Normalized defining polynomial
\( x^{6} - 3x^{5} - 9x^{4} + 14x^{3} + 27x^{2} - 3x - 10 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(170067681\) \(\medspace = 3^{8}\cdot 7^{2}\cdot 23^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.54\) | 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: | $3^{16/9}7^{2/3}23^{2/3}\approx 208.65693066268003$ | ||
Ramified primes: | \(3\), \(7\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $a+1$, $\frac{2}{3}a^{5}-\frac{8}{3}a^{4}-3a^{3}+\frac{35}{3}a^{2}+4a-\frac{13}{3}$, $\frac{1}{3}a^{5}-\frac{4}{3}a^{4}-2a^{3}+\frac{19}{3}a^{2}+4a-\frac{11}{3}$, $a^{2}-a-3$, $\frac{1}{3}a^{5}-\frac{5}{3}a^{4}-\frac{1}{3}a^{3}+\frac{20}{3}a^{2}+\frac{4}{3}a-3$ | 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: | \( 277.555419546 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 277.555419546 \cdot 1}{2\cdot\sqrt{170067681}}\cr\approx \mathstrut & 0.681065365039 \end{aligned}\]
Galois group
A non-solvable group of order 360 |
The 7 conjugacy class representatives for $A_6$ |
Character table for $A_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.6.4408324359201.1 |
Degree 6 sibling: | 6.6.4408324359201.1 |
Degree 10 sibling: | deg 10 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 sibling: | deg 20 |
Degree 30 siblings: | deg 30, deg 30 |
Degree 36 sibling: | deg 36 |
Degree 40 sibling: | 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 | ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.8.2 | $x^{6} - 12 x^{5} + 72 x^{4} + 6 x^{3} + 72 x^{2} + 90$ | $3$ | $2$ | $8$ | $C_3^2:C_4$ | $[2, 2]^{4}$ |
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
5.440...201.6t15.a.a | $5$ | $ 3^{8} \cdot 7^{4} \cdot 23^{4}$ | 6.6.170067681.1 | $A_6$ (as 6T15) | $1$ | $5$ | |
* | 5.170067681.6t15.a.a | $5$ | $ 3^{8} \cdot 7^{2} \cdot 23^{2}$ | 6.6.170067681.1 | $A_6$ (as 6T15) | $1$ | $5$ |
8.749...881.36t555.a.a | $8$ | $ 3^{16} \cdot 7^{6} \cdot 23^{6}$ | 6.6.170067681.1 | $A_6$ (as 6T15) | $1$ | $8$ | |
8.749...881.36t555.a.b | $8$ | $ 3^{16} \cdot 7^{6} \cdot 23^{6}$ | 6.6.170067681.1 | $A_6$ (as 6T15) | $1$ | $8$ | |
9.749...881.10t26.a.a | $9$ | $ 3^{16} \cdot 7^{6} \cdot 23^{6}$ | 6.6.170067681.1 | $A_6$ (as 6T15) | $1$ | $9$ | |
10.749...881.30t88.a.a | $10$ | $ 3^{16} \cdot 7^{6} \cdot 23^{6}$ | 6.6.170067681.1 | $A_6$ (as 6T15) | $1$ | $10$ |