Normalized defining polynomial
\( x^{8} - 8x^{6} - 12x^{4} + 112x^{2} - 47 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-63870861312\)
\(\medspace = -\,2^{24}\cdot 3^{4}\cdot 47\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.42\) | 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: | $2^{3}3^{1/2}47^{1/2}\approx 94.99473669630333$ | ||
Ramified primes: |
\(2\), \(3\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
$\Aut(K/\Q)$: | $C_2$ | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{4}-\frac{1}{9}a^{2}-\frac{2}{9}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{3}-\frac{2}{9}a$, $\frac{1}{27}a^{6}-\frac{1}{9}a^{2}-\frac{2}{27}$, $\frac{1}{27}a^{7}-\frac{1}{9}a^{3}-\frac{2}{27}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $6$ | 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: |
$\frac{1}{27}a^{6}-\frac{2}{9}a^{4}-\frac{8}{9}a^{2}+\frac{64}{27}$, $\frac{1}{27}a^{6}-\frac{2}{9}a^{4}-\frac{5}{9}a^{2}+\frac{19}{27}$, $\frac{1}{27}a^{6}-\frac{1}{3}a^{4}-\frac{1}{9}a^{2}+\frac{88}{27}$, $\frac{1}{3}a^{2}-a+\frac{1}{3}$, $\frac{1}{27}a^{7}+\frac{1}{27}a^{6}-\frac{1}{3}a^{5}-\frac{2}{9}a^{4}-\frac{4}{9}a^{3}-\frac{5}{9}a^{2}+\frac{133}{27}a+\frac{100}{27}$, $\frac{2}{27}a^{7}+\frac{1}{27}a^{6}-\frac{5}{9}a^{5}-\frac{1}{3}a^{4}-a^{3}-\frac{1}{9}a^{2}+\frac{179}{27}a+\frac{115}{27}$
| 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: | \( 327.706910567 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 327.706910567 \cdot 2}{2\cdot\sqrt{63870861312}}\cr\approx \mathstrut & 0.521427648444 \end{aligned}\]
Galois group
$C_2\wr C_2^2$ (as 8T31):
A solvable group of order 64 |
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{3})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 8.4.83386957824.4 |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.1.8.24c1.61 | $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $$[2, 3, 4]$$ |
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
47.1.2.1a1.2 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
47.2.1.0a1.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |