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Properties

Label 20.0.12081089504...1901.1
Degree $20$
Signature $[0, 10]$
Discriminant $101^{19}$
Root discriminant $80.19$
Ramified prime $101$
Class number $125$ (GRH)
Class group $[5, 25]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![120577, 197892, 125333, -21866, -16152, 144019, 112662, -6930, -31618, 7237, 24000, 1958, -80, -1991, 316, 300, 44, -11, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 11*x^17 + 44*x^16 + 300*x^15 + 316*x^14 - 1991*x^13 - 80*x^12 + 1958*x^11 + 24000*x^10 + 7237*x^9 - 31618*x^8 - 6930*x^7 + 112662*x^6 + 144019*x^5 - 16152*x^4 - 21866*x^3 + 125333*x^2 + 197892*x + 120577)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 - 11*x^17 + 44*x^16 + 300*x^15 + 316*x^14 - 1991*x^13 - 80*x^12 + 1958*x^11 + 24000*x^10 + 7237*x^9 - 31618*x^8 - 6930*x^7 + 112662*x^6 + 144019*x^5 - 16152*x^4 - 21866*x^3 + 125333*x^2 + 197892*x + 120577, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} - 11 x^{17} + 44 x^{16} + 300 x^{15} + 316 x^{14} - 1991 x^{13} - 80 x^{12} + 1958 x^{11} + 24000 x^{10} + 7237 x^{9} - 31618 x^{8} - 6930 x^{7} + 112662 x^{6} + 144019 x^{5} - 16152 x^{4} - 21866 x^{3} + 125333 x^{2} + 197892 x + 120577 \)

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:  \(120810895044353150938886048668570711901=101^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.19$
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:  $101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(101\)
Dirichlet character group:    $\lbrace$$\chi_{101}(1,·)$, $\chi_{101}(69,·)$, $\chi_{101}(6,·)$, $\chi_{101}(65,·)$, $\chi_{101}(10,·)$, $\chi_{101}(14,·)$, $\chi_{101}(17,·)$, $\chi_{101}(84,·)$, $\chi_{101}(87,·)$, $\chi_{101}(36,·)$, $\chi_{101}(91,·)$, $\chi_{101}(95,·)$, $\chi_{101}(32,·)$, $\chi_{101}(100,·)$, $\chi_{101}(39,·)$, $\chi_{101}(41,·)$, $\chi_{101}(44,·)$, $\chi_{101}(57,·)$, $\chi_{101}(60,·)$, $\chi_{101}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{137} a^{16} + \frac{32}{137} a^{15} + \frac{55}{137} a^{14} - \frac{25}{137} a^{13} + \frac{60}{137} a^{12} - \frac{29}{137} a^{11} + \frac{29}{137} a^{10} - \frac{4}{137} a^{9} + \frac{29}{137} a^{8} + \frac{47}{137} a^{7} + \frac{33}{137} a^{6} - \frac{37}{137} a^{5} - \frac{12}{137} a^{4} + \frac{9}{137} a^{3} - \frac{43}{137} a^{2} - \frac{60}{137} a + \frac{59}{137}$, $\frac{1}{137} a^{17} - \frac{10}{137} a^{15} - \frac{4}{137} a^{14} + \frac{38}{137} a^{13} - \frac{31}{137} a^{12} - \frac{2}{137} a^{11} + \frac{27}{137} a^{10} + \frac{20}{137} a^{9} - \frac{59}{137} a^{8} + \frac{36}{137} a^{7} + \frac{3}{137} a^{6} - \frac{61}{137} a^{5} - \frac{18}{137} a^{4} - \frac{57}{137} a^{3} - \frac{54}{137} a^{2} + \frac{61}{137} a + \frac{30}{137}$, $\frac{1}{137} a^{18} + \frac{42}{137} a^{15} + \frac{40}{137} a^{14} - \frac{7}{137} a^{13} + \frac{50}{137} a^{12} + \frac{11}{137} a^{11} + \frac{36}{137} a^{10} + \frac{38}{137} a^{9} + \frac{52}{137} a^{8} + \frac{62}{137} a^{7} - \frac{5}{137} a^{6} + \frac{23}{137} a^{5} - \frac{40}{137} a^{4} + \frac{36}{137} a^{3} + \frac{42}{137} a^{2} - \frac{22}{137} a + \frac{42}{137}$, $\frac{1}{1702944503900739822645157359537520171634168375827091} a^{19} - \frac{4706001448249966422037222583900545776686781759510}{1702944503900739822645157359537520171634168375827091} a^{18} + \frac{2619246499171055380215762982873959720167958590618}{1702944503900739822645157359537520171634168375827091} a^{17} - \frac{2367452929034828184713596065259410472424943348358}{1702944503900739822645157359537520171634168375827091} a^{16} - \frac{514469708060388436879747926938869215046995497604990}{1702944503900739822645157359537520171634168375827091} a^{15} - \frac{320743368417835206425661984178435353904278905834007}{1702944503900739822645157359537520171634168375827091} a^{14} - \frac{365419644232726948062329832852717563364935154610784}{1702944503900739822645157359537520171634168375827091} a^{13} + \frac{165835714214323341103209366196098988773482340892080}{1702944503900739822645157359537520171634168375827091} a^{12} - \frac{287469315074357243527805007529216181238916483201949}{1702944503900739822645157359537520171634168375827091} a^{11} + \frac{665075394630126663083710659065992854667434461713089}{1702944503900739822645157359537520171634168375827091} a^{10} - \frac{371884245462220023259139902574544901263984369379065}{1702944503900739822645157359537520171634168375827091} a^{9} + \frac{338335990917246473404383541704894215777669016838059}{1702944503900739822645157359537520171634168375827091} a^{8} - \frac{223502389903375918344389151150941367443460005530899}{1702944503900739822645157359537520171634168375827091} a^{7} - \frac{298634127146833656602598905534298363734786028692032}{1702944503900739822645157359537520171634168375827091} a^{6} + \frac{202889874120114986373476528898422765909443510153523}{1702944503900739822645157359537520171634168375827091} a^{5} + \frac{445478317647467557888276042122156116555610825387687}{1702944503900739822645157359537520171634168375827091} a^{4} + \frac{367323576372529537645215299945853037132244506234380}{1702944503900739822645157359537520171634168375827091} a^{3} + \frac{187914247209720508447705668221634605533154974178239}{1702944503900739822645157359537520171634168375827091} a^{2} - \frac{367615006781511950090116834515780710552718539208368}{1702944503900739822645157359537520171634168375827091} a + \frac{181145827312838811031973508414133938566638119253641}{1702944503900739822645157359537520171634168375827091}$

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

Class group and class number

$C_{5}\times C_{25}$, which has order $125$ (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:  \( 224544554.426 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{101}) \), 4.0.1030301.1, 5.5.104060401.1, 10.10.1093685272684360901.1

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 $20$ $20$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
101Data not computed