Normalized defining polynomial
\( x^{12} - 6x^{10} + 28x^{8} + 6x^{6} + 28x^{4} - 6x^{2} + 1 \)
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: | \(144555105949057024\) \(\medspace = 2^{20}\cdot 13^{10}\) | 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: | \(26.92\) | 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: | $2^{2}13^{5/6}\approx 33.911431129417196$ | ||
Ramified primes: | \(2\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}+\frac{1}{11}a^{6}+\frac{1}{11}a^{4}+\frac{1}{11}a^{2}+\frac{1}{11}$, $\frac{1}{11}a^{9}+\frac{1}{11}a^{7}+\frac{1}{11}a^{5}+\frac{1}{11}a^{3}+\frac{1}{11}a$, $\frac{1}{176}a^{10}-\frac{1}{22}a^{9}-\frac{7}{176}a^{8}+\frac{5}{11}a^{7}-\frac{29}{176}a^{6}-\frac{1}{22}a^{5}-\frac{29}{176}a^{4}+\frac{5}{11}a^{3}-\frac{7}{176}a^{2}-\frac{1}{22}a-\frac{63}{176}$, $\frac{1}{176}a^{11}-\frac{7}{176}a^{9}-\frac{1}{22}a^{8}-\frac{29}{176}a^{7}+\frac{5}{11}a^{6}-\frac{29}{176}a^{5}-\frac{1}{22}a^{4}-\frac{7}{176}a^{3}+\frac{5}{11}a^{2}-\frac{63}{176}a-\frac{1}{22}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{12}$, which has order $12$
Unit group
Rank: | $5$ | 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: | $a$, $\frac{9}{16}a^{11}+\frac{2}{11}a^{10}-\frac{581}{176}a^{9}-\frac{25}{22}a^{8}+\frac{2697}{176}a^{7}+\frac{59}{11}a^{6}+\frac{937}{176}a^{5}-\frac{3}{22}a^{4}+\frac{2939}{176}a^{3}+\frac{48}{11}a^{2}-\frac{317}{176}a-\frac{29}{22}$, $\frac{1}{8}a^{11}-\frac{53}{88}a^{9}+\frac{233}{88}a^{7}+\frac{409}{88}a^{5}+\frac{475}{88}a^{3}+\frac{299}{88}a$, $\frac{1}{16}a^{11}-\frac{2}{11}a^{10}-\frac{7}{16}a^{9}+\frac{25}{22}a^{8}+\frac{35}{16}a^{7}-\frac{59}{11}a^{6}-\frac{29}{16}a^{5}+\frac{3}{22}a^{4}+\frac{57}{16}a^{3}-\frac{48}{11}a^{2}-\frac{63}{16}a-\frac{15}{22}$, $\frac{7}{8}a^{11}-\frac{1}{44}a^{10}-\frac{475}{88}a^{9}+\frac{1}{4}a^{8}+\frac{2231}{88}a^{7}-\frac{5}{4}a^{6}+\frac{119}{88}a^{5}+\frac{11}{4}a^{4}+\frac{1989}{88}a^{3}+\frac{5}{4}a^{2}-\frac{915}{88}a+\frac{155}{44}$ | 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: | \( 1054.1560013 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1054.1560013 \cdot 12}{2\cdot\sqrt{144555105949057024}}\cr\approx \mathstrut & 1.0235725438 \end{aligned}\]
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
\(\Q(\sqrt{-13}) \), 3.1.676.1, 6.0.380204032.1, 6.0.23762752.2, 6.2.7311616.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.380204032.1, 6.2.95051008.1 |
Degree 8 siblings: | 8.4.19770609664.3, 8.0.316329754624.9 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.95051008.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.8.16.4 | $x^{8} + 12 x^{7} + 58 x^{6} + 160 x^{5} + 329 x^{4} + 500 x^{3} + 408 x^{2} + 68 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ | |
\(13\) | 13.6.5.1 | $x^{6} + 52$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
13.6.5.1 | $x^{6} + 52$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |