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Properties

Label 10.0.43913358828288.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{8}\cdot 3^{5}\cdot 163^{4}$
Root discriminant $23.13$
Ramified primes $2, 3, 163$
Class number $11$
Class group $[11]$
Galois group $D_{10}$ (as 10T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![324, -1026, 1389, -984, 337, 8, -52, 14, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 5*x^8 + 14*x^7 - 52*x^6 + 8*x^5 + 337*x^4 - 984*x^3 + 1389*x^2 - 1026*x + 324)
 
gp: K = bnfinit(x^10 - 4*x^9 + 5*x^8 + 14*x^7 - 52*x^6 + 8*x^5 + 337*x^4 - 984*x^3 + 1389*x^2 - 1026*x + 324, 1)
 

Normalized defining polynomial

\( x^{10} - 4 x^{9} + 5 x^{8} + 14 x^{7} - 52 x^{6} + 8 x^{5} + 337 x^{4} - 984 x^{3} + 1389 x^{2} - 1026 x + 324 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-43913358828288=-\,2^{8}\cdot 3^{5}\cdot 163^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.13$
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:  $2, 3, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{105534} a^{9} + \frac{109}{5863} a^{8} - \frac{1070}{17589} a^{7} - \frac{755}{17589} a^{6} + \frac{2909}{17589} a^{5} - \frac{5147}{17589} a^{4} - \frac{3941}{9594} a^{3} + \frac{5116}{17589} a^{2} - \frac{5552}{17589} a + \frac{501}{5863}$

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

Class group and class number

$C_{11}$, which has order $11$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{325}{4059} a^{9} - \frac{322}{1353} a^{8} + \frac{335}{2706} a^{7} + \frac{1742}{1353} a^{6} - \frac{1267}{451} a^{5} - \frac{3640}{1353} a^{4} + \frac{9076}{369} a^{3} - \frac{23393}{451} a^{2} + \frac{138661}{2706} a - \frac{8992}{451} \) (order $6$)
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:  \( 1113.39252057 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 10T3):

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

\(\Q(\sqrt{-3}) \), 5.1.3825936.1

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

Sibling fields

Galois closure: data not computed
Degree 10 sibling: 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 ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$163$$\Q_{163}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{163}$$x + 4$$1$$1$$0$Trivial$[\ ]$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$