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Properties

Label 8.0.1135929640000.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{6}\cdot 5^{4}\cdot 73^{4}$
Root discriminant $32.13$
Ramified primes $2, 5, 73$
Class number $4$
Class group $[2, 2]$
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![64, -80, 80, -56, 56, -20, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 16*x^6 - 20*x^5 + 56*x^4 - 56*x^3 + 80*x^2 - 80*x + 64)
 
gp: K = bnfinit(x^8 - 2*x^7 + 16*x^6 - 20*x^5 + 56*x^4 - 56*x^3 + 80*x^2 - 80*x + 64, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 16 x^{6} - 20 x^{5} + 56 x^{4} - 56 x^{3} + 80 x^{2} - 80 x + 64 \)

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:  \(1135929640000=2^{6}\cdot 5^{4}\cdot 73^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.13$
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:  $2, 5, 73$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{80} a^{7} + \frac{1}{40} a^{6} + \frac{1}{20} a^{5} - \frac{1}{20} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{7}{40} a^{7} + \frac{1}{10} a^{6} + \frac{49}{20} a^{5} + \frac{61}{20} a^{4} + 9 a^{3} + \frac{56}{5} a^{2} + \frac{69}{5} a + \frac{51}{5} \),  \( \frac{1}{40} a^{7} - \frac{1}{5} a^{6} + \frac{17}{20} a^{5} - \frac{57}{20} a^{4} + \frac{11}{2} a^{3} - \frac{32}{5} a^{2} + \frac{22}{5} a + \frac{3}{5} \),  \( \frac{1}{2} a^{7} - a^{6} + \frac{15}{2} a^{5} - 8 a^{4} + 18 a^{3} - 4 a^{2} - 2 a + 9 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 689.315412453 \)
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 R ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.6.4.1$x^{6} + 2336 x^{3} + 7092899$$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$1$ $1$ $x$ $C_1$ $1$ $1$
1.73.3t1.1c1$1$ $ 73 $ $x^{3} - x^{2} - 24 x + 27$ $C_3$ (as 3T1) $0$ $1$
1.73.3t1.1c2$1$ $ 73 $ $x^{3} - x^{2} - 24 x + 27$ $C_3$ (as 3T1) $0$ $1$
3.2e3_73e2.7t3.1c1$3$ $ 2^{3} \cdot 73^{2}$ $x^{7} - 8 x^{5} - 2 x^{4} + 16 x^{3} + 6 x^{2} - 6 x - 2$ $C_7:C_3$ (as 7T3) $0$ $3$
3.2e3_73e2.7t3.1c2$3$ $ 2^{3} \cdot 73^{2}$ $x^{7} - 8 x^{5} - 2 x^{4} + 16 x^{3} + 6 x^{2} - 6 x - 2$ $C_7:C_3$ (as 7T3) $0$ $3$
* 7.2e6_5e4_73e4.8t36.1c1$7$ $ 2^{6} \cdot 5^{4} \cdot 73^{4}$ $x^{8} - 2 x^{7} + 16 x^{6} - 20 x^{5} + 56 x^{4} - 56 x^{3} + 80 x^{2} - 80 x + 64$ $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.2e6_5e4_73e5.24t283.1c1$7$ $ 2^{6} \cdot 5^{4} \cdot 73^{5}$ $x^{8} - 2 x^{7} + 16 x^{6} - 20 x^{5} + 56 x^{4} - 56 x^{3} + 80 x^{2} - 80 x + 64$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.2e6_5e4_73e5.24t283.1c2$7$ $ 2^{6} \cdot 5^{4} \cdot 73^{5}$ $x^{8} - 2 x^{7} + 16 x^{6} - 20 x^{5} + 56 x^{4} - 56 x^{3} + 80 x^{2} - 80 x + 64$ $C_2^3:(C_7: C_3)$ (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 *.