Normalized defining polynomial
\( x^{12} - x^{11} + 4x^{10} + 3x^{9} + 6x^{8} + 10x^{7} + 15x^{6} + 23x^{5} + 12x^{4} + 12x^{3} + 13x^{2} + 4x + 1 \)
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: | \(433626201009\) \(\medspace = 3^{6}\cdot 29^{6}\) | 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: | \(9.33\) | 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: | $3^{1/2}29^{1/2}\approx 9.327379053088816$ | ||
Ramified primes: | \(3\), \(29\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{30}a^{9}+\frac{2}{15}a^{8}+\frac{2}{15}a^{7}+\frac{1}{3}a^{6}-\frac{1}{5}a^{5}+\frac{4}{15}a^{4}-\frac{1}{6}a^{3}-\frac{2}{15}a^{2}-\frac{1}{15}a-\frac{1}{10}$, $\frac{1}{30}a^{10}-\frac{1}{15}a^{8}+\frac{2}{15}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{7}{30}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{6}a-\frac{4}{15}$, $\frac{1}{30}a^{11}+\frac{1}{15}a^{8}-\frac{4}{15}a^{7}-\frac{4}{15}a^{6}+\frac{1}{30}a^{5}-\frac{4}{15}a^{4}-\frac{1}{5}a^{3}+\frac{7}{30}a^{2}-\frac{1}{15}a+\frac{7}{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( \frac{7}{10} a^{11} - \frac{17}{15} a^{10} + \frac{18}{5} a^{9} - \frac{4}{15} a^{8} + \frac{74}{15} a^{7} + \frac{58}{15} a^{6} + \frac{55}{6} a^{5} + \frac{167}{15} a^{4} + \frac{10}{3} a^{3} + \frac{239}{30} a^{2} + \frac{27}{5} a + \frac{26}{15} \) (order $6$) | 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: | $\frac{7}{30}a^{11}-\frac{8}{15}a^{10}+\frac{8}{5}a^{9}-\frac{16}{15}a^{8}+\frac{12}{5}a^{7}+\frac{1}{3}a^{6}+\frac{97}{30}a^{5}+\frac{8}{3}a^{4}+\frac{2}{5}a^{3}+\frac{133}{30}a^{2}-\frac{4}{15}$, $\frac{1}{5}a^{11}-\frac{3}{10}a^{10}+\frac{14}{15}a^{9}+\frac{1}{15}a^{8}+\frac{19}{15}a^{7}+\frac{13}{15}a^{6}+\frac{7}{3}a^{5}+\frac{89}{30}a^{4}+\frac{4}{5}a^{2}+\frac{9}{10}a+\frac{11}{15}$, $\frac{7}{30}a^{11}-\frac{8}{15}a^{10}+\frac{8}{5}a^{9}-\frac{16}{15}a^{8}+\frac{12}{5}a^{7}+\frac{1}{3}a^{6}+\frac{97}{30}a^{5}+\frac{8}{3}a^{4}+\frac{2}{5}a^{3}+\frac{133}{30}a^{2}+\frac{11}{15}$, $\frac{4}{15}a^{11}-\frac{3}{10}a^{10}+\frac{7}{6}a^{9}+\frac{7}{15}a^{8}+2a^{7}+2a^{6}+\frac{13}{3}a^{5}+\frac{53}{10}a^{4}+\frac{31}{10}a^{3}+\frac{11}{3}a^{2}+\frac{59}{30}a+\frac{13}{10}$, $\frac{2}{15}a^{11}-\frac{1}{6}a^{10}+\frac{2}{3}a^{9}-\frac{1}{15}a^{8}+\frac{8}{5}a^{7}+\frac{4}{15}a^{6}+\frac{14}{5}a^{5}+\frac{73}{30}a^{4}+\frac{11}{5}a^{3}+\frac{29}{15}a^{2}+\frac{17}{30}a+\frac{28}{15}$ | 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: | \( 15.5922746816 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 15.5922746816 \cdot 1}{6\cdot\sqrt{433626201009}}\cr\approx \mathstrut & 0.242817329726 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 6 conjugacy class representatives for $D_6$ |
Character table for $D_6$ |
Intermediate fields
\(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{29}) \), 3.1.87.1 x3, \(\Q(\sqrt{-3}, \sqrt{29})\), 6.0.658503.1, 6.0.22707.1 x3, 6.2.219501.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.22707.1, 6.2.219501.1 |
Minimal sibling: | 6.0.22707.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(29\) | 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |