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Properties

Label 8.8.31437473310108649.1
Degree $8$
Signature $[8, 0]$
Discriminant $7^{4}\cdot 19^{2}\cdot 43^{4}\cdot 103^{2}$
Root discriminant $115.39$
Ramified primes $7, 19, 43, 103$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $V_4^2:(S_3\times C_2)$ (as 8T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-77, -546, -650, 360, 477, -48, -63, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 - 63*x^6 - 48*x^5 + 477*x^4 + 360*x^3 - 650*x^2 - 546*x - 77)
 
gp: K = bnfinit(x^8 - 3*x^7 - 63*x^6 - 48*x^5 + 477*x^4 + 360*x^3 - 650*x^2 - 546*x - 77, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} - 63 x^{6} - 48 x^{5} + 477 x^{4} + 360 x^{3} - 650 x^{2} - 546 x - 77 \)

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:  \(31437473310108649=7^{4}\cdot 19^{2}\cdot 43^{4}\cdot 103^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.39$
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:  $7, 19, 43, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{316025} a^{7} + \frac{29376}{316025} a^{6} - \frac{26834}{316025} a^{5} + \frac{63036}{316025} a^{4} + \frac{91826}{316025} a^{3} - \frac{149011}{316025} a^{2} - \frac{153314}{316025} a - \frac{97842}{316025}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 826201.083964 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4:C_2$ (as 8T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$
Character table for $V_4^2:(S_3\times C_2)$

Intermediate fields

\(\Q(\sqrt{301}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.2.2$x^{4} - 103 x^{2} + 53045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.7_43.2t1.1c1$1$ $ 7 \cdot 43 $ $x^{2} - x - 75$ $C_2$ (as 2T1) $1$ $1$
1.7_19_43_103.2t1.1c1$1$ $ 7 \cdot 19 \cdot 43 \cdot 103 $ $x^{2} - x - 147264$ $C_2$ (as 2T1) $1$ $1$
1.19_103.2t1.1c1$1$ $ 19 \cdot 103 $ $x^{2} - x - 489$ $C_2$ (as 2T1) $1$ $1$
2.7e2_19_43e2_103.6t3.1c1$2$ $ 7^{2} \cdot 19 \cdot 43^{2} \cdot 103 $ $x^{6} - x^{5} - 244 x^{4} + 149 x^{3} + 17041 x^{2} - 8686 x - 325940$ $D_{6}$ (as 6T3) $1$ $2$
2.19_103.3t2.1c1$2$ $ 19 \cdot 103 $ $x^{3} - x^{2} - 9 x + 10$ $S_3$ (as 3T2) $1$ $2$
3.19e2_103e2.6t8.1c1$3$ $ 19^{2} \cdot 103^{2}$ $x^{4} - 4 x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $3$
3.7e3_19_43e3_103.6t11.1c1$3$ $ 7^{3} \cdot 19 \cdot 43^{3} \cdot 103 $ $x^{6} - 3 x^{5} - 6543 x^{4} + 13091 x^{3} + 7934756 x^{2} - 7941302 x - 831907285$ $S_4\times C_2$ (as 6T11) $1$ $3$
3.7e3_19e2_43e3_103e2.6t11.1c1$3$ $ 7^{3} \cdot 19^{2} \cdot 43^{3} \cdot 103^{2}$ $x^{6} - 3 x^{5} - 6543 x^{4} + 13091 x^{3} + 7934756 x^{2} - 7941302 x - 831907285$ $S_4\times C_2$ (as 6T11) $1$ $3$
3.19_103.4t5.1c1$3$ $ 19 \cdot 103 $ $x^{4} - 4 x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $3$
6.7e3_19e3_43e3_103e3.8t41.1c1$6$ $ 7^{3} \cdot 19^{3} \cdot 43^{3} \cdot 103^{3}$ $x^{8} - 3 x^{7} - 63 x^{6} - 48 x^{5} + 477 x^{4} + 360 x^{3} - 650 x^{2} - 546 x - 77$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $6$
6.7e3_19e4_43e3_103e4.12t111.1c1$6$ $ 7^{3} \cdot 19^{4} \cdot 43^{3} \cdot 103^{4}$ $x^{8} - 3 x^{7} - 63 x^{6} - 48 x^{5} + 477 x^{4} + 360 x^{3} - 650 x^{2} - 546 x - 77$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $6$
* 6.7e3_19e2_43e3_103e2.8t41.1c1$6$ $ 7^{3} \cdot 19^{2} \cdot 43^{3} \cdot 103^{2}$ $x^{8} - 3 x^{7} - 63 x^{6} - 48 x^{5} + 477 x^{4} + 360 x^{3} - 650 x^{2} - 546 x - 77$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $6$
6.7e3_19e3_43e3_103e3.12t108.1c1$6$ $ 7^{3} \cdot 19^{3} \cdot 43^{3} \cdot 103^{3}$ $x^{8} - 3 x^{7} - 63 x^{6} - 48 x^{5} + 477 x^{4} + 360 x^{3} - 650 x^{2} - 546 x - 77$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $6$

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 *.