Properties

Label 8.2.32941720000.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{6}\cdot 5^{4}\cdot 7^{7}$
Root discriminant $20.64$
Ramified primes $2, 5, 7$
Class number $1$
Class group Trivial
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![2, -10, -14, 0, 14, 0, 0, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 14*x^4 - 14*x^2 - 10*x + 2)
gp: K = bnfinit(x^8 - 3*x^7 + 14*x^4 - 14*x^2 - 10*x + 2, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut 14 x^{4} \) \(\mathstrut -\mathstrut 14 x^{2} \) \(\mathstrut -\mathstrut 10 x \) \(\mathstrut +\mathstrut 2 \)

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:  \(-32941720000=-\,2^{6}\cdot 5^{4}\cdot 7^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $20.64$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{119} a^{7} - \frac{4}{17} a^{6} - \frac{2}{17} a^{5} - \frac{1}{17} a^{4} - \frac{7}{17} a^{3} + \frac{5}{17} a^{2} - \frac{8}{17} a - \frac{38}{119}$

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:  $4$
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:  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:  \( 588.295128619 \)
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 R ${\href{/LocalNumberField/3.8.0.1}{8} }$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.7.7.4$x^{7} + 14 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$

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) $1$ $-1$
6.2e6_5e4_7e7.42t82.1c1$6$ $ 2^{6} \cdot 5^{4} \cdot 7^{7}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (as 8T43) $1$ $0$
6.2e6_5e4_7e7.14t16.1c1$6$ $ 2^{6} \cdot 5^{4} \cdot 7^{7}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (as 8T43) $1$ $0$
6.2e6_5e4_7e7.14t16.1c2$6$ $ 2^{6} \cdot 5^{4} \cdot 7^{7}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (as 8T43) $1$ $0$
* 7.2e6_5e4_7e7.8t43.1c1$7$ $ 2^{6} \cdot 5^{4} \cdot 7^{7}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (as 8T43) $1$ $1$
7.2e6_5e4_7e8.16t713.1c1$7$ $ 2^{6} \cdot 5^{4} \cdot 7^{8}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (as 8T43) $1$ $-1$
8.2e6_5e6_7e9.42t81.1c1$8$ $ 2^{6} \cdot 5^{6} \cdot 7^{9}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (as 8T43) $1$ $-2$
8.2e6_5e6_7e9.21t20.1c1$8$ $ 2^{6} \cdot 5^{6} \cdot 7^{9}$ $x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ $\PGL(2,7)$ (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 *.