Normalized defining polynomial
\( x^{12} + 18x^{8} - 8x^{6} + 120x^{4} + 216 \)
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: | \(1521681143169024\) \(\medspace = 2^{33}\cdot 3^{11}\) | 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: | \(18.42\) | 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))
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Galois root discriminant: | $2^{115/32}3^{3/2}\approx 62.734787173617846$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q(\sqrt{6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{12}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{12}a^{9}+\frac{1}{3}a^{3}$, $\frac{1}{6444}a^{10}-\frac{6}{179}a^{8}+\frac{87}{358}a^{6}+\frac{23}{3222}a^{4}+\frac{256}{537}a^{2}+\frac{5}{179}$, $\frac{1}{6444}a^{11}-\frac{6}{179}a^{9}+\frac{87}{358}a^{7}+\frac{23}{3222}a^{5}+\frac{256}{537}a^{3}+\frac{5}{179}a$
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: | \( -1 \) (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: | $\frac{85}{6444}a^{10}-\frac{17}{1074}a^{8}+\frac{28}{179}a^{6}-\frac{1267}{3222}a^{4}+\frac{153}{179}a^{2}-\frac{112}{179}$, $\frac{29}{6444}a^{10}+\frac{5}{179}a^{8}+\frac{17}{358}a^{6}+\frac{667}{3222}a^{4}-\frac{94}{537}a^{2}+\frac{145}{179}$, $\frac{1}{1611}a^{11}+\frac{83}{3222}a^{10}+\frac{35}{1074}a^{9}+\frac{41}{2148}a^{8}-\frac{5}{179}a^{7}+\frac{61}{179}a^{6}+\frac{1703}{3222}a^{5}+\frac{298}{1611}a^{4}-\frac{229}{537}a^{3}+\frac{263}{179}a^{2}+\frac{557}{179}a+\frac{472}{179}$, $\frac{29}{2148}a^{11}+\frac{31}{2148}a^{10}+\frac{1}{2148}a^{9}-\frac{73}{2148}a^{8}+\frac{51}{358}a^{7}+\frac{18}{179}a^{6}-\frac{407}{1074}a^{5}-\frac{449}{537}a^{4}+\frac{76}{537}a^{3}-\frac{178}{537}a^{2}-\frac{281}{179}a-\frac{251}{179}$, $\frac{25}{1074}a^{11}+\frac{31}{1611}a^{10}-\frac{5}{179}a^{9}-\frac{157}{2148}a^{8}+\frac{81}{179}a^{7}+\frac{24}{179}a^{6}-\frac{461}{1074}a^{5}-\frac{1796}{1611}a^{4}+\frac{449}{179}a^{3}+\frac{259}{179}a^{2}-\frac{145}{179}a-\frac{812}{179}$ | 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: | \( 363.849724744 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 363.849724744 \cdot 1}{2\cdot\sqrt{1521681143169024}}\cr\approx \mathstrut & 0.286952118759 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-2}) \), 6.0.41472.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.169075682574336.5 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.33.221 | $x^{12} + 24 x^{11} + 212 x^{10} + 832 x^{9} + 1442 x^{8} + 1888 x^{7} + 6704 x^{6} + 12480 x^{5} + 15948 x^{4} + 25312 x^{3} + 26576 x^{2} - 1664 x + 14808$ | $4$ | $3$ | $33$ | 12T134 | $[2, 2, 3, 7/2, 7/2, 4]^{6}$ |
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.8.6 | $x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |