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Properties

Label 8.0.167268538673...9153.5
Degree $8$
Signature $[0, 4]$
Discriminant $313^{5}\cdot 421^{6}$
Root discriminant $3372.30$
Ramified primes $313, 421$
Class number $364383760$ (GRH)
Class group $[2, 2, 2, 45547970]$ (GRH)
Galois group $C_4\wr C_2$ (as 8T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39093600296275, -4559653374927, 477022089020, -1502459109, 239357760, -98793, 32912, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 32912*x^6 - 98793*x^5 + 239357760*x^4 - 1502459109*x^3 + 477022089020*x^2 - 4559653374927*x + 39093600296275)
 
gp: K = bnfinit(x^8 - 3*x^7 + 32912*x^6 - 98793*x^5 + 239357760*x^4 - 1502459109*x^3 + 477022089020*x^2 - 4559653374927*x + 39093600296275, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 32912 x^{6} - 98793 x^{5} + 239357760 x^{4} - 1502459109 x^{3} + 477022089020 x^{2} - 4559653374927 x + 39093600296275 \)

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:  \(16726853867309592405065509153=313^{5}\cdot 421^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3372.30$
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:  $313, 421$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{36} a^{4} - \frac{1}{12} a^{3} - \frac{1}{18} a^{2} - \frac{5}{12} a - \frac{17}{36}$, $\frac{1}{864} a^{5} - \frac{1}{108} a^{4} - \frac{59}{864} a^{3} - \frac{41}{864} a^{2} + \frac{83}{432} a - \frac{275}{864}$, $\frac{1}{139968} a^{6} - \frac{7}{46656} a^{5} + \frac{511}{46656} a^{4} + \frac{1633}{23328} a^{3} + \frac{163}{15552} a^{2} + \frac{7973}{46656} a - \frac{23737}{139968}$, $\frac{1}{1587500312278323724398919751040} a^{7} + \frac{493295075712017795979487}{198437539034790465549864968880} a^{6} + \frac{5384624933885618465200699}{66145846344930155183288322960} a^{5} + \frac{1127831396249886725295417397}{529166770759441241466306583680} a^{4} + \frac{4590689844339666957513146023}{529166770759441241466306583680} a^{3} + \frac{402287122802449919443004467}{264583385379720620733153291840} a^{2} + \frac{8026712234288533455035142493}{61057704318397066323035375040} a - \frac{5556204682381741950604386451}{45357151779380677839969135744}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{45547970}$, which has order $364383760$ (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:  \( 23106.5184098 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 8T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{421}) \), 4.4.55476433.1

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
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
313Data not computed
421Data not computed