↑

Properties

Label 8.0.11590971882213376.2
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 1297^{4}$
Root discriminant $101.86$
Ramified primes $2, 1297$
Class number $2475$
Class group $[15, 165]$
Galois group $D_4$ (as 8T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9216, 0, -4040, 0, 1281, 0, 66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 66*x^6 + 1281*x^4 - 4040*x^2 + 9216)
 
gp: K = bnfinit(x^8 + 66*x^6 + 1281*x^4 - 4040*x^2 + 9216, 1)
 

Normalized defining polynomial

\( x^{8} + 66 x^{6} + 1281 x^{4} - 4040 x^{2} + 9216 \)

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:  \(11590971882213376=2^{12}\cdot 1297^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.86$
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:  $2, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{12} a^{4} - \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{15624} a^{6} - \frac{5}{279} a^{4} - \frac{1043}{2232} a^{2} - \frac{1}{2} a + \frac{38}{217}$, $\frac{1}{124992} a^{7} - \frac{113}{8928} a^{5} - \frac{857}{17856} a^{3} - \frac{1}{2} a^{2} - \frac{103}{5208} a$

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

Class group and class number

$C_{15}\times C_{165}$, which has order $2475$

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:  \( \frac{19}{124992} a^{7} + \frac{85}{8928} a^{5} + \frac{3061}{17856} a^{3} - \frac{5429}{5208} a + 1 \),  \( \frac{4}{1953} a^{6} + \frac{26}{279} a^{4} - \frac{80}{279} a^{2} + \frac{131}{217} \),  \( \frac{17279}{124992} a^{7} - \frac{2081}{7812} a^{6} + \frac{85289}{8928} a^{5} - \frac{9809}{558} a^{4} + \frac{3662441}{17856} a^{3} - \frac{400967}{1116} a^{2} - \frac{312817}{5208} a + \frac{110273}{217} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 839.804203214 \)
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

\(\Q(\sqrt{2594}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{1297}) \), \(\Q(\sqrt{2}, \sqrt{1297})\), 4.0.83008.2 x2, 4.0.13457672.1 x2

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

Sibling fields

Degree 4 siblings: 4.0.13457672.1, 4.0.83008.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
1297Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.1297.2t1.1c1$1$ $ 1297 $ $x^{2} - x - 324$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3_1297.2t1.1c1$1$ $ 2^{3} \cdot 1297 $ $x^{2} - 2594$ $C_2$ (as 2T1) $1$ $1$
*2 2.2e3_1297.4t3.6c1$2$ $ 2^{3} \cdot 1297 $ $x^{8} + 66 x^{6} + 1281 x^{4} - 4040 x^{2} + 9216$ $D_4$ (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 *.