Normalized defining polynomial
\( x^{6} + 5x^{4} - 2x^{3} + 9x^{2} - 8x + 16 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-12446784\) \(\medspace = -\,2^{6}\cdot 3^{4}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.22\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2\cdot 3^{4/3}7^{4/5}\approx 41.045930044873074$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{32}a^{5}-\frac{3}{8}a^{4}-\frac{11}{32}a^{3}+\frac{1}{16}a^{2}-\frac{15}{32}a+\frac{3}{8}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $2$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1}{4}a^{5}+\frac{5}{4}a^{3}-\frac{1}{2}a^{2}+\frac{5}{4}a-2$, $\frac{1}{32}a^{5}-\frac{3}{8}a^{4}-\frac{11}{32}a^{3}-\frac{31}{16}a^{2}-\frac{47}{32}a-\frac{29}{8}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 17.0352826889 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 17.0352826889 \cdot 2}{2\cdot\sqrt{12446784}}\cr\approx \mathstrut & 1.19773398725 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.4.63011844.2 |
Degree 6 sibling: | 6.4.63011844.2 |
Degree 10 sibling: | 10.4.196073702927424.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 siblings: | deg 20, deg 20, 20.0.9841893626795960030057286598656.1 |
Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
Degree 36 sibling: | deg 36 |
Degree 40 siblings: | deg 40, deg 40, deg 40 |
Degree 45 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.5.4.1 | $x^{5} + 7$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 5.12446784.6t16.b.a | $5$ | $ 2^{6} \cdot 3^{4} \cdot 7^{4}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $-1$ |
5.4032758016.12t183.b.a | $5$ | $ 2^{8} \cdot 3^{8} \cdot 7^{4}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
5.3111696.12t183.b.a | $5$ | $ 2^{4} \cdot 3^{4} \cdot 7^{4}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.63011844.6t16.b.a | $5$ | $ 2^{2} \cdot 3^{8} \cdot 7^{4}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.196...424.10t32.b.a | $9$ | $ 2^{6} \cdot 3^{12} \cdot 7^{8}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.125...136.20t145.b.a | $9$ | $ 2^{12} \cdot 3^{12} \cdot 7^{8}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
10.784...696.30t164.b.a | $10$ | $ 2^{8} \cdot 3^{12} \cdot 7^{8}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $2$ | |
10.125...136.30t164.b.a | $10$ | $ 2^{12} \cdot 3^{12} \cdot 7^{8}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
16.256...016.36t1252.b.a | $16$ | $ 2^{16} \cdot 3^{24} \cdot 7^{12}$ | 6.0.12446784.1 | $S_6$ (as 6T16) | $1$ | $0$ |