Properties

Label 8.0.777431921841.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{4}\cdot 313^{4}$
Root discriminant $30.64$
Ramified primes $3, 313$
Class number $2$
Class group $[2]$
Galois Group $C_2^3:(C_7: C_3)$ (as 8T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, 22, 21, 24, -4, -9, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 9*x^6 - 4*x^5 + 24*x^4 + 21*x^3 + 22*x^2 - 3*x + 3)
gp: K = bnfinit(x^8 - 9*x^6 - 4*x^5 + 24*x^4 + 21*x^3 + 22*x^2 - 3*x + 3, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 9 x^{6} \) \(\mathstrut -\mathstrut 4 x^{5} \) \(\mathstrut +\mathstrut 24 x^{4} \) \(\mathstrut +\mathstrut 21 x^{3} \) \(\mathstrut +\mathstrut 22 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 3 \)

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:  \(777431921841=3^{4}\cdot 313^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $30.64$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 313$
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 \(a\))

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

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:  \( \frac{5}{9} a^{7} - \frac{2}{3} a^{6} - \frac{47}{9} a^{5} + \frac{35}{9} a^{4} + 18 a^{3} - \frac{41}{9} a^{2} - \frac{22}{3} a - \frac{53}{3} \),  \( \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{19}{9} a^{5} + \frac{14}{9} a^{4} + \frac{61}{9} a^{3} - \frac{40}{9} a^{2} + \frac{5}{3} a - \frac{1}{3} \),  \( \frac{1}{3} a^{7} + \frac{1}{9} a^{6} - \frac{26}{9} a^{5} - \frac{8}{3} a^{4} + \frac{67}{9} a^{3} + \frac{91}{9} a^{2} + 10 a + \frac{4}{3} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 407.692725067 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$F_8:C_3$ (as 8T36):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 168
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$
Character table for $C_2^3:(C_7: C_3)$

Intermediate fields

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

Sibling fields

Degree 14 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
313Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.313.3t1.1c1$1$ $ 313 $ $x^{3} - x^{2} - 104 x - 371$ $C_3$ (as 3T1) $0$ $1$
1.313.3t1.1c2$1$ $ 313 $ $x^{3} - x^{2} - 104 x - 371$ $C_3$ (as 3T1) $0$ $1$
3.313e2.7t3.1c1$3$ $ 313^{2}$ $x^{7} - x^{6} - 15 x^{5} + 20 x^{4} + 33 x^{3} - 22 x^{2} - 32 x - 8$ $C_7:C_3$ (as 7T3) $0$ $3$
3.313e2.7t3.1c2$3$ $ 313^{2}$ $x^{7} - x^{6} - 15 x^{5} + 20 x^{4} + 33 x^{3} - 22 x^{2} - 32 x - 8$ $C_7:C_3$ (as 7T3) $0$ $3$
* 7.3e4_313e4.8t36.1c1$7$ $ 3^{4} \cdot 313^{4}$ $x^{8} - 9 x^{6} - 4 x^{5} + 24 x^{4} + 21 x^{3} + 22 x^{2} - 3 x + 3$ $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.3e4_313e5.24t283.1c1$7$ $ 3^{4} \cdot 313^{5}$ $x^{8} - 9 x^{6} - 4 x^{5} + 24 x^{4} + 21 x^{3} + 22 x^{2} - 3 x + 3$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.3e4_313e5.24t283.1c2$7$ $ 3^{4} \cdot 313^{5}$ $x^{8} - 9 x^{6} - 4 x^{5} + 24 x^{4} + 21 x^{3} + 22 x^{2} - 3 x + 3$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.