Normalized defining polynomial
\( x^{12} + 2x^{10} + 2x^{8} - 24x^{6} - 4x^{4} + 8x^{2} - 8 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-209715200000000\) \(\medspace = -\,2^{29}\cdot 5^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(15.61\) | 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))
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Galois root discriminant: | $2^{11/4}5^{2/3}\approx 19.670368273611786$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{8}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{1}{8}$, $\frac{1}{32}a^{9}-\frac{1}{16}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{32}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{64}a^{11}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{16}a^{5}+\frac{5}{16}a^{3}-\frac{1}{4}a^{2}-\frac{1}{8}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{3}{64}a^{11}+\frac{1}{32}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{6}-\frac{19}{16}a^{5}+\frac{25}{16}a^{3}+\frac{7}{8}a^{2}+\frac{1}{2}a+\frac{1}{8}$, $\frac{3}{64}a^{11}+\frac{1}{32}a^{9}+\frac{1}{32}a^{8}+\frac{1}{16}a^{6}-\frac{19}{16}a^{5}+\frac{25}{16}a^{3}-\frac{7}{8}a^{2}+\frac{1}{2}a-\frac{1}{8}$, $\frac{3}{64}a^{11}+\frac{3}{32}a^{9}+\frac{1}{32}a^{8}+\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{19}{16}a^{5}-\frac{3}{16}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{64}a^{11}+\frac{3}{32}a^{9}-\frac{1}{32}a^{8}+\frac{1}{4}a^{7}-\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{21}{16}a^{3}+\frac{7}{8}a^{2}-\frac{5}{2}a+\frac{1}{8}$, $\frac{1}{8}a^{10}+\frac{1}{32}a^{9}+\frac{5}{32}a^{8}+\frac{1}{16}a^{7}+\frac{3}{16}a^{6}-3a^{4}-\frac{7}{8}a^{3}+\frac{15}{8}a^{2}-\frac{1}{8}a-\frac{9}{8}$, $\frac{5}{32}a^{11}+\frac{1}{16}a^{10}+\frac{7}{16}a^{9}+\frac{5}{32}a^{8}+\frac{5}{8}a^{7}+\frac{3}{16}a^{6}-\frac{27}{8}a^{5}-\frac{3}{2}a^{4}-\frac{27}{8}a^{3}-\frac{11}{8}a^{2}-\frac{1}{8}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 532.390119928 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 532.390119928 \cdot 1}{2\cdot\sqrt{209715200000000}}\cr\approx \mathstrut & 0.720019685401 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
3.1.200.1, 6.2.2560000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.5120000.2, 6.0.640000.2 |
Degree 8 siblings: | 8.4.2621440000.5, 8.0.163840000.5 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.640000.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.9.6 | $x^{4} + 4 x^{3} + 10 x^{2} + 14$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ |
2.8.20.8 | $x^{8} + 12 x^{7} + 78 x^{6} + 308 x^{5} + 795 x^{4} + 1332 x^{3} + 1450 x^{2} + 936 x + 291$ | $4$ | $2$ | $20$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ | |
\(5\) | 5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |