Normalized defining polynomial
\( x^{12} - 3x^{11} + 4x^{10} - 5x^{9} + 5x^{8} + 2x^{7} - 7x^{6} + 2x^{5} + 5x^{4} - 5x^{3} + 4x^{2} - 3x + 1 \)
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: | \(107918163081\) \(\medspace = 3^{6}\cdot 23^{6}\) | 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: | \(8.31\) | 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: | $3^{1/2}23^{1/2}\approx 8.306623862918075$ | ||
Ramified primes: | \(3\), \(23\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}+\frac{5}{19}a^{9}+\frac{5}{19}a^{8}-\frac{8}{19}a^{7}-\frac{7}{19}a^{6}-\frac{8}{19}a^{5}-\frac{7}{19}a^{4}-\frac{8}{19}a^{3}+\frac{5}{19}a^{2}+\frac{5}{19}a+\frac{1}{19}$, $\frac{1}{95}a^{11}+\frac{1}{95}a^{10}+\frac{23}{95}a^{9}-\frac{28}{95}a^{8}-\frac{32}{95}a^{7}+\frac{39}{95}a^{6}+\frac{44}{95}a^{5}-\frac{37}{95}a^{4}+\frac{37}{95}a^{3}+\frac{23}{95}a^{2}-\frac{1}{5}a-\frac{4}{95}$
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: | \( \frac{8}{5} a^{11} - \frac{358}{95} a^{10} + \frac{376}{95} a^{9} - \frac{536}{95} a^{8} + \frac{451}{95} a^{7} + \frac{568}{95} a^{6} - \frac{632}{95} a^{5} - \frac{154}{95} a^{4} + \frac{584}{95} a^{3} - \frac{384}{95} a^{2} + \frac{452}{95} a - \frac{168}{95} \) (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: | $\frac{134}{95}a^{11}-\frac{306}{95}a^{10}+\frac{312}{95}a^{9}-\frac{442}{95}a^{8}+\frac{372}{95}a^{7}+\frac{516}{95}a^{6}-\frac{559}{95}a^{5}-\frac{168}{95}a^{4}+\frac{498}{95}a^{3}-\frac{163}{95}a^{2}+\frac{384}{95}a-\frac{216}{95}$, $\frac{3}{95}a^{11}-\frac{52}{95}a^{10}+\frac{79}{95}a^{9}-\frac{74}{95}a^{8}+\frac{154}{95}a^{7}-\frac{68}{95}a^{6}-\frac{188}{95}a^{5}-\frac{11}{95}a^{4}+\frac{4}{5}a^{3}-\frac{16}{95}a^{2}+\frac{48}{95}a-\frac{162}{95}$, $\frac{23}{95}a^{11}-\frac{27}{95}a^{10}-\frac{6}{95}a^{9}-\frac{39}{95}a^{8}+\frac{44}{95}a^{7}+\frac{107}{95}a^{6}+\frac{82}{95}a^{5}-\frac{216}{95}a^{4}-\frac{79}{95}a^{3}+\frac{184}{95}a^{2}+\frac{73}{95}a-\frac{47}{95}$, $\frac{58}{19}a^{11}-\frac{144}{19}a^{10}+\frac{153}{19}a^{9}-\frac{202}{19}a^{8}+\frac{178}{19}a^{7}+\frac{218}{19}a^{6}-\frac{297}{19}a^{5}-\frac{67}{19}a^{4}+15a^{3}-\frac{151}{19}a^{2}+\frac{149}{19}a-\frac{92}{19}$, $\frac{6}{5}a^{11}-\frac{331}{95}a^{10}+\frac{397}{95}a^{9}-\frac{477}{95}a^{8}+\frac{482}{95}a^{7}+\frac{341}{95}a^{6}-\frac{829}{95}a^{5}+\frac{37}{95}a^{4}+\frac{653}{95}a^{3}-\frac{363}{95}a^{2}+\frac{359}{95}a-\frac{331}{95}$ | 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: | \( 6.2323227807 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 6.2323227807 \cdot 1}{6\cdot\sqrt{107918163081}}\cr\approx \mathstrut & 0.19454972691 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 6 conjugacy class representatives for $D_6$ |
Character table for $D_6$ |
Intermediate fields
\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-3}) \), 3.1.23.1 x3, \(\Q(\sqrt{-3}, \sqrt{-23})\), 6.0.12167.1, 6.2.328509.1 x3, 6.0.14283.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.328509.1, 6.0.14283.1 |
Minimal sibling: | 6.0.14283.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(23\) | 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |