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Properties

Label 10.4.72758649087.1
Degree $10$
Signature $[4, 3]$
Discriminant $-\,3^{8}\cdot 223^{3}$
Root discriminant $12.20$
Ramified primes $3, 223$
Class number $1$
Class group Trivial
Galois group $S_5$ (as 10T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 0, 3, -6, -3, -3, -6, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 3*x^8 - 6*x^7 - 3*x^6 - 3*x^5 - 6*x^4 + 3*x^3 + 2*x + 1)
 
gp: K = bnfinit(x^10 - x^9 + 3*x^8 - 6*x^7 - 3*x^6 - 3*x^5 - 6*x^4 + 3*x^3 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 3 x^{8} - 6 x^{7} - 3 x^{6} - 3 x^{5} - 6 x^{4} + 3 x^{3} + 2 x + 1 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-72758649087=-\,3^{8}\cdot 223^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.20$
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:  $3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3}$

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:  $6$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 30.6901611785 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 10T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
223Data not computed