Normalized defining polynomial
\( x^{12} - 6x^{10} + 11x^{8} - 4x^{6} - x^{4} - 6x^{2} - 11 \)
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: | \(-43231798624256\) \(\medspace = -\,2^{28}\cdot 11^{5}\) | 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: | \(13.69\) | 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: | $2^{31/12}11^{1/2}\approx 19.877289578936544$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a+\frac{3}{8}$
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)
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Fundamental units: | $\frac{1}{8}a^{10}-\frac{5}{8}a^{8}+a^{6}-\frac{1}{2}a^{4}-\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{5}{8}a-\frac{5}{8}$, $\frac{1}{4}a^{7}-\frac{3}{4}a^{5}+\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{4}a^{7}+\frac{1}{2}a^{6}+\frac{3}{2}a^{5}-\frac{1}{4}a^{4}-\frac{5}{4}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{5}{8}$, $\frac{1}{8}a^{10}-\frac{5}{8}a^{8}+\frac{3}{4}a^{6}-\frac{1}{4}a^{4}+\frac{9}{8}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{11}-\frac{7}{8}a^{9}+\frac{1}{4}a^{8}+2a^{7}-\frac{5}{4}a^{6}-a^{5}+\frac{7}{4}a^{4}-\frac{9}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{8}a-1$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 141.402896817 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 141.402896817 \cdot 1}{2\cdot\sqrt{43231798624256}}\cr\approx \mathstrut & 0.421197700856 \end{aligned}\]
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
3.1.44.1, 6.2.123904.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.2725888.1, 6.0.247808.2 |
Degree 8 siblings: | 8.0.126877696.2, 8.4.15352201216.4 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.247808.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.28.10 | $x^{12} + 6 x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{4} + 2$ | $12$ | $1$ | $28$ | $C_2 \times S_4$ | $[8/3, 8/3, 3]_{3}^{2}$ |
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |