Normalized defining polynomial
\( x^{12} - 3x^{10} - 4x^{9} + 12x^{7} + x^{6} - 3x^{5} + 3x^{4} - 7x^{3} - 3x^{2} + 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-4770738948267\) \(\medspace = -\,3^{16}\cdot 19^{2}\cdot 307\) | 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: | \(11.39\) | 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: | $3^{4/3}19^{1/2}307^{1/2}\approx 330.45144947292306$ | ||
Ramified primes: | \(3\), \(19\), \(307\) | 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: | \(\Q(\sqrt{-307}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3293}a^{11}+\frac{1228}{3293}a^{10}-\frac{213}{3293}a^{9}-\frac{1421}{3293}a^{8}+\frac{302}{3293}a^{7}-\frac{1241}{3293}a^{6}+\frac{8}{37}a^{5}-\frac{1605}{3293}a^{4}+\frac{1570}{3293}a^{3}+\frac{1548}{3293}a^{2}+\frac{880}{3293}a+\frac{539}{3293}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{474}{3293}a^{11}-\frac{4082}{3293}a^{10}-\frac{2172}{3293}a^{9}+\frac{8097}{3293}a^{8}+\frac{18014}{3293}a^{7}+\frac{14385}{3293}a^{6}-\frac{426}{37}a^{5}-\frac{9966}{3293}a^{4}-\frac{9917}{3293}a^{3}-\frac{20345}{3293}a^{2}+\frac{21960}{3293}a+\frac{11804}{3293}$, $\frac{5620}{3293}a^{11}-\frac{768}{3293}a^{10}-\frac{14873}{3293}a^{9}-\frac{20253}{3293}a^{8}-\frac{1948}{3293}a^{7}+\frac{59428}{3293}a^{6}-\frac{69}{37}a^{5}+\frac{2720}{3293}a^{4}+\frac{21211}{3293}a^{3}-\frac{36569}{3293}a^{2}-\frac{3779}{3293}a+\frac{6206}{3293}$, $a$, $\frac{3931}{3293}a^{11}-\frac{270}{3293}a^{10}-\frac{10760}{3293}a^{9}-\frac{14195}{3293}a^{8}-\frac{1611}{3293}a^{7}+\frac{41371}{3293}a^{6}-\frac{39}{37}a^{5}-\frac{3160}{3293}a^{4}+\frac{20346}{3293}a^{3}-\frac{26620}{3293}a^{2}-\frac{1663}{3293}a+\frac{7996}{3293}$, $\frac{660}{3293}a^{11}-\frac{2891}{3293}a^{10}-\frac{2274}{3293}a^{9}+\frac{3938}{3293}a^{8}+\frac{11619}{3293}a^{7}+\frac{14069}{3293}a^{6}-\frac{270}{37}a^{5}-\frac{2247}{3293}a^{4}-\frac{7681}{3293}a^{3}-\frac{18908}{3293}a^{2}+\frac{11111}{3293}a+\frac{3389}{3293}$, $\frac{2123}{3293}a^{11}+\frac{2281}{3293}a^{10}-\frac{4351}{3293}a^{9}-\frac{13567}{3293}a^{8}-\frac{14161}{3293}a^{7}+\frac{12929}{3293}a^{6}+\frac{223}{37}a^{5}+\frac{14012}{3293}a^{4}+\frac{17059}{3293}a^{3}-\frac{3303}{3293}a^{2}-\frac{12063}{3293}a-\frac{4960}{3293}$, $\frac{3823}{3293}a^{11}-\frac{1174}{3293}a^{10}-\frac{10807}{3293}a^{9}-\frac{12205}{3293}a^{8}+\frac{1996}{3293}a^{7}+\frac{43679}{3293}a^{6}-\frac{89}{37}a^{5}-\frac{1056}{3293}a^{4}+\frac{15436}{3293}a^{3}-\frac{32447}{3293}a^{2}-\frac{1206}{3293}a+\frac{5765}{3293}$, $\frac{4556}{3293}a^{11}-\frac{3332}{3293}a^{10}-\frac{12165}{3293}a^{9}-\frac{9917}{3293}a^{8}+\frac{9317}{3293}a^{7}+\frac{52773}{3293}a^{6}-\frac{330}{37}a^{5}+\frac{1373}{3293}a^{4}+\frac{7110}{3293}a^{3}-\frac{43727}{3293}a^{2}+\frac{11578}{3293}a+\frac{12278}{3293}$ | 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: | \( 43.9848803763 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 43.9848803763 \cdot 1}{2\cdot\sqrt{4770738948267}}\cr\approx \mathstrut & 0.159845410813 \end{aligned}\]
Galois group
$C_2\wr (C_2\times A_4)$ (as 12T222):
A solvable group of order 1536 |
The 55 conjugacy class representatives for $D_4\wr C_3$ |
Character table for $D_4\wr C_3$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 6.4.124659.1 |
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 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.12.16.14 | $x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
\(19\) | 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(307\) | $\Q_{307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |