Properties

Label 8.0.8208541201.1
Degree $8$
Signature $[0, 4]$
Discriminant $7^{4}\cdot 43^{4}$
Root discriminant $17.35$
Ramified primes $7, 43$
Class number $2$
Class group $[2]$
Galois Group $A_4\times C_2$ (as 8T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 7, -3, 27, 3, 7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 7*x^6 + 3*x^5 + 27*x^4 - 3*x^3 + 7*x^2 + x + 1)
gp: K = bnfinit(x^8 - x^7 + 7*x^6 + 3*x^5 + 27*x^4 - 3*x^3 + 7*x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut 7 x^{6} \) \(\mathstrut +\mathstrut 3 x^{5} \) \(\mathstrut +\mathstrut 27 x^{4} \) \(\mathstrut -\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut 7 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(8208541201=7^{4}\cdot 43^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $17.35$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{3} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{119} a^{7} - \frac{5}{119} a^{6} - \frac{1}{17} a^{5} - \frac{37}{119} a^{4} + \frac{8}{17} a^{3} - \frac{40}{119} a^{2} + \frac{48}{119} a - \frac{3}{17}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a \),  \( \frac{18}{119} a^{7} + \frac{12}{119} a^{6} + \frac{78}{119} a^{5} + \frac{286}{119} a^{4} + \frac{447}{119} a^{3} + \frac{708}{119} a^{2} - \frac{360}{119} a + \frac{234}{119} \),  \( \frac{87}{119} a^{7} - \frac{78}{119} a^{6} + \frac{83}{17} a^{5} + \frac{351}{119} a^{4} + \frac{322}{17} a^{3} - \frac{29}{119} a^{2} + \frac{130}{119} a + \frac{45}{17} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 25.061520038 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times A_4$ (as 8T13):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 24
The 8 conjugacy class representatives for $A_4\times C_2$
Character table for $A_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.90601.1

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 6 sibling: data not computed
Degree 12 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$