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Properties

Label 8.0.865639163373...4209.1
Degree $8$
Signature $[0, 4]$
Discriminant $113^{6}\cdot 401^{6}$
Root discriminant $3105.75$
Ramified primes $113, 401$
Class number $1840363560$ (GRH)
Class group $[2, 4522, 203490]$ (GRH)
Galois group $Q_8$ (as 8T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28742837126528, -848649683424, -32011251778, 396498613, 22418861, -150738, 8468, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 8468*x^6 - 150738*x^5 + 22418861*x^4 + 396498613*x^3 - 32011251778*x^2 - 848649683424*x + 28742837126528)
 
gp: K = bnfinit(x^8 - 3*x^7 + 8468*x^6 - 150738*x^5 + 22418861*x^4 + 396498613*x^3 - 32011251778*x^2 - 848649683424*x + 28742837126528, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 8468 x^{6} - 150738 x^{5} + 22418861 x^{4} + 396498613 x^{3} - 32011251778 x^{2} - 848649683424 x + 28742837126528 \)

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:  \(8656391633739689127939614209=113^{6}\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3105.75$
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:  $113, 401$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{112} a^{5} - \frac{1}{16} a^{4} - \frac{19}{112} a^{3} + \frac{1}{16} a^{2} + \frac{9}{56} a$, $\frac{1}{3136} a^{6} + \frac{3}{1568} a^{5} - \frac{55}{1568} a^{4} - \frac{29}{196} a^{3} + \frac{221}{3136} a^{2} + \frac{117}{1568} a + \frac{1}{28}$, $\frac{1}{19279557512084546593946863670144} a^{7} - \frac{135545511005376623443285219}{876343523276570299724857439552} a^{6} + \frac{20627678872014387742974391579}{9639778756042273296973431835072} a^{5} + \frac{26425498517643162288496392049}{438171761638285149862428719776} a^{4} - \frac{3914125601072771295834706453455}{19279557512084546593946863670144} a^{3} - \frac{771011358694950930956305354837}{9639778756042273296973431835072} a^{2} + \frac{222380831651346326754806970803}{2409944689010568324243357958768} a - \frac{18295432433157194007328915855}{43034726589474434361488534978}$

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

Class group and class number

$C_{2}\times C_{4522}\times C_{203490}$, which has order $1840363560$ (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:  \( 8223.52656633 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8$ (as 8T5):

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 $Q_8$
Character table for $Q_8$

Intermediate fields

\(\Q(\sqrt{45313}) \), \(\Q(\sqrt{401}) \), \(\Q(\sqrt{113}) \), \(\Q(\sqrt{113}, \sqrt{401})\)

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$113$113.4.3.2$x^{4} - 1017$$4$$1$$3$$C_4$$[\ ]_{4}$
113.4.3.2$x^{4} - 1017$$4$$1$$3$$C_4$$[\ ]_{4}$
401Data not computed