Normalized defining polynomial
\( x^{12} + 3x^{10} - 5x^{9} + 6x^{8} + 3x^{7} - 14x^{6} + 48x^{5} - 24x^{4} - 33x^{3} + 27x^{2} - 3 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(32357746661769\) \(\medspace = 3^{18}\cdot 17^{4}\) | 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: | \(13.36\) | 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: | $3^{3/2}17^{1/2}\approx 21.42428528562855$ | ||
Ramified primes: | \(3\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{4}{9}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{45}a^{10}-\frac{1}{45}a^{9}+\frac{1}{3}a^{8}-\frac{16}{45}a^{7}+\frac{7}{45}a^{6}+\frac{1}{15}a^{4}+\frac{1}{5}a^{2}+\frac{1}{15}a-\frac{4}{15}$, $\frac{1}{267075}a^{11}+\frac{2537}{267075}a^{10}-\frac{3103}{267075}a^{9}+\frac{50909}{267075}a^{8}+\frac{69889}{267075}a^{7}+\frac{89296}{267075}a^{6}+\frac{21446}{89025}a^{5}+\frac{43918}{89025}a^{4}-\frac{13114}{29675}a^{3}-\frac{2728}{17805}a^{2}+\frac{26054}{89025}a+\frac{12773}{89025}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{29909}{89025}a^{11}+\frac{59824}{267075}a^{10}+\frac{104798}{89025}a^{9}-\frac{75994}{89025}a^{8}+\frac{395078}{267075}a^{7}+\frac{182114}{89025}a^{6}-\frac{316558}{89025}a^{5}+\frac{424712}{29675}a^{4}+\frac{22897}{29675}a^{3}-\frac{56722}{5935}a^{2}+\frac{366208}{89025}a-\frac{8293}{29675}$, $\frac{158624}{267075}a^{11}+\frac{107308}{267075}a^{10}+\frac{560908}{267075}a^{9}-\frac{424634}{267075}a^{8}+\frac{697541}{267075}a^{7}+\frac{842669}{267075}a^{6}-\frac{507146}{89025}a^{5}+\frac{2183327}{89025}a^{4}+\frac{52039}{29675}a^{3}-\frac{57823}{3561}a^{2}+\frac{231391}{89025}a+\frac{55597}{89025}$, $\frac{158624}{267075}a^{11}+\frac{107308}{267075}a^{10}+\frac{560908}{267075}a^{9}-\frac{424634}{267075}a^{8}+\frac{697541}{267075}a^{7}+\frac{842669}{267075}a^{6}-\frac{507146}{89025}a^{5}+\frac{2183327}{89025}a^{4}+\frac{52039}{29675}a^{3}-\frac{57823}{3561}a^{2}+\frac{231391}{89025}a+\frac{144622}{89025}$, $\frac{46336}{267075}a^{11}-\frac{113}{267075}a^{10}+\frac{41779}{89025}a^{9}-\frac{243076}{267075}a^{8}+\frac{222899}{267075}a^{7}+\frac{17649}{29675}a^{6}-\frac{242269}{89025}a^{5}+\frac{240551}{29675}a^{4}-\frac{114029}{29675}a^{3}-\frac{27734}{3561}a^{2}+\frac{373699}{89025}a+\frac{29561}{29675}$, $\frac{192412}{267075}a^{11}+\frac{54844}{267075}a^{10}+\frac{600064}{267075}a^{9}-\frac{808492}{267075}a^{8}+\frac{938543}{267075}a^{7}+\frac{746552}{267075}a^{6}-\frac{790598}{89025}a^{5}+\frac{2827316}{89025}a^{4}-\frac{263118}{29675}a^{3}-\frac{435856}{17805}a^{2}+\frac{757348}{89025}a+\frac{173026}{89025}$, $\frac{8179}{89025}a^{11}-\frac{43391}{267075}a^{10}+\frac{4558}{89025}a^{9}-\frac{34638}{29675}a^{8}+\frac{132128}{267075}a^{7}-\frac{10952}{29675}a^{6}-\frac{243673}{89025}a^{5}+\frac{155627}{29675}a^{4}-\frac{249593}{29675}a^{3}-\frac{45379}{5935}a^{2}+\frac{378313}{89025}a+\frac{32172}{29675}$, $\frac{274193}{267075}a^{11}+\frac{13644}{29675}a^{10}+\frac{922841}{267075}a^{9}-\frac{940763}{267075}a^{8}+\frac{458374}{89025}a^{7}+\frac{1282063}{267075}a^{6}-\frac{1004522}{89025}a^{5}+\frac{4006129}{89025}a^{4}-\frac{195627}{29675}a^{3}-\frac{516116}{17805}a^{2}+\frac{406034}{29675}a+\frac{100844}{89025}$ | 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: | \( 169.955119049 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 169.955119049 \cdot 1}{2\cdot\sqrt{32357746661769}}\cr\approx \mathstrut & 0.372524272501 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4\times C_2$ |
Character table for $A_4\times C_2$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 6.4.334611.1 x2, 6.2.1896129.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.334611.1 |
Degree 8 sibling: | 8.0.4931831529.1 |
Degree 12 sibling: | 12.0.9351388785251241.3 |
Minimal sibling: | 6.4.334611.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |