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Properties

Label 8.0.255924290200...0313.1
Degree $8$
Signature $[0, 4]$
Discriminant $37^{6}\cdot 193^{7}$
Root discriminant $1499.73$
Ramified primes $37, 193$
Class number $241115732$ (GRH)
Class group $[11, 21919612]$ (GRH)
Galois group $C_8$ (as 8T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33591285504, 5333705760, 95433112, -5894476, 1109222, 20093, 881, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 881*x^6 + 20093*x^5 + 1109222*x^4 - 5894476*x^3 + 95433112*x^2 + 5333705760*x + 33591285504)
 
gp: K = bnfinit(x^8 - x^7 + 881*x^6 + 20093*x^5 + 1109222*x^4 - 5894476*x^3 + 95433112*x^2 + 5333705760*x + 33591285504, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 881 x^{6} + 20093 x^{5} + 1109222 x^{4} - 5894476 x^{3} + 95433112 x^{2} + 5333705760 x + 33591285504 \)

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:  \(25592429020084354212450313=37^{6}\cdot 193^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1499.73$
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:  $37, 193$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(7141=37\cdot 193\)
Dirichlet character group:    $\lbrace$$\chi_{7141}(1,·)$, $\chi_{7141}(3817,·)$, $\chi_{7141}(43,·)$, $\chi_{7141}(7029,·)$, $\chi_{7141}(1849,·)$, $\chi_{7141}(5403,·)$, $\chi_{7141}(956,·)$, $\chi_{7141}(2325,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{504} a^{4} - \frac{13}{252} a^{3} - \frac{85}{504} a^{2} - \frac{113}{252} a - \frac{1}{3}$, $\frac{1}{7056} a^{5} - \frac{1}{1764} a^{4} + \frac{39}{784} a^{3} - \frac{68}{441} a^{2} + \frac{605}{1764} a + \frac{1}{21}$, $\frac{1}{49392} a^{6} - \frac{1}{16464} a^{5} - \frac{5}{5488} a^{4} + \frac{2399}{49392} a^{3} - \frac{157}{12348} a^{2} + \frac{3433}{12348} a + \frac{50}{147}$, $\frac{1}{256189825332758016} a^{7} + \frac{2194183696549}{256189825332758016} a^{6} + \frac{11079999122095}{256189825332758016} a^{5} - \frac{165137207740937}{256189825332758016} a^{4} - \frac{64325263508693}{2287409154756768} a^{3} - \frac{112935201581}{3810760774272} a^{2} + \frac{329433957495715}{5337288027765792} a - \frac{8521100403055}{31769571593844}$

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

Class group and class number

$C_{11}\times C_{21919612}$, which has order $241115732$ (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:  \( 92656.6271982 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8$ (as 8T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 8
The 8 conjugacy class representatives for $C_8$
Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{193}) \), 4.4.9841819033.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$37$37.8.6.3$x^{8} - 37 x^{4} + 6845$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$193$193.8.7.2$x^{8} - 4825$$8$$1$$7$$C_8$$[\ ]_{8}$