Normalized defining polynomial
\( x^{6} - 3 x^{5} + 9 x^{4} - 9 x^{3} + 9 x^{2} + 12 x + 1 \)
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: | \(12425625=3^{2}\cdot 5^{4}\cdot 47^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
Root discriminant: | $15.22$ | 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: | $3, 5, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{6} a - \frac{1}{2}$
Class group and class number
$C_{3}$, which has order $3$
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: | \( \frac{1}{6} a^{5} - \frac{2}{3} a^{4} + \frac{11}{6} a^{3} - \frac{7}{3} a^{2} + \frac{13}{6} a + \frac{11}{6} \), \( \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{5}{6} a^{3} - \frac{2}{3} a^{2} + \frac{1}{6} a + \frac{1}{6} \), \( \frac{2}{3} a^{5} - \frac{7}{3} a^{4} + \frac{22}{3} a^{3} - \frac{29}{3} a^{2} + \frac{35}{3} a + \frac{8}{3} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
Regulator: | \( 8.1176351334 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3:S_3.C_2$ (as 6T10):
A solvable group of order 36 |
The 6 conjugacy class representatives for $C_3^2:C_4$ |
Character table for $C_3^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Galois closure: | data not computed |
Twin sextic algebra: | 6.2.12425625.1 |
Degree 6 sibling: | 6.2.12425625.1 |
Degree 9 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 18 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
$3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
$5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
$47$ | 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
47.4.2.2 | $x^{4} - 47 x^{2} + 28717$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ |
1.3_5_47.4t1.1c1 | $1$ | $ 3 \cdot 5 \cdot 47 $ | $x^{4} - x^{3} + 176 x^{2} - 176 x + 6301$ | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.3_5_47.4t1.1c2 | $1$ | $ 3 \cdot 5 \cdot 47 $ | $x^{4} - x^{3} + 176 x^{2} - 176 x + 6301$ | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 4.3e2_5e3_47e2.6t10.3c1 | $4$ | $ 3^{2} \cdot 5^{3} \cdot 47^{2}$ | $x^{6} - 3 x^{5} + 9 x^{4} - 9 x^{3} + 9 x^{2} + 12 x + 1$ | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |
4.3e2_5e3_47e2.6t10.4c1 | $4$ | $ 3^{2} \cdot 5^{3} \cdot 47^{2}$ | $x^{6} - 3 x^{5} + 9 x^{4} - 9 x^{3} + 9 x^{2} + 12 x + 1$ | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ |