Normalized defining polynomial
\( x^{6} - x^{5} - 2x^{4} + 2x + 1 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-269395\) \(\medspace = -\,5\cdot 7\cdot 43\cdot 179\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.04\) | 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: | $5^{1/2}7^{1/2}43^{1/2}179^{1/2}\approx 519.0327542650849$ | ||
Ramified primes: | \(5\), \(7\), \(43\), \(179\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-269395}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $a^{5}-a^{4}-2a^{3}+2$, $a^{5}-a^{4}-2a^{3}+a+1$, $a^{2}-1$, $a^{5}-a^{4}-2a^{3}+a^{2}-a+2$ | 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: | \( 2.74371049773 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 2.74371049773 \cdot 1}{2\cdot\sqrt{269395}}\cr\approx \mathstrut & 0.265713349993 \end{aligned}\]
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: | 6.0.19550982758804875.2 |
Degree 6 sibling: | 6.0.19550982758804875.2 |
Degree 10 sibling: | 10.4.19550982758804875.2 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
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: | deg 45 |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | R | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(43\) | $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(179\) | 179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.269395.2t1.a.a | $1$ | $ 5 \cdot 7 \cdot 43 \cdot 179 $ | \(\Q(\sqrt{-269395}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 5.269395.6t16.a.a | $5$ | $ 5 \cdot 7 \cdot 43 \cdot 179 $ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $3$ |
5.72573666025.12t183.a.a | $5$ | $ 5^{2} \cdot 7^{2} \cdot 43^{2} \cdot 179^{2}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.526...625.12t183.a.a | $5$ | $ 5^{4} \cdot 7^{4} \cdot 43^{4} \cdot 179^{4}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
5.195...875.6t16.a.a | $5$ | $ 5^{3} \cdot 7^{3} \cdot 43^{3} \cdot 179^{3}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $-1$ | |
9.195...875.10t32.a.a | $9$ | $ 5^{3} \cdot 7^{3} \cdot 43^{3} \cdot 179^{3}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.382...625.20t145.a.a | $9$ | $ 5^{6} \cdot 7^{6} \cdot 43^{6} \cdot 179^{6}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
10.382...625.30t164.a.a | $10$ | $ 5^{6} \cdot 7^{6} \cdot 43^{6} \cdot 179^{6}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
10.526...625.30t164.a.a | $10$ | $ 5^{4} \cdot 7^{4} \cdot 43^{4} \cdot 179^{4}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $2$ | |
16.277...625.36t1252.a.a | $16$ | $ 5^{8} \cdot 7^{8} \cdot 43^{8} \cdot 179^{8}$ | 6.4.269395.1 | $S_6$ (as 6T16) | $1$ | $0$ |