Normalized defining polynomial
\( x^{12} + x^{10} + 4x^{8} + 4x^{6} + 4x^{4} + x^{2} + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(194389282816\) \(\medspace = 2^{12}\cdot 83^{4}\) | 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: | \(8.72\) | 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: | $2\cdot 83^{1/2}\approx 18.2208671582886$ | ||
Ramified primes: | \(2\), \(83\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{3}{8}a-\frac{3}{8}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{4} a^{11} + \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{3}{2} a^{5} + \frac{3}{2} a^{3} + \frac{5}{4} a \) (order $4$) | 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: | $\frac{1}{4}a^{10}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}-\frac{3}{2}a^{4}-\frac{3}{2}a^{2}-\frac{5}{4}$, $\frac{1}{2}a^{11}+a^{7}-a^{3}-\frac{3}{2}a$, $\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{3}{2}a^{5}+\frac{3}{2}a^{4}+\frac{3}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a+\frac{1}{2}$, $\frac{7}{8}a^{11}-\frac{3}{8}a^{10}+\frac{3}{4}a^{9}-\frac{3}{4}a^{8}+\frac{13}{4}a^{7}-\frac{5}{4}a^{6}+\frac{11}{4}a^{5}-\frac{11}{4}a^{4}+\frac{9}{4}a^{3}-\frac{5}{4}a^{2}+\frac{1}{8}a-\frac{5}{8}$, $\frac{5}{8}a^{11}+\frac{1}{8}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{7}{4}a^{7}+\frac{3}{4}a^{6}+\frac{5}{4}a^{5}+\frac{5}{4}a^{4}-\frac{1}{4}a^{3}+\frac{3}{4}a^{2}-\frac{5}{8}a+\frac{7}{8}$ | 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: | \( 6.84932920201 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 6.84932920201 \cdot 1}{4\cdot\sqrt{194389282816}}\cr\approx \mathstrut & 0.238963241329 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T23):
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
\(\Q(\sqrt{-1}) \), 3.1.83.1, 6.0.27556.1, 6.0.440896.1, 6.2.110224.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.2287148.1, 6.0.27556.1 |
Degree 8 siblings: | 8.4.12149330176.1, 8.0.1763584.1 |
Degree 12 siblings: | 12.2.1008394404608.1, 12.0.5231045973904.1, 12.4.1339147769319424.1, 12.0.1339147769319424.1, 12.0.1339147769319424.3 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.27556.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(83\) | 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |