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Properties

Label 20.0.68204734581...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 11^{9}$
Root discriminant $34.81$
Ramified primes $2, 5, 7, 11$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![539, 0, -1617, 0, 2277, 0, -1925, 0, 1001, 0, -259, 0, -27, 0, 49, 0, -14, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^16 + 49*x^14 - 27*x^12 - 259*x^10 + 1001*x^8 - 1925*x^6 + 2277*x^4 - 1617*x^2 + 539)
 
gp: K = bnfinit(x^20 - 14*x^16 + 49*x^14 - 27*x^12 - 259*x^10 + 1001*x^8 - 1925*x^6 + 2277*x^4 - 1617*x^2 + 539, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{16} + 49 x^{14} - 27 x^{12} - 259 x^{10} + 1001 x^{8} - 1925 x^{6} + 2277 x^{4} - 1617 x^{2} + 539 \)

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:  \(6820473458116608604160000000000=2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 11^{9}\)
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:  $2, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{8} + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{9} + \frac{1}{5} a$, $\frac{1}{17040625} a^{18} - \frac{168809}{2434375} a^{16} + \frac{45498}{486875} a^{14} - \frac{8927}{128125} a^{12} - \frac{668219}{17040625} a^{10} - \frac{740116}{2434375} a^{8} - \frac{513824}{2434375} a^{6} + \frac{89662}{2434375} a^{4} + \frac{1449762}{3408125} a^{2} + \frac{30604}{2434375}$, $\frac{1}{17040625} a^{19} - \frac{168809}{2434375} a^{17} + \frac{45498}{486875} a^{15} - \frac{8927}{128125} a^{13} - \frac{668219}{17040625} a^{11} - \frac{740116}{2434375} a^{9} - \frac{513824}{2434375} a^{7} + \frac{89662}{2434375} a^{5} + \frac{1449762}{3408125} a^{3} + \frac{30604}{2434375} a$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 1906128.20829 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.215600.5, 10.2.109853253125.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R R R $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$