Normalized defining polynomial
\( x^{12} - 6x^{10} - 11x^{9} + 21x^{8} + 39x^{7} + 14x^{6} - 27x^{5} + 72x^{4} + 20x^{3} - 129x^{2} - 297x + 327 \)
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: | \(2025170913606201\) \(\medspace = 3^{16}\cdot 19^{6}\) | 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: | \(18.86\) | 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))
| |
Galois root discriminant: | $3^{4/3}19^{1/2}\approx 18.859860385004858$ | ||
Ramified primes: | \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{6}a^{9}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2280}a^{10}-\frac{17}{1140}a^{9}-\frac{93}{760}a^{8}-\frac{153}{760}a^{7}-\frac{107}{380}a^{6}-\frac{87}{380}a^{5}+\frac{1}{6}a^{4}+\frac{1111}{2280}a^{3}+\frac{113}{380}a^{2}+\frac{183}{760}a-\frac{139}{760}$, $\frac{1}{1198459200}a^{11}+\frac{156083}{1198459200}a^{10}-\frac{80357117}{1198459200}a^{9}+\frac{41613}{199743200}a^{8}+\frac{11883953}{79897280}a^{7}+\frac{3126927}{99871600}a^{6}+\frac{178133653}{599229600}a^{5}+\frac{24578371}{1198459200}a^{4}-\frac{403667}{239691840}a^{3}-\frac{33021239}{79897280}a^{2}-\frac{13590957}{99871600}a+\frac{137154177}{399486400}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{4}$, which has order $4$
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)
| |
Fundamental units: | $\frac{40647}{23499200}a^{11}-\frac{74417}{70497600}a^{10}-\frac{800417}{70497600}a^{9}-\frac{135887}{11749600}a^{8}+\frac{32961}{939968}a^{7}+\frac{298027}{5874800}a^{6}+\frac{724531}{11749600}a^{5}-\frac{329089}{70497600}a^{4}-\frac{624431}{14099520}a^{3}-\frac{540963}{4699840}a^{2}-\frac{1501377}{5874800}a-\frac{28493163}{23499200}$, $\frac{40647}{23499200}a^{11}-\frac{74417}{70497600}a^{10}-\frac{800417}{70497600}a^{9}-\frac{135887}{11749600}a^{8}+\frac{32961}{939968}a^{7}+\frac{298027}{5874800}a^{6}+\frac{724531}{11749600}a^{5}-\frac{329089}{70497600}a^{4}-\frac{624431}{14099520}a^{3}-\frac{540963}{4699840}a^{2}-\frac{1501377}{5874800}a-\frac{4993963}{23499200}$, $\frac{1931397}{399486400}a^{11}+\frac{581053}{1198459200}a^{10}-\frac{36195347}{1198459200}a^{9}-\frac{13167117}{199743200}a^{8}+\frac{8079263}{79897280}a^{7}+\frac{26735857}{99871600}a^{6}+\frac{32264041}{199743200}a^{5}-\frac{331720339}{1198459200}a^{4}+\frac{22917283}{239691840}a^{3}-\frac{16383609}{79897280}a^{2}-\frac{104322587}{99871600}a-\frac{1175994993}{399486400}$, $\frac{461963}{239691840}a^{11}+\frac{1375721}{239691840}a^{10}-\frac{366319}{239691840}a^{9}-\frac{2510269}{39948640}a^{8}-\frac{4430481}{79897280}a^{7}+\frac{2097569}{19974320}a^{6}+\frac{2257973}{6307680}a^{5}+\frac{55592833}{239691840}a^{4}+\frac{125958067}{239691840}a^{3}+\frac{34203327}{79897280}a^{2}+\frac{6555603}{19974320}a-\frac{8227823}{4205120}$, $\frac{252041}{23499200}a^{11}+\frac{820529}{70497600}a^{10}-\frac{1330757}{23499200}a^{9}-\frac{2088281}{11749600}a^{8}+\frac{240507}{4699840}a^{7}+\frac{3087901}{5874800}a^{6}+\frac{7973133}{11749600}a^{5}+\frac{23292433}{70497600}a^{4}+\frac{3245861}{4699840}a^{3}+\frac{3313427}{4699840}a^{2}+\frac{1348229}{5874800}a-\frac{43019509}{23499200}$ | 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: | \( 538.58951047 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 538.58951047 \cdot 4}{2\cdot\sqrt{2025170913606201}}\cr\approx \mathstrut & 1.4727745000 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 4 conjugacy class representatives for $A_4$ |
Character table for $A_4$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 4.0.29241.1 x4, 6.2.2368521.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 sibling: | 4.0.29241.1 |
Degree 6 sibling: | 6.2.2368521.1 |
Minimal sibling: | 4.0.29241.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(19\) | 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |