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Properties

Label 8.0.902901108195...9393.2
Degree $8$
Signature $[0, 4]$
Discriminant $193^{7}\cdot 449^{3}$
Root discriminant $987.31$
Ramified primes $193, 449$
Class number $11425152$ (GRH)
Class group $[2, 4, 1428144]$ (GRH)
Galois group $D_{8}$ (as 8T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3541166208, -243102432, 64537136, -1759088, 386816, -4078, 1015, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 1015*x^6 - 4078*x^5 + 386816*x^4 - 1759088*x^3 + 64537136*x^2 - 243102432*x + 3541166208)
 
gp: K = bnfinit(x^8 - 2*x^7 + 1015*x^6 - 4078*x^5 + 386816*x^4 - 1759088*x^3 + 64537136*x^2 - 243102432*x + 3541166208, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 1015 x^{6} - 4078 x^{5} + 386816 x^{4} - 1759088 x^{3} + 64537136 x^{2} - 243102432 x + 3541166208 \)

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:  \(902901108195372527819393=193^{7}\cdot 449^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $987.31$
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:  $193, 449$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{192} a^{5} + \frac{1}{48} a^{4} - \frac{15}{64} a^{3} + \frac{7}{48} a^{2} - \frac{19}{48} a + \frac{1}{4}$, $\frac{1}{22272} a^{6} + \frac{5}{3712} a^{5} + \frac{251}{22272} a^{4} + \frac{245}{11136} a^{3} - \frac{1061}{5568} a^{2} - \frac{25}{2784} a - \frac{99}{232}$, $\frac{1}{166818057203712} a^{7} + \frac{112191461}{13901504766976} a^{6} - \frac{6135239987}{2827424698368} a^{5} - \frac{1669445001515}{41704514300928} a^{4} - \frac{2443701698605}{20852257150464} a^{3} + \frac{509561439095}{5213064287616} a^{2} + \frac{6861459085}{204433893632} a - \frac{311017344341}{868844047936}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{1428144}$, which has order $11425152$ (assuming GRH)

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:  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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16144.3010814 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 8T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{193}) \), 4.4.3227886593.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$193$193.8.7.1$x^{8} - 193$$8$$1$$7$$C_8$$[\ ]_{8}$
449Data not computed