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Properties

Label 20.0.68395290468...8224.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 101^{10}$
Root discriminant $34.81$
Ramified primes $2, 3, 101$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 0, -7695, 0, 18288, 0, 13080, 0, 1282, 0, -3348, 0, 1188, 0, -167, 0, 54, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^18 + 54*x^16 - 167*x^14 + 1188*x^12 - 3348*x^10 + 1282*x^8 + 13080*x^6 + 18288*x^4 - 7695*x^2 + 729)
 
gp: K = bnfinit(x^20 - 3*x^18 + 54*x^16 - 167*x^14 + 1188*x^12 - 3348*x^10 + 1282*x^8 + 13080*x^6 + 18288*x^4 - 7695*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{18} + 54 x^{16} - 167 x^{14} + 1188 x^{12} - 3348 x^{10} + 1282 x^{8} + 13080 x^{6} + 18288 x^{4} - 7695 x^{2} + 729 \)

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:  \(6839529046897455541684062388224=2^{20}\cdot 3^{10}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.81$
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:  $2, 3, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{1}{27} a^{10} - \frac{4}{27} a^{8} - \frac{1}{27} a^{6} + \frac{8}{27} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{7} - \frac{1}{27} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{135} a^{16} - \frac{2}{135} a^{14} + \frac{4}{135} a^{12} - \frac{7}{135} a^{10} - \frac{16}{135} a^{8} - \frac{13}{135} a^{6} - \frac{1}{45} a^{4} + \frac{4}{15} a^{2} - \frac{1}{5}$, $\frac{1}{405} a^{17} + \frac{1}{135} a^{15} - \frac{2}{135} a^{13} - \frac{2}{405} a^{11} + \frac{1}{45} a^{9} + \frac{14}{135} a^{7} + \frac{37}{405} a^{5} - \frac{28}{135} a^{3} - \frac{2}{5} a$, $\frac{1}{8526197626003665} a^{18} - \frac{863928491908}{315785097259395} a^{16} + \frac{26987669470949}{2842065875334555} a^{14} + \frac{94027521644137}{8526197626003665} a^{12} + \frac{1328056343158}{947355291778185} a^{10} + \frac{416843174875087}{2842065875334555} a^{8} - \frac{800395040488016}{8526197626003665} a^{6} - \frac{2992736723221}{6191864652145} a^{4} - \frac{101372509964539}{947355291778185} a^{2} + \frac{4341766375538}{105261699086465}$, $\frac{1}{25578592878010995} a^{19} - \frac{863928491908}{947355291778185} a^{17} - \frac{966346044636}{105261699086465} a^{15} + \frac{725597716162927}{25578592878010995} a^{13} + \frac{214507567202404}{8526197626003665} a^{11} - \frac{36488440185746}{2842065875334555} a^{9} + \frac{1410100640327749}{25578592878010995} a^{7} + \frac{18265942907353}{501541036823745} a^{5} - \frac{31044607539622}{105261699086465} a^{3} + \frac{4341766375538}{315785097259395} a$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 44040090.2384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-303}) \), \(\Q(\sqrt{-101}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{-101})\), 5.1.91809.1 x5, 10.0.2553954421743.1, 10.0.871749775954944.1 x5, 10.2.25893557701632.1 x5

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

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$