Normalized defining polynomial
\( x^{12} - 6 x^{11} + 20 x^{10} - 42 x^{9} + 66 x^{8} - 82 x^{7} + 101 x^{6} - 98 x^{5} + 90 x^{4} + \cdots + 16 \)
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: | \(252088148770816\) \(\medspace = 2^{14}\cdot 109^{5}\) | 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: | \(15.85\) | 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: | $2^{3/2}109^{5/6}\approx 141.05887259432856$ | ||
Ramified primes: | \(2\), \(109\) | 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: | \(\Q(\sqrt{109}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{76}a^{10}-\frac{5}{38}a^{9}-\frac{2}{19}a^{8}-\frac{7}{38}a^{7}-\frac{9}{38}a^{6}+\frac{15}{38}a^{5}-\frac{11}{76}a^{4}+\frac{17}{38}a^{3}+\frac{9}{38}a^{2}-\frac{4}{19}a+\frac{9}{19}$, $\frac{1}{5624}a^{11}-\frac{3}{703}a^{10}+\frac{261}{1406}a^{9}+\frac{277}{2812}a^{8}+\frac{1115}{2812}a^{7}+\frac{1053}{2812}a^{6}+\frac{2153}{5624}a^{5}+\frac{351}{1406}a^{4}+\frac{1025}{2812}a^{3}-\frac{347}{703}a^{2}-\frac{4}{19}a+\frac{70}{703}$
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: | \( -\frac{15}{76} a^{11} + \frac{93}{76} a^{10} - \frac{149}{38} a^{9} + \frac{295}{38} a^{8} - \frac{208}{19} a^{7} + \frac{239}{19} a^{6} - \frac{1165}{76} a^{5} + \frac{1181}{76} a^{4} - \frac{210}{19} a^{3} + \frac{367}{38} a^{2} - \frac{135}{19} a + 5 \) (order $4$) | 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: | $\frac{1533}{5624}a^{11}-\frac{4151}{2812}a^{10}+\frac{6211}{1406}a^{9}-\frac{22541}{2812}a^{8}+\frac{30753}{2812}a^{7}-\frac{34101}{2812}a^{6}+\frac{89097}{5624}a^{5}-\frac{36609}{2812}a^{4}+\frac{29609}{2812}a^{3}-\frac{4923}{703}a^{2}+\frac{137}{19}a-\frac{2099}{703}$, $\frac{177}{2812}a^{11}+\frac{248}{703}a^{10}-\frac{42}{37}a^{9}+\frac{177}{74}a^{8}-\frac{5843}{1406}a^{7}+\frac{8831}{1406}a^{6}-\frac{26029}{2812}a^{5}+\frac{7285}{703}a^{4}-\frac{14629}{1406}a^{3}+\frac{329}{37}a^{2}-\frac{113}{19}a+\frac{1453}{703}$, $\frac{71}{74}a^{11}+\frac{7253}{1406}a^{10}-\frac{21781}{1406}a^{9}+\frac{19970}{703}a^{8}-\frac{27724}{703}a^{7}+\frac{30939}{703}a^{6}-\frac{78411}{1406}a^{5}+\frac{62527}{1406}a^{4}-\frac{53313}{1406}a^{3}+\frac{19563}{703}a^{2}-\frac{420}{19}a+\frac{7339}{703}$, $\frac{559}{2812}a^{11}-\frac{1417}{1406}a^{10}+\frac{2043}{703}a^{9}-\frac{6995}{1406}a^{8}+\frac{9307}{1406}a^{7}-\frac{9959}{1406}a^{6}+\frac{27815}{2812}a^{5}-\frac{10253}{1406}a^{4}+\frac{10501}{1406}a^{3}-\frac{3590}{703}a^{2}+\frac{103}{19}a-\frac{1549}{703}$, $\frac{375}{1406}a^{11}-\frac{1785}{1406}a^{10}+\frac{5109}{1406}a^{9}-\frac{4424}{703}a^{8}+\frac{6278}{703}a^{7}-\frac{6797}{703}a^{6}+\frac{17129}{1406}a^{5}-\frac{8409}{1406}a^{4}+\frac{12285}{1406}a^{3}-\frac{3536}{703}a^{2}+\frac{78}{19}a+\frac{771}{703}$ | 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: | \( 2052.5093627792508 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2052.5093627792508 \cdot 1}{4\cdot\sqrt{252088148770816}}\cr\approx \mathstrut & 1.98851170263669 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 6.0.760384.2 |
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 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(109\) | 109.6.0.1 | $x^{6} + 107 x^{3} + 102 x^{2} + 66 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
109.6.5.6 | $x^{6} + 1199$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |