Properties

Label 8.0.6411541765625.2
Degree $8$
Signature $[0, 4]$
Discriminant $5^{6}\cdot 17^{7}$
Root discriminant $39.89$
Ramified primes $5, 17$
Class number $292$
Class group $[292]$
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![1616, 2044, 500, -129, 49, 6, 10, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 10*x^6 + 6*x^5 + 49*x^4 - 129*x^3 + 500*x^2 + 2044*x + 1616)
gp: K = bnfinit(x^8 - x^7 + 10*x^6 + 6*x^5 + 49*x^4 - 129*x^3 + 500*x^2 + 2044*x + 1616, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut 10 x^{6} \) \(\mathstrut +\mathstrut 6 x^{5} \) \(\mathstrut +\mathstrut 49 x^{4} \) \(\mathstrut -\mathstrut 129 x^{3} \) \(\mathstrut +\mathstrut 500 x^{2} \) \(\mathstrut +\mathstrut 2044 x \) \(\mathstrut +\mathstrut 1616 \)

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:  \(6411541765625=5^{6}\cdot 17^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $39.89$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 17$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(64,·)$, $\chi_{85}(1,·)$, $\chi_{85}(4,·)$, $\chi_{85}(42,·)$, $\chi_{85}(77,·)$, $\chi_{85}(16,·)$, $\chi_{85}(83,·)$, $\chi_{85}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{16} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{128} a^{6} - \frac{1}{64} a^{5} + \frac{3}{64} a^{4} - \frac{1}{32} a^{3} + \frac{1}{128} a^{2} + \frac{3}{64} a - \frac{1}{16}$, $\frac{1}{58496} a^{7} - \frac{13}{14624} a^{6} + \frac{417}{29248} a^{5} - \frac{289}{7312} a^{4} - \frac{2687}{58496} a^{3} + \frac{1323}{14624} a^{2} + \frac{3487}{14624} a + \frac{1597}{3656}$

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

Class group and class number

$C_{292}$, which has order $292$

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
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 489.802404202 \)
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

\(\Q(\sqrt{17}) \), 4.4.122825.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.8.6.4$x^{8} - 5 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5_17.8t1.3c1$1$ $ 5 \cdot 17 $ $x^{8} - x^{7} + 10 x^{6} + 6 x^{5} + 49 x^{4} - 129 x^{3} + 500 x^{2} + 2044 x + 1616$ $C_8$ (as 8T1) $0$ $-1$
* 1.5_17.4t1.1c1$1$ $ 5 \cdot 17 $ $x^{4} - x^{3} - 23 x^{2} + x + 86$ $C_4$ (as 4T1) $0$ $1$
* 1.5_17.8t1.3c2$1$ $ 5 \cdot 17 $ $x^{8} - x^{7} + 10 x^{6} + 6 x^{5} + 49 x^{4} - 129 x^{3} + 500 x^{2} + 2044 x + 1616$ $C_8$ (as 8T1) $0$ $-1$
* 1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
* 1.5_17.8t1.3c3$1$ $ 5 \cdot 17 $ $x^{8} - x^{7} + 10 x^{6} + 6 x^{5} + 49 x^{4} - 129 x^{3} + 500 x^{2} + 2044 x + 1616$ $C_8$ (as 8T1) $0$ $-1$
* 1.5_17.4t1.1c2$1$ $ 5 \cdot 17 $ $x^{4} - x^{3} - 23 x^{2} + x + 86$ $C_4$ (as 4T1) $0$ $1$
* 1.5_17.8t1.3c4$1$ $ 5 \cdot 17 $ $x^{8} - x^{7} + 10 x^{6} + 6 x^{5} + 49 x^{4} - 129 x^{3} + 500 x^{2} + 2044 x + 1616$ $C_8$ (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 *.