Properties

Label 8.4.211922578125.1
Degree $8$
Signature $[4, 2]$
Discriminant $3^{6}\cdot 5^{7}\cdot 61^{2}$
Root discriminant $26.05$
Ramified primes $3, 5, 61$
Class number $2$
Class group $[2]$
Galois Group $C_8:C_2$ (as 8T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![271, 672, 13, 84, -135, -21, 13, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 13*x^6 - 21*x^5 - 135*x^4 + 84*x^3 + 13*x^2 + 672*x + 271)
gp: K = bnfinit(x^8 - 3*x^7 + 13*x^6 - 21*x^5 - 135*x^4 + 84*x^3 + 13*x^2 + 672*x + 271, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut 13 x^{6} \) \(\mathstrut -\mathstrut 21 x^{5} \) \(\mathstrut -\mathstrut 135 x^{4} \) \(\mathstrut +\mathstrut 84 x^{3} \) \(\mathstrut +\mathstrut 13 x^{2} \) \(\mathstrut +\mathstrut 672 x \) \(\mathstrut +\mathstrut 271 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[4, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(211922578125=3^{6}\cdot 5^{7}\cdot 61^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $26.05$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 61$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4185394529} a^{7} - \frac{63616348}{4185394529} a^{6} + \frac{1599953226}{4185394529} a^{5} + \frac{1736743091}{4185394529} a^{4} + \frac{1898158960}{4185394529} a^{3} - \frac{207682873}{4185394529} a^{2} - \frac{1239546044}{4185394529} a - \frac{1423927367}{4185394529}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C2, order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $5$
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:  \( \frac{36110}{17512111} a^{7} - \frac{141633}{17512111} a^{6} + \frac{518094}{17512111} a^{5} - \frac{997527}{17512111} a^{4} - \frac{4993177}{17512111} a^{3} + \frac{10406943}{17512111} a^{2} + \frac{2413166}{17512111} a + \frac{27393392}{17512111} \),  \( \frac{2007026}{4185394529} a^{7} - \frac{18959374}{4185394529} a^{6} + \frac{35064793}{4185394529} a^{5} - \frac{103473472}{4185394529} a^{4} - \frac{535700594}{4185394529} a^{3} + \frac{2700671941}{4185394529} a^{2} + \frac{1554926985}{4185394529} a - \frac{4896966878}{4185394529} \),  \( \frac{13257910}{4185394529} a^{7} - \frac{37801245}{4185394529} a^{6} + \frac{193541999}{4185394529} a^{5} - \frac{278369667}{4185394529} a^{4} - \frac{1476613396}{4185394529} a^{3} + \frac{474623271}{4185394529} a^{2} - \frac{1504476171}{4185394529} a + \frac{667327581}{4185394529} \),  \( \frac{31797338}{4185394529} a^{7} - \frac{46054221}{4185394529} a^{6} + \frac{342540465}{4185394529} a^{5} - \frac{158296603}{4185394529} a^{4} - \frac{4748786929}{4185394529} a^{3} - \frac{5167790584}{4185394529} a^{2} - \frac{10820952146}{4185394529} a - \frac{4008570526}{4185394529} \),  \( \frac{1656162}{4185394529} a^{7} - \frac{41657859}{4185394529} a^{6} + \frac{87579654}{4185394529} a^{5} - \frac{771087928}{4185394529} a^{4} + \frac{537990562}{4185394529} a^{3} - \frac{759920206}{4185394529} a^{2} + \frac{5107206082}{4185394529} a + \frac{1965985067}{4185394529} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 152.927931617 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$OD_{16}$ (as 8T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $C_8:C_2$
Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_5_61.2t1.1c1$1$ $ 3 \cdot 5 \cdot 61 $ $x^{2} - x + 229$ $C_2$ (as 2T1) $1$ $-1$
1.3_61.2t1.1c1$1$ $ 3 \cdot 61 $ $x^{2} - x + 46$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.3_5.4t1.1c1$1$ $ 3 \cdot 5 $ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
1.5_61.4t1.3c1$1$ $ 5 \cdot 61 $ $x^{4} - x^{3} + 76 x^{2} - 76 x + 1201$ $C_4$ (as 4T1) $0$ $-1$
1.5_61.4t1.3c2$1$ $ 5 \cdot 61 $ $x^{4} - x^{3} + 76 x^{2} - 76 x + 1201$ $C_4$ (as 4T1) $0$ $-1$
* 1.3_5.4t1.1c2$1$ $ 3 \cdot 5 $ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
* 2.3e2_5e2_61.8t7.1c1$2$ $ 3^{2} \cdot 5^{2} \cdot 61 $ $x^{8} - 3 x^{7} + 13 x^{6} - 21 x^{5} - 135 x^{4} + 84 x^{3} + 13 x^{2} + 672 x + 271$ $C_8:C_2$ (as 8T7) $0$ $0$
* 2.3e2_5e2_61.8t7.1c2$2$ $ 3^{2} \cdot 5^{2} \cdot 61 $ $x^{8} - 3 x^{7} + 13 x^{6} - 21 x^{5} - 135 x^{4} + 84 x^{3} + 13 x^{2} + 672 x + 271$ $C_8:C_2$ (as 8T7) $0$ $0$

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 *.