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Properties

Label 8.0.1625943765625.5
Degree $8$
Signature $[0, 4]$
Discriminant $5^{6}\cdot 101^{4}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $16$
Class group $[2, 2, 4]$
Galois group $D_4$ (as 8T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, -1089, 0, 136, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 9*x^6 + 136*x^4 - 1089*x^2 + 14641)
 
gp: K = bnfinit(x^8 - 9*x^6 + 136*x^4 - 1089*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{8} - 9 x^{6} + 136 x^{4} - 1089 x^{2} + 14641 \)

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:  \(1625943765625=5^{6}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{4} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{66} a^{5} - \frac{3}{22} a^{3} - \frac{1}{2} a^{2} + \frac{13}{33} a - \frac{1}{2}$, $\frac{1}{5808} a^{6} - \frac{31}{484} a^{4} + \frac{155}{1452} a^{2} - \frac{1}{2} a + \frac{1}{16}$, $\frac{1}{127776} a^{7} - \frac{1}{11616} a^{6} + \frac{149}{31944} a^{5} - \frac{149}{2904} a^{4} - \frac{5653}{31944} a^{3} - \frac{155}{2904} a^{2} - \frac{7}{96} a - \frac{19}{96}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$

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:  \( \frac{13}{63888} a^{7} + \frac{1}{15972} a^{5} - \frac{163}{15972} a^{3} - \frac{25}{528} a - \frac{1}{2} \),  \( \frac{35}{31944} a^{7} - \frac{109}{7986} a^{5} + \frac{1795}{7986} a^{3} - \frac{85}{24} a - 10 \),  \( \frac{47}{31944} a^{7} + \frac{227}{7986} a^{5} - \frac{5057}{7986} a^{3} + \frac{1645}{264} a \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 170.570009535 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4$ (as 8T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8
The 5 conjugacy class representatives for $D_4$
Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.0.12625.1 x2, 4.0.1275125.1 x2

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

Sibling fields

Degree 4 siblings: 4.0.12625.1, 4.0.1275125.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5_101.2t1.1c1$1$ $ 5 \cdot 101 $ $x^{2} - x - 126$ $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.101.2t1.1c1$1$ $ 101 $ $x^{2} - x - 25$ $C_2$ (as 2T1) $1$ $1$
*2 2.5e2_101.4t3.3c1$2$ $ 5^{2} \cdot 101 $ $x^{8} - 9 x^{6} + 136 x^{4} - 1089 x^{2} + 14641$ $D_4$ (as 8T4) $1$ $-2$

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 *.