Normalized defining polynomial
\( x^{12} - 8x^{10} - 3x^{8} + 44x^{6} + 133x^{4} - 228x^{2} - 361 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-202060520306507776\) \(\medspace = -\,2^{32}\cdot 19^{6}\) | 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: | \(27.68\) | 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: | $2^{41/12}19^{3/4}\approx 97.18166881943587$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1806121}a^{10}+\frac{732594}{1806121}a^{8}+\frac{137709}{1806121}a^{6}-\frac{18172}{78527}a^{4}-\frac{5053}{95059}a^{2}+\frac{44719}{95059}$, $\frac{1}{1806121}a^{11}+\frac{732594}{1806121}a^{9}+\frac{137709}{1806121}a^{7}-\frac{18172}{78527}a^{5}-\frac{5053}{95059}a^{3}+\frac{44719}{95059}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{2808}{1806121}a^{10}-\frac{47867}{1806121}a^{8}+\frac{176978}{1806121}a^{6}+\frac{15574}{78527}a^{4}-\frac{25033}{95059}a^{2}-\frac{192105}{95059}$, $\frac{12275}{1806121}a^{10}-\frac{85109}{1806121}a^{8}-\frac{151281}{1806121}a^{6}+\frac{33907}{78527}a^{4}+\frac{47952}{95059}a^{2}-\frac{135059}{95059}$, $\frac{58}{95059}a^{10}-\frac{921}{95059}a^{8}+\frac{2166}{95059}a^{6}-\frac{61}{4133}a^{4}+\frac{40075}{95059}a^{2}+\frac{39776}{95059}$, $\frac{2808}{1806121}a^{11}+\frac{25011}{1806121}a^{10}-\frac{47867}{1806121}a^{9}-\frac{189011}{1806121}a^{8}+\frac{176978}{1806121}a^{7}-\frac{32948}{1806121}a^{6}+\frac{15574}{78527}a^{5}+\frac{14384}{78527}a^{4}-\frac{25033}{95059}a^{3}+\frac{142946}{95059}a^{2}-\frac{192105}{95059}a-\frac{92344}{95059}$, $\frac{27819}{1806121}a^{11}-\frac{25472}{1806121}a^{10}-\frac{236878}{1806121}a^{9}+\frac{207804}{1806121}a^{8}+\frac{144030}{1806121}a^{7}-\frac{236666}{1806121}a^{6}+\frac{29958}{78527}a^{5}+\frac{39046}{78527}a^{4}+\frac{22854}{95059}a^{3}-\frac{94929}{95059}a^{2}-\frac{189390}{95059}a+\frac{9629}{95059}$, $\frac{140121}{1806121}a^{11}-\frac{9898}{95059}a^{10}-\frac{889282}{1806121}a^{9}+\frac{75226}{95059}a^{8}-\frac{2480096}{1806121}a^{7}+\frac{102378}{95059}a^{6}+\frac{194744}{78527}a^{5}-\frac{18236}{4133}a^{4}+\frac{1964258}{95059}a^{3}-\frac{2689189}{95059}a^{2}+\frac{1968076}{95059}a-\frac{2641441}{95059}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 9700.33266527 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 9700.33266527 \cdot 1}{2\cdot\sqrt{202060520306507776}}\cr\approx \mathstrut & 0.422644510255 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr S_4$ (as 12T227):
A solvable group of order 1536 |
The 40 conjugacy class representatives for $C_2\wr S_4$ |
Character table for $C_2\wr S_4$ |
Intermediate fields
3.1.76.1, 6.2.369664.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 12.4.202060520306507776.11 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.32.381 | $x^{12} + 2 x^{10} + 4 x^{9} + 8 x^{8} + 12 x^{6} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 6$ | $12$ | $1$ | $32$ | 12T140 | $[2, 8/3, 8/3, 3, 11/3, 11/3]_{3}^{2}$ |
\(19\) | 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.3.1 | $x^{4} + 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |