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Properties

Label 8.2.9380669484375.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,3^{6}\cdot 5^{6}\cdot 7^{7}$
Root discriminant $41.83$
Ramified primes $3, 5, 7$
Class number $3$
Class group $[3]$
Galois group $\PGL(2,7)$ (as 8T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -64, 112, -112, 70, -28, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 7*x^6 - 28*x^5 + 70*x^4 - 112*x^3 + 112*x^2 - 64*x + 1)
 
gp: K = bnfinit(x^8 - x^7 + 7*x^6 - 28*x^5 + 70*x^4 - 112*x^3 + 112*x^2 - 64*x + 1, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9380669484375=-\,3^{6}\cdot 5^{6}\cdot 7^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.83$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3615} a^{7} + \frac{451}{3615} a^{6} + \frac{473}{1205} a^{5} + \frac{301}{723} a^{4} + \frac{142}{723} a^{3} - \frac{309}{1205} a^{2} + \frac{448}{3615} a - \frac{8}{3615}$

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

Class group and class number

$C_{3}$, which has order $3$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
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:  \( a \),  \( a^{2} - 2 a + 2 \),  \( \frac{117}{241} a^{7} - \frac{12}{241} a^{6} + \frac{697}{241} a^{5} - \frac{2737}{241} a^{4} + \frac{4745}{241} a^{3} - \frac{6516}{241} a^{2} + \frac{3734}{241} a + \frac{269}{241} \),  \( a^{2} - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3762.85592164 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SO(3,7)$ (as 8T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 336
The 9 conjugacy class representatives for $\PGL(2,7)$
Character table for $\PGL(2,7)$

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 16 sibling: data not computed
Degree 21 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: 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 ${\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R R R ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$7$7.8.7.2$x^{8} - 7$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
6.3e6_5e6_7e5.42t82.1c1$6$ $ 3^{6} \cdot 5^{6} \cdot 7^{5}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $0$
6.3e6_5e6_7e5.14t16.1c1$6$ $ 3^{6} \cdot 5^{6} \cdot 7^{5}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $0$
6.3e6_5e6_7e5.14t16.1c2$6$ $ 3^{6} \cdot 5^{6} \cdot 7^{5}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $0$
* 7.3e6_5e6_7e7.8t43.1c1$7$ $ 3^{6} \cdot 5^{6} \cdot 7^{7}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $1$
7.3e6_5e6_7e6.16t713.1c1$7$ $ 3^{6} \cdot 5^{6} \cdot 7^{6}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $-1$
8.3e6_5e6_7e7.42t81.1c1$8$ $ 3^{6} \cdot 5^{6} \cdot 7^{7}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $-2$
8.3e6_5e6_7e7.21t20.1c1$8$ $ 3^{6} \cdot 5^{6} \cdot 7^{7}$ $x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1$ $\PGL(2,7)$ (as 8T43) $1$ $2$

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 *.