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Properties

Label 20.4.50522262278...3936.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{40}\cdot 11^{16}$
Root discriminant $27.24$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -230, 0, -465, 0, -198, 0, 287, 0, 70, 0, 20, 0, -22, 0, 13, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 + 13*x^16 - 22*x^14 + 20*x^12 + 70*x^10 + 287*x^8 - 198*x^6 - 465*x^4 - 230*x^2 + 1)
 
gp: K = bnfinit(x^20 - 6*x^18 + 13*x^16 - 22*x^14 + 20*x^12 + 70*x^10 + 287*x^8 - 198*x^6 - 465*x^4 - 230*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} + 13 x^{16} - 22 x^{14} + 20 x^{12} + 70 x^{10} + 287 x^{8} - 198 x^{6} - 465 x^{4} - 230 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50522262278163705147147943936=2^{40}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.24$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{4}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{4}{9} a^{7} - \frac{1}{3} a^{5} + \frac{1}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{621} a^{16} - \frac{2}{621} a^{14} - \frac{44}{621} a^{12} - \frac{100}{621} a^{10} - \frac{29}{207} a^{8} + \frac{275}{621} a^{6} + \frac{61}{621} a^{4} + \frac{233}{621} a^{2} - \frac{38}{621}$, $\frac{1}{621} a^{17} - \frac{2}{621} a^{15} - \frac{44}{621} a^{13} - \frac{100}{621} a^{11} - \frac{29}{207} a^{9} + \frac{275}{621} a^{7} + \frac{61}{621} a^{5} + \frac{233}{621} a^{3} - \frac{38}{621} a$, $\frac{1}{1767891987} a^{18} - \frac{803293}{1767891987} a^{16} - \frac{39913315}{1767891987} a^{14} + \frac{5203936}{589297329} a^{12} + \frac{398586131}{1767891987} a^{10} + \frac{794298824}{1767891987} a^{8} + \frac{119096}{65477481} a^{6} + \frac{17422274}{65477481} a^{4} - \frac{59462105}{589297329} a^{2} + \frac{391783942}{1767891987}$, $\frac{1}{1767891987} a^{19} - \frac{803293}{1767891987} a^{17} - \frac{39913315}{1767891987} a^{15} + \frac{5203936}{589297329} a^{13} + \frac{398586131}{1767891987} a^{11} + \frac{794298824}{1767891987} a^{9} + \frac{119096}{65477481} a^{7} + \frac{17422274}{65477481} a^{5} - \frac{59462105}{589297329} a^{3} + \frac{391783942}{1767891987} a$

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:  $11$
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:  \( 2467785.33551 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1, 10.2.7024111812608.1, 10.2.219503494144.1

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 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$