Normalized defining polynomial
\( x^{12} - 6 x^{11} + 15 x^{10} - 20 x^{9} + 10 x^{8} + 14 x^{7} - 21 x^{6} - 4 x^{5} + 17 x^{4} + \cdots + 1 \)
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: | \(2111929749504\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 29^{4}\) | 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: | \(10.64\) | 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: | $2\cdot 3^{1/2}29^{1/2}\approx 18.65475810617763$ | ||
Ramified primes: | \(2\), \(3\), \(29\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{71}a^{11}+\frac{30}{71}a^{10}+\frac{30}{71}a^{9}-\frac{5}{71}a^{8}-\frac{28}{71}a^{7}-\frac{21}{71}a^{5}+\frac{21}{71}a^{4}-\frac{8}{71}a^{3}-\frac{8}{71}a^{2}-\frac{8}{71}a-\frac{2}{71}$
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: | \( -\frac{34}{71} a^{11} + \frac{187}{71} a^{10} - \frac{452}{71} a^{9} + \frac{596}{71} a^{8} - \frac{255}{71} a^{7} - 7 a^{6} + \frac{714}{71} a^{5} - \frac{4}{71} a^{4} - \frac{580}{71} a^{3} + \frac{272}{71} a^{2} + \frac{130}{71} a - \frac{74}{71} \) (order $12$) | 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: | $\frac{17}{71}a^{11}-\frac{129}{71}a^{10}+\frac{368}{71}a^{9}-\frac{511}{71}a^{8}+\frac{234}{71}a^{7}+7a^{6}-\frac{854}{71}a^{5}+\frac{2}{71}a^{4}+\frac{716}{71}a^{3}-\frac{278}{71}a^{2}-\frac{136}{71}a+\frac{108}{71}$, $\frac{34}{71}a^{11}-\frac{116}{71}a^{10}+\frac{97}{71}a^{9}+\frac{114}{71}a^{8}-\frac{455}{71}a^{7}+7a^{6}+\frac{280}{71}a^{5}-\frac{493}{71}a^{4}-\frac{201}{71}a^{3}+\frac{225}{71}a^{2}-\frac{59}{71}a-\frac{68}{71}$, $\frac{50}{71}a^{11}-\frac{204}{71}a^{10}+\frac{293}{71}a^{9}-\frac{108}{71}a^{8}-\frac{406}{71}a^{7}+10a^{6}+\frac{157}{71}a^{5}-\frac{796}{71}a^{4}+\frac{97}{71}a^{3}+\frac{310}{71}a^{2}-\frac{187}{71}a-\frac{29}{71}$, $\frac{34}{71}a^{11}-\frac{187}{71}a^{10}+\frac{452}{71}a^{9}-\frac{596}{71}a^{8}+\frac{255}{71}a^{7}+7a^{6}-\frac{714}{71}a^{5}+\frac{4}{71}a^{4}+\frac{580}{71}a^{3}-\frac{272}{71}a^{2}-\frac{130}{71}a+\frac{145}{71}$, $\frac{66}{71}a^{11}-\frac{434}{71}a^{10}+\frac{1199}{71}a^{9}-\frac{1821}{71}a^{8}+\frac{1347}{71}a^{7}+7a^{6}-\frac{1670}{71}a^{5}+\frac{250}{71}a^{4}+\frac{1105}{71}a^{3}-\frac{528}{71}a^{2}-\frac{31}{71}a+\frac{81}{71}$ | 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: | \( 72.1781797986 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 72.1781797986 \cdot 1}{12\cdot\sqrt{2111929749504}}\cr\approx \mathstrut & 0.254662006101 \end{aligned}\]
Galois group
$S_3\times D_6$ (as 12T37):
A solvable group of order 72 |
The 18 conjugacy class representatives for $S_3\times D_6$ |
Character table for $S_3\times D_6$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 6.2.1453248.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |