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Properties

Label 20.0.45582459238...3584.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{18}\cdot 1429^{8}$
Root discriminant $34.11$
Ramified primes $2, 1429$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, -1024, 0, 576, 0, -96, 0, -64, 0, 64, 0, -16, 0, -6, 0, 9, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 + 9*x^16 - 6*x^14 - 16*x^12 + 64*x^10 - 64*x^8 - 96*x^6 + 576*x^4 - 1024*x^2 + 1024)
 
gp: K = bnfinit(x^20 - 4*x^18 + 9*x^16 - 6*x^14 - 16*x^12 + 64*x^10 - 64*x^8 - 96*x^6 + 576*x^4 - 1024*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{18} + 9 x^{16} - 6 x^{14} - 16 x^{12} + 64 x^{10} - 64 x^{8} - 96 x^{6} + 576 x^{4} - 1024 x^{2} + 1024 \)

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:  \(4558245923829641289021197123584=2^{18}\cdot 1429^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.11$
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:  $2, 1429$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{8} - \frac{1}{8} a^{6}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{10} - \frac{1}{16} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{64} a^{14} + \frac{1}{64} a^{13} - \frac{1}{32} a^{12} + \frac{5}{128} a^{11} + \frac{3}{64} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{512} a^{16} - \frac{1}{256} a^{14} - \frac{1}{32} a^{13} + \frac{13}{512} a^{12} + \frac{1}{128} a^{10} - \frac{1}{32} a^{9} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{1024} a^{17} - \frac{1}{512} a^{15} - \frac{1}{64} a^{14} + \frac{13}{1024} a^{13} + \frac{1}{256} a^{11} + \frac{3}{64} a^{10} - \frac{1}{32} a^{9} - \frac{3}{32} a^{8} - \frac{3}{32} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{5}{32} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a$, $\frac{1}{1024} a^{18} + \frac{9}{1024} a^{14} + \frac{15}{512} a^{12} - \frac{3}{128} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{5}{32} a^{4} - \frac{5}{16} a^{2} - \frac{1}{4}$, $\frac{1}{4096} a^{19} - \frac{1}{2048} a^{18} + \frac{9}{4096} a^{15} - \frac{9}{2048} a^{14} + \frac{15}{2048} a^{13} - \frac{15}{1024} a^{12} + \frac{13}{512} a^{11} - \frac{13}{256} a^{10} - \frac{1}{128} a^{9} - \frac{7}{64} a^{8} + \frac{5}{64} a^{7} - \frac{1}{32} a^{6} - \frac{27}{128} a^{5} - \frac{5}{64} a^{4} - \frac{13}{64} a^{3} - \frac{3}{32} a^{2} - \frac{1}{16} a + \frac{1}{8}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 5690597.15793 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.2042041.1, 10.0.33359451565448.1, 10.0.2135004900188672.2, 10.10.266875612523584.2

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
Degree 20 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
1429Data not computed