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Properties

Label 8.0.454971136951...9001.1
Degree $8$
Signature $[0, 4]$
Discriminant $149^{6}\cdot 401^{6}$
Root discriminant $3821.62$
Ramified primes $149, 401$
Class number $304867800$ (GRH)
Class group $[6, 270, 188190]$ (GRH)
Galois group $C_4\times C_2$ (as 8T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![273542293828125, -172383112500, 132729039125, 203660475, 69712610, 45771, 11169, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 11169*x^6 + 45771*x^5 + 69712610*x^4 + 203660475*x^3 + 132729039125*x^2 - 172383112500*x + 273542293828125)
 
gp: K = bnfinit(x^8 - x^7 + 11169*x^6 + 45771*x^5 + 69712610*x^4 + 203660475*x^3 + 132729039125*x^2 - 172383112500*x + 273542293828125, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 11169 x^{6} + 45771 x^{5} + 69712610 x^{4} + 203660475 x^{3} + 132729039125 x^{2} - 172383112500 x + 273542293828125 \)

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:  \(45497113695130083887456829001=149^{6}\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3821.62$
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:  $149, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(59749=149\cdot 401\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{15} a^{4} - \frac{1}{15} a^{3} - \frac{2}{5} a^{2} - \frac{4}{15} a$, $\frac{1}{525} a^{5} - \frac{16}{525} a^{4} - \frac{1}{25} a^{3} - \frac{139}{525} a^{2} + \frac{1}{7} a$, $\frac{1}{15750} a^{6} - \frac{1}{2625} a^{5} - \frac{38}{7875} a^{4} - \frac{349}{15750} a^{3} + \frac{451}{3150} a^{2} + \frac{101}{210} a - \frac{1}{2}$, $\frac{1}{11095354767566998884809573606250} a^{7} - \frac{94028293234380056708296451}{11095354767566998884809573606250} a^{6} + \frac{5109327341573837622584491747}{5547677383783499442404786803125} a^{5} + \frac{35863211205524258899419677971}{11095354767566998884809573606250} a^{4} + \frac{14391869976425073364372288441}{1109535476756699888480957360625} a^{3} - \frac{10231594207096287777857378963}{221907095351339977696191472125} a^{2} + \frac{2122219599205601118625090813}{14793806356755998513079431475} a - \frac{343395196749587123866493}{1943357156880919344903702}$

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

Class group and class number

$C_{6}\times C_{270}\times C_{188190}$, which has order $304867800$ (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:  \( 6617.93570943 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4$ (as 8T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 8
The 8 conjugacy class representatives for $C_4\times C_2$
Character table for $C_4\times C_2$

Intermediate fields

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{59749}) \), \(\Q(\sqrt{401}) \), \(\Q(\sqrt{149}, \sqrt{401})\), 4.0.213300524366749.2, 4.0.213300524366749.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$149$149.4.3.2$x^{4} - 596$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.2$x^{4} - 596$$4$$1$$3$$C_4$$[\ ]_{4}$
401Data not computed