Normalized defining polynomial
\( x^{12} - x^{9} + 3x^{8} - 3x^{7} + 14x^{6} - 15x^{5} + 12x^{4} - 10x^{3} + 9x^{2} - 3x + 1 \)
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: | \(3595305184641\) \(\medspace = 3^{16}\cdot 17^{4}\) | 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: | \(11.13\) | 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: | $3^{4/3}17^{1/2}\approx 17.839641950637386$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{18}a^{9}+\frac{1}{6}a^{8}-\frac{1}{9}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{9}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{4}{9}$, $\frac{1}{18}a^{10}-\frac{1}{9}a^{7}+\frac{1}{3}a^{5}-\frac{1}{9}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}-\frac{4}{9}a+\frac{1}{3}$, $\frac{1}{36}a^{11}-\frac{1}{36}a^{10}-\frac{1}{36}a^{9}+\frac{1}{9}a^{8}-\frac{7}{36}a^{7}+\frac{2}{9}a^{6}-\frac{1}{18}a^{5}-\frac{13}{36}a^{4}+\frac{17}{36}a^{3}-\frac{17}{36}a^{2}-\frac{5}{18}a-\frac{7}{36}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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)
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Fundamental units: | $\frac{5}{36}a^{11}-\frac{1}{4}a^{10}-\frac{7}{36}a^{9}-\frac{1}{9}a^{8}+\frac{3}{4}a^{7}-\frac{7}{9}a^{6}+\frac{43}{18}a^{5}-\frac{59}{12}a^{4}+\frac{89}{36}a^{3}-\frac{43}{36}a^{2}+\frac{13}{6}a+\frac{11}{36}$, $\frac{5}{12}a^{11}-\frac{1}{36}a^{10}-\frac{1}{12}a^{9}-\frac{1}{3}a^{8}+\frac{47}{36}a^{7}-\frac{4}{3}a^{6}+\frac{11}{2}a^{5}-\frac{223}{36}a^{4}+\frac{47}{12}a^{3}-\frac{11}{4}a^{2}+\frac{49}{18}a-\frac{1}{4}$, $\frac{1}{4}a^{11}+\frac{5}{12}a^{10}-\frac{1}{36}a^{9}-\frac{1}{3}a^{8}+\frac{5}{12}a^{7}+\frac{5}{9}a^{6}+\frac{13}{6}a^{5}+\frac{7}{4}a^{4}-\frac{115}{36}a^{3}+\frac{17}{12}a^{2}-\frac{1}{2}a+\frac{71}{36}$, $\frac{11}{36}a^{11}+\frac{7}{36}a^{10}-\frac{1}{36}a^{9}-\frac{4}{9}a^{8}+\frac{31}{36}a^{7}-\frac{1}{9}a^{6}+\frac{67}{18}a^{5}-\frac{83}{36}a^{4}+\frac{41}{36}a^{3}-\frac{85}{36}a^{2}+\frac{65}{18}a-\frac{13}{36}$, $\frac{1}{12}a^{11}+\frac{1}{36}a^{10}-\frac{1}{12}a^{9}-\frac{1}{6}a^{8}+\frac{7}{36}a^{7}-\frac{1}{3}a^{6}+\frac{1}{2}a^{5}-\frac{47}{36}a^{4}-\frac{7}{12}a^{3}-\frac{3}{4}a^{2}+\frac{7}{9}a-\frac{11}{12}$ | 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: | \( 20.5548733802 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 20.5548733802 \cdot 1}{2\cdot\sqrt{3595305184641}}\cr\approx \mathstrut & 0.333500231565 \end{aligned}\]
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
\(\Q(\zeta_{9})^+\), 6.2.111537.1 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: | 6.2.111537.1 |
Degree 8 sibling: | 8.0.547981281.1 |
Degree 12 sibling: | 12.4.1039043198361249.1 |
Minimal sibling: | 6.2.111537.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |
3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |