Properties

Label 8.4.4546939761.1
Degree $8$
Signature $[4, 2]$
Discriminant $3^{2}\cdot 7^{2}\cdot 13^{4}\cdot 19^{2}$
Root discriminant $16.11$
Ramified primes $3, 7, 13, 19$
Class number $1$
Class group Trivial
Galois Group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53, 58, -64, -38, 41, 10, -11, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 11*x^6 + 10*x^5 + 41*x^4 - 38*x^3 - 64*x^2 + 58*x + 53)
gp: K = bnfinit(x^8 - x^7 - 11*x^6 + 10*x^5 + 41*x^4 - 38*x^3 - 64*x^2 + 58*x + 53, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut -\mathstrut 11 x^{6} \) \(\mathstrut +\mathstrut 10 x^{5} \) \(\mathstrut +\mathstrut 41 x^{4} \) \(\mathstrut -\mathstrut 38 x^{3} \) \(\mathstrut -\mathstrut 64 x^{2} \) \(\mathstrut +\mathstrut 58 x \) \(\mathstrut +\mathstrut 53 \)

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:  \(4546939761=3^{2}\cdot 7^{2}\cdot 13^{4}\cdot 19^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $16.11$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 7, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{21} a^{7} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{21} a^{3} - \frac{5}{21} a^{2} + \frac{1}{21} a - \frac{4}{21}$

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:  $5$
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:  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:  \( 64.5079170739 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\wr C_2^2$ (as 8T29):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 64
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.2.507.1

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$

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_13.2t1.1c1$1$ $ 3 \cdot 13 $ $x^{2} - x + 10$ $C_2$ (as 2T1) $1$ $-1$
1.3_7_19.2t1.1c1$1$ $ 3 \cdot 7 \cdot 19 $ $x^{2} - x + 100$ $C_2$ (as 2T1) $1$ $-1$
1.7_13_19.2t1.1c1$1$ $ 7 \cdot 13 \cdot 19 $ $x^{2} - x - 432$ $C_2$ (as 2T1) $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
1.7_19.2t1.1c1$1$ $ 7 \cdot 19 $ $x^{2} - x - 33$ $C_2$ (as 2T1) $1$ $1$
1.3_7_13_19.2t1.1c1$1$ $ 3 \cdot 7 \cdot 13 \cdot 19 $ $x^{2} - x + 1297$ $C_2$ (as 2T1) $1$ $-1$
* 2.3_13.4t3.1c1$2$ $ 3 \cdot 13 $ $x^{4} - x^{3} - x^{2} - x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7_19.4t3.2c1$2$ $ 3 \cdot 7 \cdot 19 $ $x^{4} - 11 x^{2} - 3$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7_13_19.4t3.2c1$2$ $ 3 \cdot 7 \cdot 13 \cdot 19 $ $x^{4} + 23 x^{2} - 300$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7_13_19.4t3.1c1$2$ $ 3 \cdot 7 \cdot 13 \cdot 19 $ $x^{4} - x^{2} - 432$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7_13e2_19.4t3.1c1$2$ $ 3 \cdot 7 \cdot 13^{2} \cdot 19 $ $x^{4} - x^{3} - 52 x^{2} - 190 x - 209$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7e2_13_19e2.4t3.1c1$2$ $ 3 \cdot 7^{2} \cdot 13 \cdot 19^{2}$ $x^{4} - x^{3} + 32 x^{2} - 232 x - 3233$ $D_{4}$ (as 4T3) $1$ $0$
4.3e3_7e2_13e2_19e2.8t29.4c1$4$ $ 3^{3} \cdot 7^{2} \cdot 13^{2} \cdot 19^{2}$ $x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 41 x^{4} - 38 x^{3} - 64 x^{2} + 58 x + 53$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $-2$
* 4.3_7e2_13e2_19e2.8t29.2c1$4$ $ 3 \cdot 7^{2} \cdot 13^{2} \cdot 19^{2}$ $x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 41 x^{4} - 38 x^{3} - 64 x^{2} + 58 x + 53$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $2$

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 *.