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Properties

Label 20.20.1784375128...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{35}\cdot 19^{10}$
Root discriminant $72.87$
Ramified primes $5, 19$
Class number $1$ (GRH)
Class group Trivial (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![-6502099, 68478000, -84750000, -210961225, 278040625, 118907199, -200058500, -28406250, 66677450, 3471875, -12505424, -227250, 1421875, 7575, -100000, -101, 4250, 0, -100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099)
 
gp: K = bnfinit(x^20 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099, 1)
 

Normalized defining polynomial

\( x^{20} - 100 x^{18} + 4250 x^{16} - 101 x^{15} - 100000 x^{14} + 7575 x^{13} + 1421875 x^{12} - 227250 x^{11} - 12505424 x^{10} + 3471875 x^{9} + 66677450 x^{8} - 28406250 x^{7} - 200058500 x^{6} + 118907199 x^{5} + 278040625 x^{4} - 210961225 x^{3} - 84750000 x^{2} + 68478000 x - 6502099 \)

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:  \(17843751288604107685387134552001953125=5^{35}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.87$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(475=5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{475}(1,·)$, $\chi_{475}(322,·)$, $\chi_{475}(132,·)$, $\chi_{475}(134,·)$, $\chi_{475}(398,·)$, $\chi_{475}(208,·)$, $\chi_{475}(18,·)$, $\chi_{475}(227,·)$, $\chi_{475}(324,·)$, $\chi_{475}(286,·)$, $\chi_{475}(229,·)$, $\chi_{475}(96,·)$, $\chi_{475}(417,·)$, $\chi_{475}(419,·)$, $\chi_{475}(37,·)$, $\chi_{475}(39,·)$, $\chi_{475}(303,·)$, $\chi_{475}(113,·)$, $\chi_{475}(381,·)$, $\chi_{475}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{5}{11} a^{8} - \frac{5}{11} a^{6} - \frac{1}{11} a^{5} - \frac{2}{11} a^{4} + \frac{3}{11} a^{3} + \frac{5}{11} a^{2} - \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{5}{11} a^{9} - \frac{5}{11} a^{7} - \frac{1}{11} a^{6} - \frac{2}{11} a^{5} + \frac{3}{11} a^{4} + \frac{5}{11} a^{3} - \frac{4}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{11} a^{12} + \frac{3}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{3}{11} a^{5} + \frac{4}{11} a^{4} + \frac{3}{11} a^{3} - \frac{4}{11} a^{2} - \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{200108161} a^{13} + \frac{6671470}{200108161} a^{12} - \frac{65}{200108161} a^{11} - \frac{71878}{200108161} a^{10} + \frac{1625}{200108161} a^{9} - \frac{89341000}{200108161} a^{8} - \frac{36402802}{200108161} a^{7} + \frac{86314489}{200108161} a^{6} + \frac{72880354}{200108161} a^{5} - \frac{76012471}{200108161} a^{4} - \frac{36667677}{200108161} a^{3} - \frac{80196297}{200108161} a^{2} + \frac{54778078}{200108161} a - \frac{29266262}{200108161}$, $\frac{1}{200108161} a^{14} - \frac{70}{200108161} a^{12} - \frac{3025952}{200108161} a^{11} + \frac{175}{18191651} a^{10} + \frac{93660756}{200108161} a^{9} - \frac{26250}{200108161} a^{8} - \frac{54050020}{200108161} a^{7} + \frac{54758703}{200108161} a^{6} + \frac{18146400}{200108161} a^{5} + \frac{35770802}{200108161} a^{4} - \frac{33622884}{200108161} a^{3} + \frac{18957276}{200108161} a^{2} + \frac{15431233}{200108161} a - \frac{156250}{200108161}$, $\frac{1}{200108161} a^{15} - \frac{9005978}{200108161} a^{12} - \frac{2625}{200108161} a^{11} - \frac{2328959}{200108161} a^{10} + \frac{87500}{200108161} a^{9} + \frac{22774528}{200108161} a^{8} - \frac{19372901}{200108161} a^{7} + \frac{38724149}{200108161} a^{6} + \frac{43733302}{200108161} a^{5} - \frac{60725413}{200108161} a^{4} - \frac{37332276}{200108161} a^{3} + \frac{41102253}{200108161} a^{2} - \frac{58704104}{200108161} a + \frac{79784827}{200108161}$, $\frac{1}{200108161} a^{16} - \frac{3000}{200108161} a^{12} - \frac{5584697}{200108161} a^{11} + \frac{10000}{18191651} a^{10} - \frac{41365230}{200108161} a^{9} - \frac{1687500}{200108161} a^{8} - \frac{64798003}{200108161} a^{7} - \frac{60166604}{200108161} a^{6} - \frac{2595723}{200108161} a^{5} - \frac{25558349}{200108161} a^{4} - \frac{13263717}{200108161} a^{3} - \frac{34708255}{200108161} a^{2} - \frac{3864399}{18191651} a - \frac{11718750}{200108161}$, $\frac{1}{200108161} a^{17} - \frac{1990797}{200108161} a^{12} - \frac{85000}{200108161} a^{11} - \frac{210556}{18191651} a^{10} + \frac{3187500}{200108161} a^{9} - \frac{70203820}{200108161} a^{8} - \frac{9516698}{200108161} a^{7} - \frac{71855661}{200108161} a^{6} + \frac{42816886}{200108161} a^{5} - \frac{36523083}{200108161} a^{4} - \frac{14634007}{200108161} a^{3} - \frac{28623263}{200108161} a^{2} + \frac{15523418}{200108161} a - \frac{5878274}{200108161}$, $\frac{1}{200108161} a^{18} - \frac{102000}{200108161} a^{12} - \frac{4376364}{200108161} a^{11} + \frac{382500}{18191651} a^{10} - \frac{505969}{200108161} a^{9} - \frac{68850000}{200108161} a^{8} + \frac{37735979}{200108161} a^{7} + \frac{722011}{18191651} a^{6} - \frac{51227632}{200108161} a^{5} + \frac{70389959}{200108161} a^{4} - \frac{73724320}{200108161} a^{3} - \frac{26508}{18191651} a^{2} + \frac{10743923}{200108161} a + \frac{69074483}{200108161}$, $\frac{1}{200108161} a^{19} + \frac{8666330}{200108161} a^{12} - \frac{2422500}{200108161} a^{11} - \frac{826616}{200108161} a^{10} + \frac{96900000}{200108161} a^{9} + \frac{90429664}{200108161} a^{8} + \frac{38215382}{200108161} a^{7} - \frac{77534196}{200108161} a^{6} + \frac{4402270}{18191651} a^{5} - \frac{45918469}{200108161} a^{4} - \frac{27241545}{200108161} a^{3} + \frac{82621885}{200108161} a^{2} - \frac{77999214}{200108161} a - \frac{1631346}{18191651}$

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:  \( 1261324189610 \) (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

\(\Q(\sqrt{5}) \), 4.4.45125.1, 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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ R $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$19$19.10.5.1$x^{10} - 722 x^{6} + 130321 x^{2} - 61902475$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
19.10.5.1$x^{10} - 722 x^{6} + 130321 x^{2} - 61902475$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$