Properties

Label 8.0.272225149504.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{6}\cdot 7^{4}\cdot 11^{6}$
Root discriminant $26.88$
Ramified primes $2, 7, 11$
Class number $2$
Class group $[2]$
Galois Group $C_2^3:(C_7: C_3)$ (as 8T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![66, -35, -67, 21, 41, -1, -9, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 9*x^6 - x^5 + 41*x^4 + 21*x^3 - 67*x^2 - 35*x + 66)
gp: K = bnfinit(x^8 - x^7 - 9*x^6 - x^5 + 41*x^4 + 21*x^3 - 67*x^2 - 35*x + 66, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut -\mathstrut 9 x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut +\mathstrut 41 x^{4} \) \(\mathstrut +\mathstrut 21 x^{3} \) \(\mathstrut -\mathstrut 67 x^{2} \) \(\mathstrut -\mathstrut 35 x \) \(\mathstrut +\mathstrut 66 \)

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:  \(272225149504=2^{6}\cdot 7^{4}\cdot 11^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $26.88$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{56} a^{7} + \frac{1}{14} a^{6} - \frac{3}{56} a^{5} - \frac{1}{28} a^{4} + \frac{3}{56} a^{3} - \frac{5}{14} a^{2} + \frac{15}{56} a + \frac{13}{28}$

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

Class group and class number

$C_{2}$, which has 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:  $3$
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:  \( 851.951772132 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$F_8:C_3$ (as 8T36):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 168
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$
Character table for $C_2^3:(C_7: C_3)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 14 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 sibling: 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
3.2e3_7e2_11e3.7t3.1c1$3$ $ 2^{3} \cdot 7^{2} \cdot 11^{3}$ $x^{7} - 3 x^{6} - 15 x^{5} + 39 x^{4} + 43 x^{3} - 133 x^{2} + 63 x - 7$ $C_7:C_3$ (as 7T3) $0$ $3$
3.2e3_7e2_11e3.7t3.1c2$3$ $ 2^{3} \cdot 7^{2} \cdot 11^{3}$ $x^{7} - 3 x^{6} - 15 x^{5} + 39 x^{4} + 43 x^{3} - 133 x^{2} + 63 x - 7$ $C_7:C_3$ (as 7T3) $0$ $3$
* 7.2e6_7e4_11e6.8t36.1c1$7$ $ 2^{6} \cdot 7^{4} \cdot 11^{6}$ $x^{8} - x^{7} - 9 x^{6} - x^{5} + 41 x^{4} + 21 x^{3} - 67 x^{2} - 35 x + 66$ $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.2e6_7e5_11e6.24t283.1c1$7$ $ 2^{6} \cdot 7^{5} \cdot 11^{6}$ $x^{8} - x^{7} - 9 x^{6} - x^{5} + 41 x^{4} + 21 x^{3} - 67 x^{2} - 35 x + 66$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.2e6_7e5_11e6.24t283.1c2$7$ $ 2^{6} \cdot 7^{5} \cdot 11^{6}$ $x^{8} - x^{7} - 9 x^{6} - x^{5} + 41 x^{4} + 21 x^{3} - 67 x^{2} - 35 x + 66$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$

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 *.