Normalized defining polynomial
\( x^{12} + 4x^{10} + 4x^{8} + 24x^{6} + 96x^{4} + 144x^{2} + 144 \)
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: | \(167961600000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{8}\) | 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: | \(15.33\) | 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: | $2^{19/12}3^{3/4}5^{2/3}\approx 19.973389461199396$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}$, $\frac{1}{8}a^{7}-\frac{1}{2}a$, $\frac{1}{24}a^{8}-\frac{1}{12}a^{6}+\frac{1}{6}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{48}a^{9}-\frac{1}{24}a^{7}-\frac{1}{8}a^{6}+\frac{1}{12}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{144}a^{10}-\frac{1}{72}a^{8}+\frac{1}{36}a^{6}-\frac{1}{12}a^{4}-\frac{1}{6}a^{2}$, $\frac{1}{144}a^{11}+\frac{1}{144}a^{9}-\frac{1}{72}a^{7}-\frac{1}{8}a^{6}+\frac{1}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | \( -\frac{1}{48} a^{10} - \frac{5}{12} a^{4} \) (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: | $\frac{1}{18}a^{11}-\frac{1}{48}a^{10}+\frac{1}{18}a^{9}+\frac{1}{72}a^{7}+\frac{5}{4}a^{5}-\frac{5}{12}a^{4}+\frac{5}{3}a^{3}-\frac{1}{2}a^{2}+\frac{5}{2}a$, $\frac{1}{144}a^{11}-\frac{1}{48}a^{10}+\frac{1}{144}a^{9}-\frac{1}{24}a^{8}-\frac{1}{72}a^{7}-\frac{1}{24}a^{6}+\frac{1}{4}a^{5}-\frac{7}{12}a^{4}+\frac{7}{12}a^{3}-\frac{3}{2}a^{2}+\frac{1}{2}a-\frac{3}{2}$, $\frac{1}{16}a^{11}-\frac{7}{144}a^{10}+\frac{5}{48}a^{9}-\frac{1}{36}a^{8}+\frac{1}{24}a^{7}-\frac{5}{72}a^{6}+\frac{17}{12}a^{5}-\frac{11}{12}a^{4}+\frac{11}{4}a^{3}-\frac{4}{3}a^{2}+\frac{7}{2}a-\frac{5}{2}$, $\frac{7}{144}a^{10}+\frac{11}{72}a^{8}+\frac{7}{36}a^{6}+\frac{11}{12}a^{4}+\frac{23}{6}a^{2}+7$, $\frac{1}{12}a^{8}-\frac{1}{8}a^{7}+\frac{1}{12}a^{6}-\frac{1}{6}a^{4}+2a^{2}-\frac{5}{2}a+3$ | 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: | \( 1408.19992485 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1408.19992485 \cdot 1}{6\cdot\sqrt{167961600000000}}\cr\approx \mathstrut & 1.11426188489 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T21):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2\times S_4$ |
Character table for $C_2\times S_4$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 6.0.12960000.1, 6.2.4320000.1, 6.0.270000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.4320000.1, 6.0.12960000.1 |
Degree 8 siblings: | 8.0.1866240000.18, 8.4.1866240000.6 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.4320000.1 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.16.19 | $x^{12} + 4 x^{11} + 2 x^{10} + 14 x^{8} + 8 x^{6} + 4 x^{5} + 16 x^{4} + 12 x^{2} + 12$ | $6$ | $2$ | $16$ | $C_2\times S_4$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(5\) | 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |