Normalized defining polynomial
\( x^{12} - 8x^{9} + 3x^{8} + 12x^{7} - 12x^{6} - 48x^{5} + 6x^{4} + 32x^{3} - 4 \)
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: | \(-855945643032576\) \(\medspace = -\,2^{29}\cdot 3^{13}\) | 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: | \(17.55\) | 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: | $2^{11/4}3^{7/6}\approx 24.23672593327708$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}+\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{6}a^{10}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{954222}a^{11}+\frac{2264}{159037}a^{10}+\frac{21068}{477111}a^{9}+\frac{1253}{954222}a^{8}-\frac{51747}{318074}a^{7}+\frac{177481}{477111}a^{6}+\frac{219052}{477111}a^{5}+\frac{66395}{318074}a^{4}-\frac{63236}{477111}a^{3}-\frac{38971}{477111}a^{2}+\frac{70382}{159037}a+\frac{134006}{477111}$
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: | $\frac{20497}{159037}a^{11}+\frac{31420}{477111}a^{10}+\frac{22327}{318074}a^{9}-\frac{321543}{318074}a^{8}-\frac{46480}{477111}a^{7}+\frac{190475}{159037}a^{6}-\frac{253997}{318074}a^{5}-\frac{6183809}{954222}a^{4}-\frac{470595}{159037}a^{3}+\frac{267565}{159037}a^{2}+\frac{550234}{477111}a+\frac{144947}{159037}$, $\frac{428591}{477111}a^{11}+\frac{90574}{159037}a^{10}+\frac{440941}{954222}a^{9}-\frac{6607369}{954222}a^{8}-\frac{270753}{159037}a^{7}+\frac{4290637}{477111}a^{6}-\frac{4453393}{954222}a^{5}-\frac{14218149}{318074}a^{4}-\frac{12108017}{477111}a^{3}+\frac{4458053}{477111}a^{2}+\frac{1246981}{159037}a+\frac{2580731}{477111}$, $\frac{91331}{318074}a^{11}+\frac{68519}{477111}a^{10}+\frac{77572}{477111}a^{9}-\frac{705045}{318074}a^{8}-\frac{183589}{954222}a^{7}+\frac{1239476}{477111}a^{6}-\frac{298314}{159037}a^{5}-\frac{13518397}{954222}a^{4}-\frac{2957086}{477111}a^{3}+\frac{305733}{159037}a^{2}+\frac{272636}{477111}a+\frac{310879}{477111}$, $\frac{96351}{318074}a^{11}+\frac{31459}{954222}a^{10}+\frac{82837}{477111}a^{9}-\frac{776465}{318074}a^{8}+\frac{685073}{954222}a^{7}+\frac{2177917}{954222}a^{6}-\frac{398129}{159037}a^{5}-\frac{12912043}{954222}a^{4}-\frac{592135}{477111}a^{3}+\frac{286823}{159037}a^{2}+\frac{663692}{477111}a+\frac{738820}{477111}$, $\frac{1987613}{954222}a^{11}+\frac{647899}{477111}a^{10}+\frac{429547}{477111}a^{9}-\frac{15307883}{954222}a^{8}-\frac{4030331}{954222}a^{7}+\frac{10570781}{477111}a^{6}-\frac{5151061}{477111}a^{5}-\frac{101661113}{954222}a^{4}-\frac{27200488}{477111}a^{3}+\frac{13540846}{477111}a^{2}+\frac{7949491}{477111}a+\frac{5329039}{477111}$, $\frac{102725}{477111}a^{11}+\frac{25762}{477111}a^{10}+\frac{69608}{477111}a^{9}-\frac{1642423}{954222}a^{8}+\frac{144413}{477111}a^{7}+\frac{740386}{477111}a^{6}-\frac{693008}{477111}a^{5}-\frac{9589625}{954222}a^{4}-\frac{1057892}{477111}a^{3}+\frac{284852}{477111}a^{2}+\frac{695506}{477111}a+\frac{319556}{477111}$ | 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: | \( 2145.51696839 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 2145.51696839 \cdot 1}{2\cdot\sqrt{855945643032576}}\cr\approx \mathstrut & 1.43627697474 \end{aligned}\]
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
3.1.216.1, 6.2.2985984.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.2239488.2, some data not computed |
Degree 8 siblings: | 8.4.1719926784.1, 8.0.3057647616.11 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.2239488.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.9.3 | $x^{4} + 4 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ |
2.8.20.3 | $x^{8} + 8 x^{7} + 28 x^{6} + 64 x^{5} + 216 x^{4} + 128 x^{3} + 328 x^{2} + 64 x + 124$ | $4$ | $2$ | $20$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ | |
\(3\) | 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.6.7.1 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ |