Normalized defining polynomial
\( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 3x^{2} - 2x + 2 \)
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: | \(-176836\) \(\medspace = -\,2^{2}\cdot 11\cdot 4019\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.49\) | 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: | $2\cdot 11^{1/2}4019^{1/2}\approx 420.51872728809593$ | ||
Ramified primes: | \(2\), \(11\), \(4019\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-44209}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $2$ | 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: | $a^{5}-a^{4}+a^{3}-a^{2}+2a-1$, $a^{4}+a^{2}-a+1$ | 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: | \( 2.41031848571 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 2.41031848571 \cdot 1}{2\cdot\sqrt{176836}}\cr\approx \mathstrut & 0.710883934587 \end{aligned}\]
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: | 6.4.5529833409365056.1 |
Degree 6 sibling: | 6.4.5529833409365056.1 |
Degree 10 sibling: | 10.4.5529833409365056.1 |
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 | R | ${\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }$ | R | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(4019\) | $\Q_{4019}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.176836.2t1.a.a | $1$ | $ 2^{2} \cdot 11 \cdot 4019 $ | \(\Q(\sqrt{-44209}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 5.176836.6t16.a.a | $5$ | $ 2^{2} \cdot 11 \cdot 4019 $ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $-1$ |
5.31270970896.12t183.a.a | $5$ | $ 2^{4} \cdot 11^{2} \cdot 4019^{2}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
5.977...816.12t183.a.a | $5$ | $ 2^{8} \cdot 11^{4} \cdot 4019^{4}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.552...056.6t16.a.a | $5$ | $ 2^{6} \cdot 11^{3} \cdot 4019^{3}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.552...056.10t32.a.a | $9$ | $ 2^{6} \cdot 11^{3} \cdot 4019^{3}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.305...136.20t145.a.a | $9$ | $ 2^{12} \cdot 11^{6} \cdot 4019^{6}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
10.305...136.30t164.a.a | $10$ | $ 2^{12} \cdot 11^{6} \cdot 4019^{6}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $2$ | |
10.977...816.30t164.a.a | $10$ | $ 2^{8} \cdot 11^{4} \cdot 4019^{4}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
16.956...856.36t1252.a.a | $16$ | $ 2^{16} \cdot 11^{8} \cdot 4019^{8}$ | 6.0.176836.1 | $S_6$ (as 6T16) | $1$ | $0$ |