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Properties

Label 8.0.1860940880613376.2
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 821^{4}$
Root discriminant $81.04$
Ramified primes $2, 821$
Class number $120$
Class group $[2, 60]$
Galois group $V_4^2:S_3$ (as 8T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14520, -8056, 1232, -348, 365, -72, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 2*x^6 - 72*x^5 + 365*x^4 - 348*x^3 + 1232*x^2 - 8056*x + 14520)
 
gp: K = bnfinit(x^8 - 4*x^7 + 2*x^6 - 72*x^5 + 365*x^4 - 348*x^3 + 1232*x^2 - 8056*x + 14520, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 2 x^{6} - 72 x^{5} + 365 x^{4} - 348 x^{3} + 1232 x^{2} - 8056 x + 14520 \)

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:  \(1860940880613376=2^{12}\cdot 821^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.04$
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:  $2, 821$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{74318228} a^{7} + \frac{4254895}{37159114} a^{6} - \frac{6649337}{37159114} a^{5} - \frac{676873}{18579557} a^{4} + \frac{12186505}{74318228} a^{3} + \frac{12820729}{37159114} a^{2} + \frac{1439231}{18579557} a + \frac{8249785}{18579557}$

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

Class group and class number

$C_{2}\times C_{60}$, which has order $120$

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:  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:  \( 2474.20036767 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4$ (as 8T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 10 conjugacy class representatives for $V_4^2:S_3$
Character table for $V_4^2:S_3$

Intermediate fields

\(\Q(\sqrt{-821}) \)

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
821Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_821.2t1.1c1$1$ $ 2^{2} \cdot 821 $ $x^{2} + 821$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_821.3t2.1c1$2$ $ 2^{2} \cdot 821 $ $x^{3} - x^{2} + 8 x - 10$ $S_3$ (as 3T2) $1$ $0$
3.2e6_821e2.6t8.2c1$3$ $ 2^{6} \cdot 821^{2}$ $x^{4} - 3 x^{2} - 4 x + 7$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_821.4t5.1c1$3$ $ 2^{4} \cdot 821 $ $x^{4} - 2 x^{3} + 2 x^{2} + 6 x - 11$ $S_4$ (as 4T5) $1$ $1$
3.2e4_821e2.6t8.1c1$3$ $ 2^{4} \cdot 821^{2}$ $x^{4} - x^{3} - x^{2} - 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_821.4t5.2c1$3$ $ 2^{4} \cdot 821 $ $x^{4} - 3 x^{2} - 4 x + 7$ $S_4$ (as 4T5) $1$ $1$
3.2e6_821e2.6t8.1c1$3$ $ 2^{6} \cdot 821^{2}$ $x^{4} - 2 x^{3} + 2 x^{2} + 6 x - 11$ $S_4$ (as 4T5) $1$ $-1$
3.2e2_821.4t5.1c1$3$ $ 2^{2} \cdot 821 $ $x^{4} - x^{3} - x^{2} - 2$ $S_4$ (as 4T5) $1$ $1$
* 6.2e10_821e3.8t34.1c1$6$ $ 2^{10} \cdot 821^{3}$ $x^{8} - 4 x^{7} + 2 x^{6} - 72 x^{5} + 365 x^{4} - 348 x^{3} + 1232 x^{2} - 8056 x + 14520$ $V_4^2:S_3$ (as 8T34) $1$ $0$

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 *.