Properties

Label 8.0.173946175488.1
Degree $8$
Signature $[0, 4]$
Discriminant $173946175488$
Root discriminant $25.41$
Ramified primes $2, 3$
Class number $18$
Class group $[3, 6]$
Galois group $C_8$ (as 8T1)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 24*x^6 + 180*x^4 + 432*x^2 + 162)
 
gp: K = bnfinit(x^8 + 24*x^6 + 180*x^4 + 432*x^2 + 162, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![162, 0, 432, 0, 180, 0, 24, 0, 1]);
 

\(x^{8} + 24 x^{6} + 180 x^{4} + 432 x^{2} + 162\)  Toggle raw display

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(173946175488\)\(\medspace = 2^{31}\cdot 3^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.41$
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:  $2, 3$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $8$
This field is Galois and abelian over $\Q$.
Conductor:  \(96=2^{5}\cdot 3\)
Dirichlet character group:    $\lbrace$$\chi_{96}(1,·)$, $\chi_{96}(5,·)$, $\chi_{96}(73,·)$, $\chi_{96}(77,·)$, $\chi_{96}(49,·)$, $\chi_{96}(53,·)$, $\chi_{96}(25,·)$, $\chi_{96}(29,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$  Toggle raw display

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{1}{9} a^{4} + \frac{4}{3} a^{2} + 1 \),  \( \frac{1}{3} a^{2} + 1 \),  \( \frac{1}{27} a^{6} + \frac{2}{3} a^{4} + 3 a^{2} + 1 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 19.534360053 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 19.534360053 \cdot 18}{2\sqrt{173946175488}}\approx 0.65698239029$

Galois group

$C_8$ (as 8T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 8
The 8 conjugacy class representatives for $C_8$
Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\)

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

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 ${\href{/padicField/5.8.0.1}{8} }$ ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.1.0.1}{1} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }$ ${\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }$ ${\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }$ ${\href{/padicField/59.8.0.1}{8} }$

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$2.8.31.5$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.96.8t1.b.a$1$ $ 2^{5} \cdot 3 $ 8.0.173946175488.1 $C_8$ (as 8T1) $0$ $-1$
* 1.16.4t1.a.a$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.96.8t1.b.b$1$ $ 2^{5} \cdot 3 $ 8.0.173946175488.1 $C_8$ (as 8T1) $0$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
* 1.96.8t1.b.c$1$ $ 2^{5} \cdot 3 $ 8.0.173946175488.1 $C_8$ (as 8T1) $0$ $-1$
* 1.16.4t1.a.b$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.96.8t1.b.d$1$ $ 2^{5} \cdot 3 $ 8.0.173946175488.1 $C_8$ (as 8T1) $0$ $-1$

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 *.