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Properties

Label 8.8.12782719397154721.1
Degree $8$
Signature $[8, 0]$
Discriminant $7^{12}\cdot 31^{4}$
Root discriminant $103.12$
Ramified primes $7, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3:C_7$ (as 8T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![733, 1577, -987, -854, 420, 77, -42, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733)
 
gp: K = bnfinit(x^8 - x^7 - 42*x^6 + 77*x^5 + 420*x^4 - 854*x^3 - 987*x^2 + 1577*x + 733, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 42 x^{6} + 77 x^{5} + 420 x^{4} - 854 x^{3} - 987 x^{2} + 1577 x + 733 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12782719397154721=7^{12}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.12$
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:  $7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1081841} a^{7} + \frac{172360}{1081841} a^{6} - \frac{293783}{1081841} a^{5} - \frac{81740}{1081841} a^{4} + \frac{27623}{1081841} a^{3} - \frac{55192}{1081841} a^{2} - \frac{321386}{1081841} a + \frac{175795}{1081841}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 942691.948946 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_8$ (as 8T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 56
The 8 conjugacy class representatives for $C_2^3:C_7$
Character table for $C_2^3:C_7$

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 28 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 ${\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7e2.7t1.1c1$1$ $ 7^{2}$ $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ $C_7$ (as 7T1) $0$ $1$
1.7e2.7t1.1c2$1$ $ 7^{2}$ $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ $C_7$ (as 7T1) $0$ $1$
1.7e2.7t1.1c3$1$ $ 7^{2}$ $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ $C_7$ (as 7T1) $0$ $1$
1.7e2.7t1.1c4$1$ $ 7^{2}$ $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ $C_7$ (as 7T1) $0$ $1$
1.7e2.7t1.1c5$1$ $ 7^{2}$ $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ $C_7$ (as 7T1) $0$ $1$
1.7e2.7t1.1c6$1$ $ 7^{2}$ $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$ $C_7$ (as 7T1) $0$ $1$
* 7.7e12_31e4.8t25.1c1$7$ $ 7^{12} \cdot 31^{4}$ $x^{8} - x^{7} - 42 x^{6} + 77 x^{5} + 420 x^{4} - 854 x^{3} - 987 x^{2} + 1577 x + 733$ $C_2^3:C_7$ (as 8T25) $1$ $7$

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 *.