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Properties

Label 20.0.48544842802...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{35}\cdot 7^{10}$
Root discriminant $76.61$
Ramified primes $3, 5, 7$
Class number $238144$ (GRH)
Class group $[2, 2, 2, 2, 122, 122]$ (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![924979351, -725290750, 600406250, -439980025, 484303125, -182755451, 228935250, -42468750, 68327550, -5190625, 12538426, -339750, 1421875, -11325, 100000, -151, 4250, 0, 100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351)
 
gp: K = bnfinit(x^20 + 100*x^18 + 4250*x^16 - 151*x^15 + 100000*x^14 - 11325*x^13 + 1421875*x^12 - 339750*x^11 + 12538426*x^10 - 5190625*x^9 + 68327550*x^8 - 42468750*x^7 + 228935250*x^6 - 182755451*x^5 + 484303125*x^4 - 439980025*x^3 + 600406250*x^2 - 725290750*x + 924979351, 1)
 

Normalized defining polynomial

\( x^{20} + 100 x^{18} + 4250 x^{16} - 151 x^{15} + 100000 x^{14} - 11325 x^{13} + 1421875 x^{12} - 339750 x^{11} + 12538426 x^{10} - 5190625 x^{9} + 68327550 x^{8} - 42468750 x^{7} + 228935250 x^{6} - 182755451 x^{5} + 484303125 x^{4} - 439980025 x^{3} + 600406250 x^{2} - 725290750 x + 924979351 \)

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:  \(48544842802805942483246326446533203125=3^{10}\cdot 5^{35}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.61$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(525=3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{525}(64,·)$, $\chi_{525}(1,·)$, $\chi_{525}(398,·)$, $\chi_{525}(272,·)$, $\chi_{525}(274,·)$, $\chi_{525}(211,·)$, $\chi_{525}(421,·)$, $\chi_{525}(188,·)$, $\chi_{525}(482,·)$, $\chi_{525}(484,·)$, $\chi_{525}(293,·)$, $\chi_{525}(167,·)$, $\chi_{525}(169,·)$, $\chi_{525}(106,·)$, $\chi_{525}(83,·)$, $\chi_{525}(503,·)$, $\chi_{525}(377,·)$, $\chi_{525}(379,·)$, $\chi_{525}(316,·)$, $\chi_{525}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10} + \frac{9}{41} a^{8} + \frac{14}{41} a^{6} - \frac{14}{41} a^{5} + \frac{18}{41} a^{4} + \frac{19}{41} a^{3} + \frac{4}{41} a^{2} + \frac{13}{41} a + \frac{9}{41}$, $\frac{1}{41} a^{11} + \frac{9}{41} a^{9} + \frac{14}{41} a^{7} - \frac{14}{41} a^{6} + \frac{18}{41} a^{5} + \frac{19}{41} a^{4} + \frac{4}{41} a^{3} + \frac{13}{41} a^{2} + \frac{9}{41} a$, $\frac{1}{41} a^{12} + \frac{15}{41} a^{8} - \frac{14}{41} a^{7} + \frac{15}{41} a^{6} - \frac{19}{41} a^{5} + \frac{6}{41} a^{4} + \frac{6}{41} a^{3} + \frac{14}{41} a^{2} + \frac{6}{41} a + \frac{1}{41}$, $\frac{1}{30479834641} a^{13} - \frac{34265820}{30479834641} a^{12} + \frac{65}{30479834641} a^{11} + \frac{174282603}{30479834641} a^{10} + \frac{1625}{30479834641} a^{9} - \frac{1654220940}{30479834641} a^{8} - \frac{5203854707}{30479834641} a^{7} + \frac{14646569665}{30479834641} a^{6} + \frac{113750}{30479834641} a^{5} - \frac{620780076}{30479834641} a^{4} + \frac{11151443390}{30479834641} a^{3} - \frac{1064194416}{30479834641} a^{2} - \frac{14124598294}{30479834641} a - \frac{14420410979}{30479834641}$, $\frac{1}{30479834641} a^{14} + \frac{70}{30479834641} a^{12} + \frac{171329100}{30479834641} a^{11} + \frac{1925}{30479834641} a^{10} + \frac{3475815692}{30479834641} a^{9} + \frac{26250}{30479834641} a^{8} + \frac{12273697563}{30479834641} a^{7} - \frac{4460279856}{30479834641} a^{6} + \frac{12052444098}{30479834641} a^{5} + \frac{10408360914}{30479834641} a^{4} - \frac{14729180642}{30479834641} a^{3} - \frac{8920161587}{30479834641} a^{2} - \frac{8781895834}{30479834641} a + \frac{156250}{30479834641}$, $\frac{1}{30479834641} a^{15} + \frac{339704697}{30479834641} a^{12} - \frac{2625}{30479834641} a^{11} + \frac{196960694}{30479834641} a^{10} - \frac{87500}{30479834641} a^{9} - \frac{7974976620}{30479834641} a^{8} - \frac{5205055457}{30479834641} a^{7} - \frac{7372889299}{30479834641} a^{6} - \frac{11158509015}{30479834641} a^{5} - \frac{6958284170}{30479834641} a^{4} + \frac{6671554784}{30479834641} a^{3} + \frac{9212507610}{30479834641} a^{2} - \frac{5961347308}{30479834641} a - \frac{9787165441}{30479834641}$, $\frac{1}{30479834641} a^{16} - \frac{3000}{30479834641} a^{12} - \frac{324937182}{30479834641} a^{11} - \frac{110000}{30479834641} a^{10} - \frac{206926692}{30479834641} a^{9} - \frac{1687500}{30479834641} a^{8} + \frac{10759239136}{30479834641} a^{7} - \frac{13393990818}{30479834641} a^{6} - \frac{12575223349}{30479834641} a^{5} + \frac{699660601}{30479834641} a^{4} - \frac{4458135037}{30479834641} a^{3} + \frac{3660803005}{30479834641} a^{2} - \frac{10418579948}{30479834641} a - \frac{11718750}{30479834641}$, $\frac{1}{30479834641} a^{17} + \frac{211676357}{30479834641} a^{12} + \frac{85000}{30479834641} a^{11} + \frac{23229805}{30479834641} a^{10} + \frac{3187500}{30479834641} a^{9} + \frac{2184664200}{30479834641} a^{8} - \frac{2927742404}{30479834641} a^{7} - \frac{2527038}{301780541} a^{6} - \frac{10110248414}{30479834641} a^{5} + \frac{14030457493}{30479834641} a^{4} + \frac{8230981010}{30479834641} a^{3} - \frac{154542528}{743410601} a^{2} + \frac{6544941058}{30479834641} a - \frac{8117349618}{30479834641}$, $\frac{1}{30479834641} a^{18} + \frac{102000}{30479834641} a^{12} - \frac{8642502}{743410601} a^{11} + \frac{4207500}{30479834641} a^{10} - \frac{7998056076}{30479834641} a^{9} + \frac{68850000}{30479834641} a^{8} + \frac{9135802227}{30479834641} a^{7} + \frac{13173480217}{30479834641} a^{6} - \frac{3671049892}{30479834641} a^{5} + \frac{13063659015}{30479834641} a^{4} + \frac{4050667063}{30479834641} a^{3} - \frac{2693717957}{30479834641} a^{2} - \frac{8238435206}{30479834641} a + \frac{531250000}{30479834641}$, $\frac{1}{30479834641} a^{19} - \frac{13937883}{30479834641} a^{12} - \frac{2422500}{30479834641} a^{11} - \frac{211754353}{30479834641} a^{10} - \frac{96900000}{30479834641} a^{9} - \frac{11561102369}{30479834641} a^{8} + \frac{6724016611}{30479834641} a^{7} + \frac{13831227521}{30479834641} a^{6} - \frac{8203178798}{30479834641} a^{5} + \frac{11797995499}{30479834641} a^{4} + \frac{6213972694}{30479834641} a^{3} - \frac{7276676418}{30479834641} a^{2} + \frac{627996828}{30479834641} a + \frac{8362070652}{30479834641}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{122}\times C_{122}$, which has order $238144$ (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:  \( 161406.837641 \) (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

\(\Q(\sqrt{5}) \), 4.0.55125.1, 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$