Normalized defining polynomial
\( x^{12} - 3 x^{11} + 2 x^{10} - 3 x^{9} + 15 x^{8} - 24 x^{7} + 25 x^{6} - 24 x^{5} + 15 x^{4} - 3 x^{3} + \cdots + 1 \)
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: | \(646990183449\) \(\medspace = 3^{6}\cdot 31^{6}\) | 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: | \(9.64\) | 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: | $3^{1/2}31^{1/2}\approx 9.643650760992955$ | ||
Ramified primes: | \(3\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{37}a^{10}-\frac{13}{37}a^{9}-\frac{17}{37}a^{8}-\frac{5}{37}a^{7}+\frac{8}{37}a^{6}+\frac{12}{37}a^{5}+\frac{8}{37}a^{4}-\frac{5}{37}a^{3}-\frac{17}{37}a^{2}-\frac{13}{37}a+\frac{1}{37}$, $\frac{1}{407}a^{11}-\frac{4}{407}a^{10}-\frac{60}{407}a^{9}+\frac{101}{407}a^{8}-\frac{5}{11}a^{7}+\frac{84}{407}a^{6}-\frac{180}{407}a^{5}+\frac{178}{407}a^{4}+\frac{123}{407}a^{3}-\frac{203}{407}a^{2}+\frac{106}{407}a-\frac{65}{407}$
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: | \( \frac{846}{407} a^{11} - \frac{2064}{407} a^{10} + \frac{456}{407} a^{9} - \frac{2114}{407} a^{8} + \frac{11493}{407} a^{7} - \frac{13614}{407} a^{6} + \frac{12522}{407} a^{5} - \frac{12234}{407} a^{4} + \frac{126}{11} a^{3} + \frac{1182}{407} a^{2} + \frac{1698}{407} a - \frac{1167}{407} \) (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: | $\frac{921}{407}a^{11}-\frac{2254}{407}a^{10}+\frac{631}{407}a^{9}-\frac{2514}{407}a^{8}+\frac{12534}{407}a^{7}-\frac{15388}{407}a^{6}+\frac{14994}{407}a^{5}-\frac{14691}{407}a^{4}+\frac{7232}{407}a^{3}-\frac{40}{407}a^{2}+\frac{2520}{407}a-\frac{1862}{407}$, $\frac{100}{37}a^{11}-\frac{240}{37}a^{10}+\frac{60}{37}a^{9}-\frac{279}{37}a^{8}+\frac{1346}{37}a^{7}-\frac{1605}{37}a^{6}+\frac{1606}{37}a^{5}-\frac{1566}{37}a^{4}+\frac{696}{37}a^{3}-\frac{6}{37}a^{2}+\frac{269}{37}a-\frac{161}{37}$, $\frac{600}{407}a^{11}-\frac{1663}{407}a^{10}+\frac{817}{407}a^{9}-\frac{1583}{407}a^{8}+\frac{8636}{407}a^{7}-\frac{12487}{407}a^{6}+\frac{11955}{407}a^{5}-\frac{11439}{407}a^{4}+\frac{179}{11}a^{3}-\frac{19}{407}a^{2}+\frac{1109}{407}a-\frac{1633}{407}$, $\frac{1982}{407}a^{11}-\frac{4947}{407}a^{10}+\frac{1464}{407}a^{9}-\frac{5155}{407}a^{8}+\frac{27053}{407}a^{7}-\frac{33921}{407}a^{6}+\frac{32288}{407}a^{5}-\frac{30763}{407}a^{4}+\frac{13578}{407}a^{3}+\frac{1596}{407}a^{2}+\frac{4469}{407}a-\frac{3749}{407}$, $\frac{585}{407}a^{11}-\frac{1350}{407}a^{10}+\frac{56}{407}a^{9}-\frac{1294}{407}a^{8}+\frac{7704}{407}a^{7}-\frac{8060}{407}a^{6}+\frac{6702}{407}a^{5}-\frac{6387}{407}a^{4}+\frac{664}{407}a^{3}+\frac{1981}{407}a^{2}+\frac{707}{407}a-\frac{812}{407}$ | 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: | \( 19.1654785683 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 19.1654785683 \cdot 1}{6\cdot\sqrt{646990183449}}\cr\approx \mathstrut & 0.244342370170 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 6 conjugacy class representatives for $D_6$ |
Character table for $D_6$ |
Intermediate fields
\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{93}) \), 3.1.31.1 x3, \(\Q(\sqrt{-3}, \sqrt{-31})\), 6.0.29791.1, 6.0.25947.1 x3, 6.2.804357.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.804357.1, 6.0.25947.1 |
Minimal sibling: | 6.0.25947.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |