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Properties

Label 20.20.8221111755...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{35}\cdot 7^{10}$
Root discriminant $44.23$
Ramified primes $5, 7$
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![401, 8580, -15600, -92290, 175600, 132429, -325460, -79200, 272780, 24200, -128039, -3960, 36400, 330, -6400, -11, 680, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 + 680*x^16 - 11*x^15 - 6400*x^14 + 330*x^13 + 36400*x^12 - 3960*x^11 - 128039*x^10 + 24200*x^9 + 272780*x^8 - 79200*x^7 - 325460*x^6 + 132429*x^5 + 175600*x^4 - 92290*x^3 - 15600*x^2 + 8580*x + 401)
 
gp: K = bnfinit(x^20 - 40*x^18 + 680*x^16 - 11*x^15 - 6400*x^14 + 330*x^13 + 36400*x^12 - 3960*x^11 - 128039*x^10 + 24200*x^9 + 272780*x^8 - 79200*x^7 - 325460*x^6 + 132429*x^5 + 175600*x^4 - 92290*x^3 - 15600*x^2 + 8580*x + 401, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} + 680 x^{16} - 11 x^{15} - 6400 x^{14} + 330 x^{13} + 36400 x^{12} - 3960 x^{11} - 128039 x^{10} + 24200 x^{9} + 272780 x^{8} - 79200 x^{7} - 325460 x^{6} + 132429 x^{5} + 175600 x^{4} - 92290 x^{3} - 15600 x^{2} + 8580 x + 401 \)

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:  \(822111175511963665485382080078125=5^{35}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.23$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(64,·)$, $\chi_{175}(1,·)$, $\chi_{175}(132,·)$, $\chi_{175}(134,·)$, $\chi_{175}(71,·)$, $\chi_{175}(13,·)$, $\chi_{175}(141,·)$, $\chi_{175}(83,·)$, $\chi_{175}(153,·)$, $\chi_{175}(27,·)$, $\chi_{175}(29,·)$, $\chi_{175}(97,·)$, $\chi_{175}(99,·)$, $\chi_{175}(36,·)$, $\chi_{175}(167,·)$, $\chi_{175}(169,·)$, $\chi_{175}(106,·)$, $\chi_{175}(48,·)$, $\chi_{175}(118,·)$, $\chi_{175}(62,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4049} a^{13} + \frac{90}{4049} a^{12} - \frac{26}{4049} a^{11} + \frac{1889}{4049} a^{10} + \frac{260}{4049} a^{9} - \frac{805}{4049} a^{8} - \frac{1248}{4049} a^{7} + \frac{340}{4049} a^{6} - \frac{1137}{4049} a^{5} + \frac{1387}{4049} a^{4} + \frac{1137}{4049} a^{3} + \frac{1594}{4049} a^{2} + \frac{832}{4049} a - \frac{627}{4049}$, $\frac{1}{4049} a^{14} - \frac{28}{4049} a^{12} + \frac{180}{4049} a^{11} + \frac{308}{4049} a^{10} + \frac{89}{4049} a^{9} - \frac{1680}{4049} a^{8} - \frac{712}{4049} a^{7} + \frac{655}{4049} a^{6} - \frac{1557}{4049} a^{5} + \frac{1826}{4049} a^{4} + \frac{489}{4049} a^{3} - \frac{913}{4049} a^{2} + \frac{1424}{4049} a - \frac{256}{4049}$, $\frac{1}{4049} a^{15} - \frac{1349}{4049} a^{12} - \frac{420}{4049} a^{11} + \frac{344}{4049} a^{10} + \frac{1551}{4049} a^{9} + \frac{1042}{4049} a^{8} - \frac{1897}{4049} a^{7} - \frac{135}{4049} a^{6} - \frac{1667}{4049} a^{5} - \frac{1165}{4049} a^{4} - \frac{1469}{4049} a^{3} + \frac{1517}{4049} a^{2} - \frac{1254}{4049} a - \frac{1360}{4049}$, $\frac{1}{4049} a^{16} - \frac{480}{4049} a^{12} + \frac{1711}{4049} a^{11} - \frac{1058}{4049} a^{10} - \frac{481}{4049} a^{9} + \frac{1339}{4049} a^{8} + \frac{697}{4049} a^{7} - \frac{544}{4049} a^{6} - \frac{407}{4049} a^{5} - \frac{1044}{4049} a^{4} + \frac{759}{4049} a^{3} - \frac{967}{4049} a^{2} - \frac{565}{4049} a + \frac{418}{4049}$, $\frac{1}{4049} a^{17} + \frac{372}{4049} a^{12} - \frac{1391}{4049} a^{11} - \frac{737}{4049} a^{10} + \frac{620}{4049} a^{9} - \frac{1048}{4049} a^{8} - \frac{332}{4049} a^{7} + \frac{833}{4049} a^{6} - \frac{189}{4049} a^{5} - \frac{1566}{4049} a^{4} - \frac{1822}{4049} a^{3} - \frac{706}{4049} a^{2} - \frac{1073}{4049} a - \frac{1334}{4049}$, $\frac{1}{4049} a^{18} + \frac{1570}{4049} a^{12} + \frac{837}{4049} a^{11} - \frac{1611}{4049} a^{10} - \frac{592}{4049} a^{9} - \frac{498}{4049} a^{8} - \frac{546}{4049} a^{7} - \frac{1150}{4049} a^{6} + \frac{302}{4049} a^{5} + \frac{486}{4049} a^{4} + \frac{1475}{4049} a^{3} + \frac{1162}{4049} a^{2} + \frac{935}{4049} a - \frac{1598}{4049}$, $\frac{1}{4049} a^{19} + \frac{1252}{4049} a^{12} - \frac{1281}{4049} a^{11} + \frac{1595}{4049} a^{10} + \frac{251}{4049} a^{9} + \frac{16}{4049} a^{8} - \frac{1506}{4049} a^{7} + \frac{970}{4049} a^{6} - \frac{33}{4049} a^{5} - \frac{1802}{4049} a^{4} + \frac{1681}{4049} a^{3} + \frac{637}{4049} a^{2} - \frac{11}{4049} a + \frac{483}{4049}$

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:  \( 9489514790.4 \) (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

\(\Q(\sqrt{5}) \), 4.4.6125.1, 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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$