Normalized defining polynomial
\( x^{12} + 12 x^{10} - 4 x^{9} + 54 x^{8} - 36 x^{7} + 120 x^{6} - 108 x^{5} + 153 x^{4} - 148 x^{3} + \cdots + 52 \)
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: | \(43873901280755712\) \(\medspace = 2^{22}\cdot 3^{21}\) | 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: | \(24.37\) | 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: | $2^{2}3^{25/12}\approx 39.4514168808688$ | ||
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: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{3}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{3}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{104}a^{11}-\frac{5}{104}a^{10}-\frac{1}{52}a^{9}-\frac{7}{104}a^{8}-\frac{1}{52}a^{7}-\frac{1}{8}a^{6}+\frac{2}{13}a^{5}+\frac{33}{104}a^{4}-\frac{51}{104}a^{3}-\frac{9}{26}a^{2}+\frac{1}{52}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{8} a^{9} + \frac{9}{8} a^{7} - \frac{1}{4} a^{6} + \frac{27}{8} a^{5} - \frac{3}{2} a^{4} + \frac{37}{8} a^{3} - \frac{9}{4} a^{2} + \frac{15}{4} a - \frac{5}{2} \) (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: | $\frac{11}{52}a^{11}-\frac{29}{52}a^{10}+\frac{229}{104}a^{9}-\frac{68}{13}a^{8}+\frac{983}{104}a^{7}-17a^{6}+\frac{2263}{104}a^{5}-\frac{657}{26}a^{4}+\frac{2557}{104}a^{3}-\frac{1163}{52}a^{2}+\frac{1023}{52}a-\frac{15}{2}$, $\frac{7}{104}a^{11}+\frac{17}{104}a^{10}+\frac{19}{52}a^{9}+\frac{133}{104}a^{8}-\frac{5}{13}a^{7}+\frac{29}{8}a^{6}-\frac{51}{13}a^{5}+\frac{621}{104}a^{4}-\frac{487}{104}a^{3}+\frac{93}{26}a^{2}-\frac{279}{52}a+\frac{5}{2}$, $\frac{1}{104}a^{11}-\frac{5}{104}a^{10}+\frac{11}{104}a^{9}-\frac{59}{104}a^{8}+\frac{63}{104}a^{7}-\frac{21}{8}a^{6}+\frac{263}{104}a^{5}-\frac{643}{104}a^{4}+\frac{293}{52}a^{3}-\frac{89}{13}a^{2}+\frac{49}{13}a-\frac{1}{2}$, $\frac{7}{104}a^{11}-\frac{11}{52}a^{10}+\frac{77}{104}a^{9}-\frac{24}{13}a^{8}+\frac{337}{104}a^{7}-\frac{21}{4}a^{6}+\frac{697}{104}a^{5}-\frac{80}{13}a^{4}+\frac{283}{52}a^{3}-\frac{115}{26}a^{2}+\frac{70}{13}a-4$, $\frac{31}{104}a^{11}+\frac{27}{104}a^{10}+\frac{177}{52}a^{9}+\frac{225}{104}a^{8}+\frac{749}{52}a^{7}+\frac{45}{8}a^{6}+\frac{374}{13}a^{5}+\frac{477}{104}a^{4}+\frac{2995}{104}a^{3}-\frac{259}{52}a^{2}+\frac{551}{52}a-10$ (assuming GRH) | 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: | \( 19615.2324223 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 19615.2324223 \cdot 1}{6\cdot\sqrt{43873901280755712}}\cr\approx \mathstrut & 0.960325124828 \end{aligned}\] (assuming GRH)
Galois group
$\SOPlus(4,2)$ (as 12T36):
A solvable group of order 72 |
The 9 conjugacy class representatives for $\SOPlus(4,2)$ |
Character table for $\SOPlus(4,2)$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.1728.1, 6.0.3779136.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.3779136.3, 6.4.60466176.4 |
Degree 9 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 6.0.3779136.3 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.6 | $x^{4} - 4 x^{3} + 28 x^{2} - 24 x + 36$ | $2$ | $2$ | $6$ | $D_{4}$ | $[2, 3]^{2}$ |
2.8.16.5 | $x^{8} + 4 x^{7} + 14 x^{6} + 36 x^{5} + 73 x^{4} + 88 x^{3} + 48 x^{2} + 56 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ | |
\(3\) | 3.12.21.46 | $x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{6} + 12 x^{3} + 12$ | $12$ | $1$ | $21$ | 12T36 | $[9/4, 9/4]_{4}^{2}$ |