Normalized defining polynomial
\( x^{6} - 2x^{5} - 2x^{4} + 2x^{3} + 4x^{2} - 2 \)
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: | \(-490752\) \(\medspace = -\,2^{8}\cdot 3^{3}\cdot 71\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.88\) | 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: | $2^{4/3}3^{1/2}71^{1/2}\approx 36.775884711001936$ | ||
Ramified primes: | \(2\), \(3\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-213}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{2}-a-1$, $a-1$, $a^{2}-1$, $a^{4}-2a^{3}-a^{2}+a+1$ | 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: | \( 5.70407224298 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 5.70407224298 \cdot 1}{2\cdot\sqrt{490752}}\cr\approx \mathstrut & 0.409283120064 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 6T13):
A solvable group of order 72 |
The 9 conjugacy class representatives for $C_3^2:D_4$ |
Character table for $C_3^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 6.0.17179728.1 |
Degree 6 sibling: | 6.0.17179728.1 |
Degree 9 sibling: | 9.3.9895523328.1 |
Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
Degree 18 siblings: | 18.6.42302036995916628492288.1, deg 18, deg 18 |
Degree 24 siblings: | deg 24, deg 24 |
Degree 36 siblings: | deg 36, deg 36, deg 36 |
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 | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.8.3 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ |
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.852.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 71 $ | \(\Q(\sqrt{-213}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.852.4t3.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 71 $ | 4.0.60492.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.206156736.12t34.c.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 71^{3}$ | 6.4.490752.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
* | 4.40896.6t13.b.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 71 $ | 6.4.490752.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ |
4.241968.6t13.b.a | $4$ | $ 2^{4} \cdot 3 \cdot 71^{2}$ | 6.4.490752.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.8710848.12t34.b.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 71^{2}$ | 6.4.490752.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |