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Properties

Label 20.0.55154187683...0224.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 47^{10}$
Root discriminant $13.71$
Ramified primes $2, 47$
Class number $1$
Class group Trivial
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![1, 0, -11, 0, 52, 0, -139, 0, 241, 0, -287, 0, 241, 0, -139, 0, 52, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 52*x^16 - 139*x^14 + 241*x^12 - 287*x^10 + 241*x^8 - 139*x^6 + 52*x^4 - 11*x^2 + 1)
 
gp: K = bnfinit(x^20 - 11*x^18 + 52*x^16 - 139*x^14 + 241*x^12 - 287*x^10 + 241*x^8 - 139*x^6 + 52*x^4 - 11*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} + 52 x^{16} - 139 x^{14} + 241 x^{12} - 287 x^{10} + 241 x^{8} - 139 x^{6} + 52 x^{4} - 11 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(55154187683317729460224=2^{20}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.71$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{12} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{2}{5} a^{6} - \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{19} - \frac{2}{5} a^{13} - \frac{1}{5} a^{11} + \frac{2}{5} a^{7} - \frac{1}{5} a^{3} - \frac{2}{5} a$

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:  \( -\frac{28}{5} a^{19} + \frac{297}{5} a^{17} - \frac{1341}{5} a^{15} + \frac{3383}{5} a^{13} - \frac{5499}{5} a^{11} + \frac{6073}{5} a^{9} - \frac{4664}{5} a^{7} + \frac{2354}{5} a^{5} - 143 a^{3} + \frac{91}{5} a \) (order $4$)
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:  \( 1968.78648064 \)
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

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{47}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.2.234849287168.5 x5, 10.0.4996793344.3 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$