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Properties

Label 8.0.4921953125.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{7}\cdot 251^{2}$
Root discriminant $16.27$
Ramified primes $5, 251$
Class number $1$
Class group Trivial
Galois group $(C_8:C_2):C_2$ (as 8T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1441, 158, -597, -119, 140, 26, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 17*x^6 + 26*x^5 + 140*x^4 - 119*x^3 - 597*x^2 + 158*x + 1441)
 
gp: K = bnfinit(x^8 - 2*x^7 - 17*x^6 + 26*x^5 + 140*x^4 - 119*x^3 - 597*x^2 + 158*x + 1441, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 17 x^{6} + 26 x^{5} + 140 x^{4} - 119 x^{3} - 597 x^{2} + 158 x + 1441 \)

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:  \(4921953125=5^{7}\cdot 251^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.27$
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:  $5, 251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{126418559} a^{7} + \frac{48937417}{126418559} a^{6} - \frac{25655232}{126418559} a^{5} + \frac{49907636}{126418559} a^{4} - \frac{41482416}{126418559} a^{3} - \frac{4972287}{126418559} a^{2} + \frac{55729586}{126418559} a + \frac{15205085}{126418559}$

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:  $3$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{91471}{126418559} a^{7} - \frac{285224}{126418559} a^{6} - \frac{2015555}{126418559} a^{5} + \frac{788507}{126418559} a^{4} + \frac{14974449}{126418559} a^{3} + \frac{33911105}{126418559} a^{2} - \frac{61012110}{126418559} a - \frac{159074642}{126418559} \) (order $10$)
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:  \( 122.087114428 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 8T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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

Sibling fields

Galois closure: data not computed
Degree 8 sibling: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
251Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.251.2t1.1c1$1$ $ 251 $ $x^{2} - x + 63$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5_251.2t1.1c1$1$ $ 5 \cdot 251 $ $x^{2} - x + 314$ $C_2$ (as 2T1) $1$ $-1$
1.5_251.4t1.1c1$1$ $ 5 \cdot 251 $ $x^{4} - x^{3} - 314 x^{2} + 314 x + 19531$ $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
1.5_251.4t1.1c2$1$ $ 5 \cdot 251 $ $x^{4} - x^{3} - 314 x^{2} + 314 x + 19531$ $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
2.5e2_251.4t3.2c1$2$ $ 5^{2} \cdot 251 $ $x^{4} - 5 x^{2} + 320$ $D_{4}$ (as 4T3) $1$ $0$
2.5_251.4t3.2c1$2$ $ 5 \cdot 251 $ $x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 69$ $D_{4}$ (as 4T3) $1$ $0$
* 4.5e4_251e2.8t16.1c1$4$ $ 5^{4} \cdot 251^{2}$ $x^{8} - 2 x^{7} - 17 x^{6} + 26 x^{5} + 140 x^{4} - 119 x^{3} - 597 x^{2} + 158 x + 1441$ $(C_8:C_2):C_2$ (as 8T16) $1$ $0$

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 *.