Properties

Label 8.0.1513728.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 3^{4}\cdot 73$
Root discriminant $5.92$
Ramified primes $2, 3, 73$
Class number $1$
Class group Trivial
Galois Group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 4]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1513728=2^{8}\cdot 3^{4}\cdot 73\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $5.92$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 73$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

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

Class group and class number

Trivial Abelian group, 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:  $3$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( a^{7} - 2 a^{6} + 2 a^{5} - a^{3} - 2 a^{2} + 1 \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( 2 a^{7} - 6 a^{6} + 7 a^{5} - 2 a^{4} - 5 a^{3} + 2 a + 4 \),  \( 3 a^{7} - 8 a^{6} + 9 a^{5} - a^{4} - 7 a^{3} - a^{2} + 3 a + 4 \),  \( 2 a^{7} - 3 a^{6} + 2 a^{5} + 3 a^{4} - 3 a^{3} - 4 a^{2} - a + 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 2.10694405423 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 64
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\)

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$