Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 2*x^4 - 3*x^3 + 2*x^2 - 2*x + 1)
gp: K = bnfinit(x^6 - 2*x^5 + 2*x^4 - 3*x^3 + 2*x^2 - 2*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 2, -3, 2, -2, 1]);
\( x^{6} - 2 x^{5} + 2 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(31213\)\(\medspace = 7^{4}\cdot 13\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $5.61$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $7, 13$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $2$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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Fundamental units: | \( a^{5} - 2 a^{4} + 2 a^{3} - 2 a^{2} + a - 1 \), \( a^{4} - a^{3} - a \), \( a^{5} - a^{4} + a^{3} - 2 a^{2} - 1 \) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 0.517040346258 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_2\times A_4$ (as 6T6):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4\times C_2$ |
Character table for $A_4\times C_2$ |
Intermediate fields
\(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Galois closure: | data not computed |
Twin sextic algebra: | 4.0.8281.1 $\times$ \(\Q(\sqrt{13}) \) |
Degree 8 sibling: | 8.0.68574961.1 |
Degree 12 siblings: | 12.0.164648481361.1, 12.4.27825593350009.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }$ | ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
$7$ | 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
$13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
1.13.2t1.a.a | $1$ | $ 13 $ | $x^{2} - x - 3$ | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | $x^{3} - x^{2} - 2 x + 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
1.91.6t1.j.a | $1$ | $ 7 \cdot 13 $ | $x^{6} - x^{5} - 14 x^{4} + 9 x^{3} + 35 x^{2} - 16 x - 1$ | $C_6$ (as 6T1) | $0$ | $1$ | |
1.91.6t1.j.b | $1$ | $ 7 \cdot 13 $ | $x^{6} - x^{5} - 14 x^{4} + 9 x^{3} + 35 x^{2} - 16 x - 1$ | $C_6$ (as 6T1) | $0$ | $1$ | |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | $x^{3} - x^{2} - 2 x + 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
3.8281.4t4.b.a | $3$ | $ 7^{2} \cdot 13^{2}$ | $x^{4} - x^{3} + 5 x^{2} - 4 x + 3$ | $A_4$ (as 4T4) | $1$ | $-1$ | |
* | 3.637.6t6.a.a | $3$ | $ 7^{2} \cdot 13 $ | $x^{6} - 2 x^{5} + 2 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$ | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ |
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 *.