magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 4, -5, 2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 2*x^4 - 5*x^3 + 4*x^2 + x - 1)
gp: K = bnfinit(x^6 - x^5 + 2*x^4 - 5*x^3 + 4*x^2 + x - 1, 1)
Normalized defining polynomial
\( x^{6} - x^{5} + 2 x^{4} - 5 x^{3} + 4 x^{2} + x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(152000=2^{6}\cdot 5^{3}\cdot 19\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $7.31$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$
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: | $3$ | 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: | \( a^{5} - \frac{1}{2} a^{4} + 2 a^{3} - \frac{9}{2} a^{2} + 2 a + \frac{3}{2} \), \( a \), \( a - 1 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1.75942927329 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\wr C_2$ (as 6T13):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $C_3^2:D_4$ |
| Character table for $C_3^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.2.35118080.4 |
| Degree 6 sibling: | 6.2.35118080.4 |
| Degree 9 sibling: | 9.1.14047232000.1 |
| Degree 12 siblings: | Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 |
| Degree 18 siblings: | Deg 18, Deg 18, Deg 18 |
| Degree 24 siblings: | data not computed |
| Degree 36 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 | R | ${\href{/LocalNumberField/3.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.4.6.6 | $x^{4} - 20$ | $2$ | $2$ | $6$ | $D_{4}$ | $[2, 3]^{2}$ | |
| $5$ | 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2_5_19.2t1.1c1 | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | $x^{2} - 95$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.2e2_19.2t1.1c1 | $1$ | $ 2^{2} \cdot 19 $ | $x^{2} - 19$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.2e6_5_19.4t3.6c1 | $2$ | $ 2^{6} \cdot 5 \cdot 19 $ | $x^{4} + 16 x^{2} + 45$ | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| * | 4.2e6_5e2_19.6t13.1c1 | $4$ | $ 2^{6} \cdot 5^{2} \cdot 19 $ | $x^{6} - x^{5} + 2 x^{4} - 5 x^{3} + 4 x^{2} + x - 1$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
| 4.2e10_5e2_19e3.12t34.1c1 | $4$ | $ 2^{10} \cdot 5^{2} \cdot 19^{3}$ | $x^{6} - x^{5} + 2 x^{4} - 5 x^{3} + 4 x^{2} + x - 1$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.2e8_5_19e2.6t13.1c1 | $4$ | $ 2^{8} \cdot 5 \cdot 19^{2}$ | $x^{6} - x^{5} + 2 x^{4} - 5 x^{3} + 4 x^{2} + x - 1$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.2e8_5e3_19e2.12t34.1c1 | $4$ | $ 2^{8} \cdot 5^{3} \cdot 19^{2}$ | $x^{6} - x^{5} + 2 x^{4} - 5 x^{3} + 4 x^{2} + x - 1$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
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 *.