Properties

Label 8.8.235260548044817.1
Degree $8$
Signature $[8, 0]$
Discriminant $113^{7}$
Root discriminant $62.58$
Ramified prime $113$
Class number $1$
Class group Trivial
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![1372, -798, -1499, 367, 511, -16, -49, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 49*x^6 - 16*x^5 + 511*x^4 + 367*x^3 - 1499*x^2 - 798*x + 1372)
gp: K = bnfinit(x^8 - x^7 - 49*x^6 - 16*x^5 + 511*x^4 + 367*x^3 - 1499*x^2 - 798*x + 1372, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut -\mathstrut 49 x^{6} \) \(\mathstrut -\mathstrut 16 x^{5} \) \(\mathstrut +\mathstrut 511 x^{4} \) \(\mathstrut +\mathstrut 367 x^{3} \) \(\mathstrut -\mathstrut 1499 x^{2} \) \(\mathstrut -\mathstrut 798 x \) \(\mathstrut +\mathstrut 1372 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[8, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(235260548044817=113^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $62.58$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $113$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(113\)
Dirichlet character group:    $\lbrace$$\chi_{113}(1,·)$, $\chi_{113}(98,·)$, $\chi_{113}(69,·)$, $\chi_{113}(44,·)$, $\chi_{113}(15,·)$, $\chi_{113}(112,·)$, $\chi_{113}(18,·)$, $\chi_{113}(95,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{14} a^{5} + \frac{1}{7} a^{3} - \frac{1}{2} a^{2} + \frac{2}{7} a$, $\frac{1}{56} a^{6} - \frac{1}{56} a^{5} - \frac{3}{14} a^{4} + \frac{19}{56} a^{3} + \frac{11}{56} a^{2} - \frac{9}{28} a - \frac{1}{2}$, $\frac{1}{9296} a^{7} - \frac{13}{4648} a^{6} + \frac{269}{9296} a^{5} + \frac{2223}{9296} a^{4} + \frac{3}{581} a^{3} + \frac{827}{9296} a^{2} - \frac{1293}{4648} a - \frac{91}{332}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 172951.810041 \)
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

\(\Q(\sqrt{113}) \), 4.4.1442897.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$113$113.8.7.1$x^{8} - 113$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.113.8t1.1c1$1$ $ 113 $ $x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372$ $C_8$ (as 8T1) $0$ $1$
* 1.113.4t1.1c1$1$ $ 113 $ $x^{4} - x^{3} - 42 x^{2} + 120 x - 64$ $C_4$ (as 4T1) $0$ $1$
* 1.113.8t1.1c2$1$ $ 113 $ $x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372$ $C_8$ (as 8T1) $0$ $1$
* 1.113.2t1.1c1$1$ $ 113 $ $x^{2} - x - 28$ $C_2$ (as 2T1) $1$ $1$
* 1.113.8t1.1c3$1$ $ 113 $ $x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372$ $C_8$ (as 8T1) $0$ $1$
* 1.113.4t1.1c2$1$ $ 113 $ $x^{4} - x^{3} - 42 x^{2} + 120 x - 64$ $C_4$ (as 4T1) $0$ $1$
* 1.113.8t1.1c4$1$ $ 113 $ $x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372$ $C_8$ (as 8T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.