Properties

Label 8.4.65289632040.1
Degree $8$
Signature $[4, 2]$
Discriminant $65289632040$
Root discriminant $22.48$
Ramified primes $2, 3, 5, 67$
Class number $1$
Class group trivial
Galois group $S_4\wr C_2$ (as 8T47)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 + 11*x^5 - 5*x^4 - 12*x^3 - 34*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^8 - 2*x^7 - 3*x^6 + 11*x^5 - 5*x^4 - 12*x^3 - 34*x^2 - 7*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, -34, -12, -5, 11, -3, -2, 1]);
 

\( x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(65289632040\)\(\medspace = 2^{3}\cdot 3^{4}\cdot 5\cdot 67^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.48$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5, 67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4295} a^{7} - \frac{2001}{4295} a^{6} + \frac{1351}{4295} a^{5} + \frac{917}{4295} a^{4} + \frac{877}{4295} a^{3} - \frac{155}{859} a^{2} - \frac{1304}{4295} a - \frac{376}{4295}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 952.155611166 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{2}\cdot 952.155611166 \cdot 1}{2\sqrt{65289632040}}\approx 1.17688911015$

Galois group

$S_4\wr C_2$ (as 8T47):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1152
The 20 conjugacy class representatives for $S_4\wr C_2$
Character table for $S_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{201}) \)

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 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.40.2t1.a.a$1$ $ 2^{3} \cdot 5 $ $x^{2} - 10$ $C_2$ (as 2T1) $1$ $1$
1.8040.2t1.b.a$1$ $ 2^{3} \cdot 3 \cdot 5 \cdot 67 $ $x^{2} - 2010$ $C_2$ (as 2T1) $1$ $1$
* 1.201.2t1.a.a$1$ $ 3 \cdot 67 $ $x^{2} - x - 50$ $C_2$ (as 2T1) $1$ $1$
2.8040.4t3.b.a$2$ $ 2^{3} \cdot 3 \cdot 5 \cdot 67 $ $x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 15$ $D_{4}$ (as 4T3) $1$ $-2$
4.12992961600.12t34.g.a$4$ $ 2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 67^{3}$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2585664000.12t34.c.a$4$ $ 2^{9} \cdot 3^{2} \cdot 5^{3} \cdot 67^{2}$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.321600.6t13.e.a$4$ $ 2^{6} \cdot 3 \cdot 5^{2} \cdot 67 $ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.1616040.6t13.c.a$4$ $ 2^{3} \cdot 3^{2} \cdot 5 \cdot 67^{2}$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
6.207...000.12t201.a.a$6$ $ 2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.831...000.12t202.a.a$6$ $ 2^{15} \cdot 3^{3} \cdot 5^{5} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $2$
* 6.324824040.8t47.a.a$6$ $ 2^{3} \cdot 3^{3} \cdot 5 \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.12992961600.12t200.a.a$6$ $ 2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.519718464000.16t1294.a.a$9$ $ 2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.332...000.18t272.a.a$9$ $ 2^{18} \cdot 3^{3} \cdot 5^{6} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.270...000.18t273.a.a$9$ $ 2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.422...000.18t274.a.a$9$ $ 2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
12.108...000.36t1763.a.a$12$ $ 2^{21} \cdot 3^{6} \cdot 5^{7} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $0$
12.675...000.24t2821.a.a$12$ $ 2^{15} \cdot 3^{6} \cdot 5^{5} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $0$
18.140...000.36t1758.a.a$18$ $ 2^{27} \cdot 3^{9} \cdot 5^{9} \cdot 67^{9}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$

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 *.