15} - 152812 x^{14} - 117981 x^{13} + 2164979 x^{12} + 3491841 x^{11} - 15663089 x^{10} - 38883595 x^{9} + 42546574 x^{8} + 187263905 x^{7} + 52059719 x^{6} - 314665312 x^{5} - 318011368 x^{4} + 4965810 x^{3} + 106222904 x^{2} + 21505161 x - 6072119 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1114719476673733231325235265693136806001=401^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.61$
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:  $401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{19} + \frac{838101751098418998279831992863364527126977480764222798330312447868265}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{18} + \frac{15666811414237624144333373382044382647023044567583215238882897174596}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{17} + \frac{134555555275580049161834207665672889429616153227796754581901580027372}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{16} + \frac{657874188410601513767773330405490939995969708017128873855572194838398}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{15} + \frac{437464631090396205128007783239597002483418160506803001510650176603078}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{14} - \frac{203264125252576828875486813679137804451949866793325376148792538984764}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{13} - \frac{163253699271873981595721933922735839634678420836054716908473576163686}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{12} - \frac{193363557625111574717163190149584286567576808936313367189640128003518}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{11} + \frac{892939285984699505849467551338651141962837093851520450479047580591867}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{10} + \frac{368660910450792932760908189044620457801720989088404058184294066595973}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{9} + \frac{711381673812221261817056759280931716812640037091206786821022319405559}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{8} + \frac{748252382792245756472318836941158298475440382711376827401278106109506}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{7} + \frac{279097962721307147831512103058388021607241971638972440218372797282212}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{6} - \frac{179144583112746165399491278678940147131056511913760637347141510231914}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{5} + \frac{653030860193594660006888275825193700231344749434443757009297736294880}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{4} - \frac{16967899423989059712508566983754148076133325220947862725970895684986}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{3} - \frac{338287613851089250229301192156398284721402334735806988158180636431307}{1915818496374114293913576176117251578108461084223628060509903438958243} a^{2} + \frac{923555705016934347333789354894705545759148287766334352740525996545741}{1915818496374114293913576176117251578108461084223628060509903438958243} a + \frac{808570242619340480277995935628435686900397405830299640699379837083317}{1915818496374114293913576176117251578108461084223628060509903438958243}$

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:  $19$
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:  \( 10186030781000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:C_4$ (as 20T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $C_5:C_4$
Character table for $C_5:C_4$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.4.64481201.1, 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
401Data not computed