Properties

Label 8.0.618765625.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{6}\cdot 199^{2}$
Root discriminant $12.56$
Ramified primes $5, 199$
Class number $1$
Class group Trivial
Galois Group $C_2^2:C_4$ (as 8T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 66, -72, -70, 21, 20, -3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 - 3*x^6 + 20*x^5 + 21*x^4 - 70*x^3 - 72*x^2 + 66*x + 121)
gp: K = bnfinit(x^8 - 3*x^7 - 3*x^6 + 20*x^5 + 21*x^4 - 70*x^3 - 72*x^2 + 66*x + 121, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut -\mathstrut 3 x^{6} \) \(\mathstrut +\mathstrut 20 x^{5} \) \(\mathstrut +\mathstrut 21 x^{4} \) \(\mathstrut -\mathstrut 70 x^{3} \) \(\mathstrut -\mathstrut 72 x^{2} \) \(\mathstrut +\mathstrut 66 x \) \(\mathstrut +\mathstrut 121 \)

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:  \(618765625=5^{6}\cdot 199^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.56$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{18117} a^{7} + \frac{382}{18117} a^{6} + \frac{2131}{18117} a^{5} + \frac{1730}{6039} a^{4} + \frac{589}{2013} a^{3} - \frac{6406}{18117} a^{2} - \frac{2470}{18117} a - \frac{800}{1647}$

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:  $3$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{112}{18117} a^{7} + \frac{511}{18117} a^{6} - \frac{2888}{18117} a^{5} + \frac{512}{6039} a^{4} + \frac{1552}{2013} a^{3} + \frac{7208}{18117} a^{2} - \frac{35080}{18117} a - \frac{1760}{1647} \) (order $10$)
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:  \( 68.1474054485 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2^2:C_4$ (as 8T10):

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 $C_2^2:C_4$
Character table for $C_2^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.4975.1, \(\Q(\zeta_{5})\), 4.2.24875.1

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 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$199$199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$