magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1648408, 340192, 53714, 2290, 107, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 107*x^4 + 2290*x^3 + 53714*x^2 + 340192*x + 1648408)
gp: K = bnfinit(x^6 - 2*x^5 + 107*x^4 + 2290*x^3 + 53714*x^2 + 340192*x + 1648408, 1)
Normalized defining polynomial
\( x^{6} - 2 x^{5} + 107 x^{4} + 2290 x^{3} + 53714 x^{2} + 340192 x + 1648408 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-9691389429113664=-\,2^{6}\cdot 3^{3}\cdot 109^{3}\cdot 163^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $461.74$ | 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: | $2, 3, 109, 163$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{210} a^{4} + \frac{3}{7} a^{3} - \frac{19}{42} a^{2} + \frac{1}{7} a + \frac{2}{105}$, $\frac{1}{32213160} a^{5} - \frac{1571}{4026645} a^{4} - \frac{627241}{6442632} a^{3} + \frac{321491}{805329} a^{2} - \frac{926569}{2300940} a + \frac{1868837}{8053290}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{40}\times C_{440}$, which has order $140800$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $2$ | 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: | \( \frac{457}{1789620} a^{5} + \frac{3}{21305} a^{4} - \frac{3145}{357924} a^{3} + \frac{28391}{29827} a^{2} + \frac{6930629}{894810} a + \frac{7867464}{149135} \), \( \frac{1111}{1342215} a^{5} - \frac{4012}{1342215} a^{4} + \frac{13277}{268443} a^{3} + \frac{641602}{268443} a^{2} + \frac{6257332}{191745} a + \frac{231151517}{1342215} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 221.9117894285191 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| \(\Q(\sqrt{-53301}) \), 3.1.213204.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | data not computed |
| Degree 3 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 | R | ${\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $109$ | 109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $163$ | 163.2.1.2 | $x^{2} + 652$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 163.2.1.2 | $x^{2} + 652$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 163.2.1.2 | $x^{2} + 652$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |