Properties

Label 8.8.355444453125.1
Degree $8$
Signature $[8, 0]$
Discriminant $3^{6}\cdot 5^{7}\cdot 79^{2}$
Root discriminant $27.79$
Ramified primes $3, 5, 79$
Class number $1$
Class group Trivial
Galois Group $C_8:C_2$ (as 8T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![271, -962, -1097, 121, 325, 16, -32, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 32*x^6 + 16*x^5 + 325*x^4 + 121*x^3 - 1097*x^2 - 962*x + 271)
gp: K = bnfinit(x^8 - 2*x^7 - 32*x^6 + 16*x^5 + 325*x^4 + 121*x^3 - 1097*x^2 - 962*x + 271, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut -\mathstrut 32 x^{6} \) \(\mathstrut +\mathstrut 16 x^{5} \) \(\mathstrut +\mathstrut 325 x^{4} \) \(\mathstrut +\mathstrut 121 x^{3} \) \(\mathstrut -\mathstrut 1097 x^{2} \) \(\mathstrut -\mathstrut 962 x \) \(\mathstrut +\mathstrut 271 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[8, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(355444453125=3^{6}\cdot 5^{7}\cdot 79^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $27.79$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{1713} a^{7} + \frac{269}{1713} a^{6} - \frac{221}{1713} a^{5} + \frac{80}{1713} a^{4} - \frac{88}{571} a^{3} - \frac{619}{1713} a^{2} + \frac{247}{571} a - \frac{572}{1713}$

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:  $7$
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:  \( \frac{7}{571} a^{7} - \frac{61}{1713} a^{6} - \frac{644}{1713} a^{5} + \frac{1109}{1713} a^{4} + \frac{5876}{1713} a^{3} - \frac{5005}{1713} a^{2} - \frac{16415}{1713} a + \frac{1121}{1713} \),  \( \frac{332}{1713} a^{7} - \frac{1481}{1713} a^{6} - \frac{7136}{1713} a^{5} + \frac{23134}{1713} a^{4} + \frac{55102}{1713} a^{3} - \frac{95305}{1713} a^{2} - \frac{51610}{571} a + \frac{44206}{1713} \),  \( \frac{163}{1713} a^{7} - \frac{691}{1713} a^{6} - \frac{3476}{1713} a^{5} + \frac{3395}{571} a^{4} + \frac{24917}{1713} a^{3} - \frac{39229}{1713} a^{2} - \frac{61366}{1713} a + \frac{16967}{1713} \),  \( \frac{376}{1713} a^{7} - \frac{1636}{1713} a^{6} - \frac{2765}{571} a^{5} + \frac{8504}{571} a^{4} + \frac{21728}{571} a^{3} - \frac{33995}{571} a^{2} - \frac{184465}{1713} a + \frac{24748}{1713} \),  \( \frac{2}{1713} a^{7} - \frac{11}{571} a^{6} + \frac{43}{571} a^{5} + \frac{160}{1713} a^{4} - \frac{1670}{1713} a^{3} + \frac{539}{571} a^{2} + \frac{1636}{571} a - \frac{7996}{1713} \),  \( \frac{332}{1713} a^{7} - \frac{1481}{1713} a^{6} - \frac{7136}{1713} a^{5} + \frac{23134}{1713} a^{4} + \frac{55102}{1713} a^{3} - \frac{95305}{1713} a^{2} - \frac{52181}{571} a + \frac{40780}{1713} \),  \( \frac{34}{1713} a^{7} + \frac{10}{1713} a^{6} - \frac{411}{571} a^{5} - \frac{1277}{1713} a^{4} + \frac{10438}{1713} a^{3} + \frac{4595}{571} a^{2} - \frac{23912}{1713} a - \frac{34865}{1713} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1058.96238593 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$OD_{16}$ (as 8T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $C_8:C_2$
Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$79$79.4.2.2$x^{4} - 79 x^{2} + 18723$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
79.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$