magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12857136, 2097328, 197554, 6064, 169, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 169*x^4 + 6064*x^3 + 197554*x^2 + 2097328*x + 12857136)
gp: K = bnfinit(x^6 - 2*x^5 + 169*x^4 + 6064*x^3 + 197554*x^2 + 2097328*x + 12857136, 1)
Normalized defining polynomial
\( x^{6} - 2 x^{5} + 169 x^{4} + 6064 x^{3} + 197554 x^{2} + 2097328 x + 12857136 \)
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: | \(-487294769549888000=-\,2^{9}\cdot 5^{3}\cdot 103^{3}\cdot 191^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $887.09$ | 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: | $2, 5, 103, 191$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{90} a^{4} + \frac{1}{9} a^{3} - \frac{7}{90} a^{2} - \frac{4}{9} a - \frac{4}{15}$, $\frac{1}{315821700} a^{5} + \frac{88207}{26318475} a^{4} - \frac{39622307}{315821700} a^{3} - \frac{517541}{17545650} a^{2} - \frac{49105807}{157910850} a + \frac{12428252}{26318475}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{84}\times C_{756}$, which has order $1016064$ (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: | \( \frac{65}{1052739} a^{5} - \frac{4411}{5263695} a^{4} + \frac{1947}{116971} a^{3} + \frac{1583137}{5263695} a^{2} + \frac{6604762}{1052739} a + \frac{112166443}{1754565} \), \( \frac{58}{26318475} a^{5} - \frac{1967}{2924275} a^{4} + \frac{386344}{26318475} a^{3} - \frac{1723493}{8772825} a^{2} - \frac{46574312}{26318475} a - \frac{704730961}{8772825} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 188.7152501940767 \) (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
| \(\Q(\sqrt{-196730}) \), 3.1.786920.2 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 | ${\href{/LocalNumberField/3.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $103$ | 103.2.1.2 | $x^{2} + 206$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 103.2.1.2 | $x^{2} + 206$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 206$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $191$ | 191.2.1.2 | $x^{2} + 382$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 191.2.1.2 | $x^{2} + 382$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 382$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |