Normalized defining polynomial
\( x^{12} - 12x^{10} - 8x^{9} + 45x^{8} + 60x^{7} - 34x^{6} - 108x^{5} + 9x^{4} + 200x^{3} + 216x^{2} + 96x + 16 \)
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: | \(406239826673664\) \(\medspace = 2^{20}\cdot 3^{18}\) | 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: | \(16.50\) | 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: | $2^{2}3^{31/18}\approx 26.531928538998848$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{10}-\frac{1}{16}a^{9}+\frac{1}{4}a^{8}+\frac{1}{8}a^{7}+\frac{13}{64}a^{6}-\frac{3}{8}a^{5}+\frac{13}{32}a^{4}+\frac{3}{16}a^{3}+\frac{17}{64}a^{2}+\frac{5}{16}a+\frac{5}{16}$, $\frac{1}{512}a^{11}+\frac{1}{256}a^{10}-\frac{1}{64}a^{9}-\frac{3}{64}a^{8}-\frac{3}{512}a^{7}+\frac{27}{256}a^{6}+\frac{37}{256}a^{5}+\frac{5}{64}a^{4}+\frac{89}{512}a^{3}-\frac{67}{256}a^{2}-\frac{13}{128}a-\frac{1}{64}$
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: | \( -\frac{1593}{512} a^{11} + \frac{711}{256} a^{10} + \frac{2233}{64} a^{9} - \frac{405}{64} a^{8} - \frac{68949}{512} a^{7} - \frac{16899}{256} a^{6} + \frac{42435}{256} a^{5} + \frac{12003}{64} a^{4} - \frac{100817}{512} a^{3} - \frac{114453}{256} a^{2} - \frac{34587}{128} a - \frac{3591}{64} \) (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: | $\frac{27}{16}a^{11}-\frac{9}{16}a^{10}-\frac{161}{8}a^{9}-\frac{27}{4}a^{8}+\frac{1263}{16}a^{7}+\frac{1199}{16}a^{6}-\frac{171}{2}a^{5}-\frac{1245}{8}a^{4}+\frac{1147}{16}a^{3}+\frac{5103}{16}a^{2}+\frac{2025}{8}a+\frac{131}{2}$, $\frac{2187}{512}a^{11}-\frac{729}{256}a^{10}-\frac{3159}{64}a^{9}-\frac{81}{64}a^{8}+\frac{98847}{512}a^{7}+\frac{32661}{256}a^{6}-\frac{58953}{256}a^{5}-\frac{19699}{64}a^{4}+\frac{124915}{512}a^{3}+\frac{177147}{256}a^{2}+\frac{58921}{128}a+\frac{6497}{64}$, $\frac{2105}{512}a^{11}-\frac{967}{256}a^{10}-\frac{2937}{64}a^{9}+\frac{597}{64}a^{8}+\frac{90453}{512}a^{7}+\frac{21507}{256}a^{6}-\frac{55747}{256}a^{5}-\frac{15587}{64}a^{4}+\frac{134097}{512}a^{3}+\frac{149013}{256}a^{2}+\frac{44955}{128}a+\frac{4551}{64}$, $\frac{7515}{512}a^{11}-\frac{3613}{256}a^{10}-\frac{10539}{64}a^{9}+\frac{2783}{64}a^{8}+\frac{328559}{512}a^{7}+\frac{64385}{256}a^{6}-\frac{212537}{256}a^{5}-\frac{51221}{64}a^{4}+\frac{524771}{512}a^{3}+\frac{520759}{256}a^{2}+\frac{133793}{128}a+\frac{9821}{64}$, $\frac{11501}{512}a^{11}-\frac{4171}{256}a^{10}-\frac{16493}{64}a^{9}+\frac{457}{64}a^{8}+\frac{514745}{512}a^{7}+\frac{158503}{256}a^{6}-\frac{310239}{256}a^{5}-\frac{99131}{64}a^{4}+\frac{677541}{512}a^{3}+\frac{905025}{256}a^{2}+\frac{293975}{128}a+\frac{31851}{64}$ | 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: | \( 1774.17941346 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1774.17941346 \cdot 1}{12\cdot\sqrt{406239826673664}}\cr\approx \mathstrut & 0.451340442780 \end{aligned}\]
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
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.2.20155392.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 3.12.18.79 | $x^{12} - 6 x^{11} + 57 x^{10} - 30 x^{9} + 54 x^{8} + 36 x^{7} + 60 x^{6} - 126 x^{5} + 252 x^{4} + 126 x^{3} + 441$ | $6$ | $2$ | $18$ | $C_6\times S_3$ | $[3/2, 2]_{2}^{2}$ |