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Properties

Label 8.0.2528484794641.1
Degree $8$
Signature $[0, 4]$
Discriminant $13^{4}\cdot 97^{4}$
Root discriminant $35.51$
Ramified primes $13, 97$
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![9, 13, 9, 17, 31, 19, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + x^6 + 19*x^5 + 31*x^4 + 17*x^3 + 9*x^2 + 13*x + 9)
 
gp: K = bnfinit(x^8 - 2*x^7 + x^6 + 19*x^5 + 31*x^4 + 17*x^3 + 9*x^2 + 13*x + 9, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + x^{6} + 19 x^{5} + 31 x^{4} + 17 x^{3} + 9 x^{2} + 13 x + 9 \)

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:  \(2528484794641=13^{4}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.51$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 97$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{35} a^{7} - \frac{3}{7} a^{6} - \frac{2}{5} a^{5} - \frac{9}{35} a^{4} + \frac{8}{35} a^{3} - \frac{17}{35} a^{2} - \frac{3}{7} a - \frac{2}{35}$

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:  \( \frac{33}{35} a^{7} - \frac{15}{7} a^{6} + \frac{4}{5} a^{5} + \frac{718}{35} a^{4} + \frac{649}{35} a^{3} + \frac{104}{35} a^{2} + \frac{41}{7} a + \frac{389}{35} \),  \( \frac{274}{35} a^{7} - \frac{122}{7} a^{6} + \frac{32}{5} a^{5} + \frac{5864}{35} a^{4} + \frac{5902}{35} a^{3} + \frac{1327}{35} a^{2} + \frac{263}{7} a + \frac{3757}{35} \),  \( \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{11}{5} a^{4} + \frac{23}{5} a^{3} + \frac{13}{5} a^{2} - a - \frac{7}{5} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1031.97284673 \)
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 ${\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.6.4.3$x^{6} + 873 x^{3} + 235225$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.13_97.3t1.1c1$1$ $ 13 \cdot 97 $ $x^{3} - x^{2} - 420 x + 1728$ $C_3$ (as 3T1) $0$ $1$
1.13_97.3t1.1c2$1$ $ 13 \cdot 97 $ $x^{3} - x^{2} - 420 x + 1728$ $C_3$ (as 3T1) $0$ $1$
3.13e2_97e2.7t3.1c1$3$ $ 13^{2} \cdot 97^{2}$ $x^{7} - 2 x^{6} - 20 x^{5} + 29 x^{4} + 119 x^{3} - 91 x^{2} - 234 x + 13$ $C_7:C_3$ (as 7T3) $0$ $3$
3.13e2_97e2.7t3.1c2$3$ $ 13^{2} \cdot 97^{2}$ $x^{7} - 2 x^{6} - 20 x^{5} + 29 x^{4} + 119 x^{3} - 91 x^{2} - 234 x + 13$ $C_7:C_3$ (as 7T3) $0$ $3$
* 7.13e4_97e4.8t36.1c1$7$ $ 13^{4} \cdot 97^{4}$ $x^{8} - 2 x^{7} + x^{6} + 19 x^{5} + 31 x^{4} + 17 x^{3} + 9 x^{2} + 13 x + 9$ $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.13e5_97e5.24t283.1c1$7$ $ 13^{5} \cdot 97^{5}$ $x^{8} - 2 x^{7} + x^{6} + 19 x^{5} + 31 x^{4} + 17 x^{3} + 9 x^{2} + 13 x + 9$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.13e5_97e5.24t283.1c2$7$ $ 13^{5} \cdot 97^{5}$ $x^{8} - 2 x^{7} + x^{6} + 19 x^{5} + 31 x^{4} + 17 x^{3} + 9 x^{2} + 13 x + 9$ $C_2^3:(C_7: C_3)$ (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 *.