Properties

 Label 20.0.37937967949...6597.1 Degree $20$ Signature $[0, 10]$ Discriminant $3^{15}\cdot 31^{9}$ Root discriminant $10.69$ Ramified primes $3, 31$ Class number $1$ Class group Trivial Galois group $C_5\times C_5:D_4$ (as 20T53)

Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 25, -26, -46, 161, -128, -127, 308, -144, -146, 193, -8, -122, 73, 25, -46, 14, 6, -5, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1)

gp: K = bnfinit(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1, 1)

Normalizeddefining polynomial

$$x^{20} - 5 x^{19} + 6 x^{18} + 14 x^{17} - 46 x^{16} + 25 x^{15} + 73 x^{14} - 122 x^{13} - 8 x^{12} + 193 x^{11} - 146 x^{10} - 144 x^{9} + 308 x^{8} - 127 x^{7} - 128 x^{6} + 161 x^{5} - 46 x^{4} - 26 x^{3} + 25 x^{2} - 8 x + 1$$

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: $$379379679498607236597=3^{15}\cdot 31^{9}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.69$ 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: $3, 31$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6978527} a^{19} - \frac{502850}{6978527} a^{18} + \frac{2639465}{6978527} a^{17} - \frac{2706308}{6978527} a^{16} - \frac{1189948}{6978527} a^{15} - \frac{438476}{6978527} a^{14} - \frac{1096272}{6978527} a^{13} + \frac{110407}{6978527} a^{12} - \frac{3425638}{6978527} a^{11} - \frac{707323}{6978527} a^{10} - \frac{751820}{6978527} a^{9} + \frac{1184585}{6978527} a^{8} + \frac{3485122}{6978527} a^{7} - \frac{557869}{6978527} a^{6} - \frac{1191169}{6978527} a^{5} - \frac{574971}{6978527} a^{4} + \frac{918839}{6978527} a^{3} + \frac{718635}{6978527} a^{2} + \frac{68564}{6978527} a - \frac{3141208}{6978527}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

Class group and class number

Trivial group, which has order $1$

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: $$\frac{294846}{18413} a^{19} - \frac{1356362}{18413} a^{18} + \frac{1211607}{18413} a^{17} + \frac{4680751}{18413} a^{16} - \frac{11744765}{18413} a^{15} + \frac{2419699}{18413} a^{14} + \frac{23070963}{18413} a^{13} - \frac{26770987}{18413} a^{12} - \frac{14299518}{18413} a^{11} + \frac{52373757}{18413} a^{10} - \frac{21096924}{18413} a^{9} - \frac{53479978}{18413} a^{8} + \frac{69996438}{18413} a^{7} - \frac{6459642}{18413} a^{6} - \frac{43411115}{18413} a^{5} + \frac{29757726}{18413} a^{4} + \frac{526458}{18413} a^{3} - \frac{8417968}{18413} a^{2} + \frac{3705327}{18413} a - \frac{536855}{18413}$$ (order $6$) 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 magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$181.753135766$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

Galois group

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 200 The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

 Degree 20 siblings: data not computed Degree 40 siblings: data not computed

Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $20$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$31$31.10.9.3$x^{10} - 74431$$10$$1$$9$$C_{10}$$[\ ]_{10} 31.10.0.1x^{10} - x + 11$$1$$10$$0$$C_{10}$$[\ ]^{10}$