Normalized defining polynomial
\( x^{12} + 6x^{10} + 21x^{8} + 49x^{6} + 84x^{4} + 96x^{2} + 64 \)
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: | \(61585005599133696\) \(\medspace = 2^{12}\cdot 3^{18}\cdot 197^{2}\) | 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: | \(25.07\) | 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: | $2\cdot 3^{3/2}197^{1/2}\approx 145.8629493737186$ | ||
Ramified primes: | \(2\), \(3\), \(197\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.88935971164569.2$^{4}$, 8.0.2529734290903296.4$^{4}$, deg 24$^{12}$, deg 24$^{12}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{3}{8}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{8}+\frac{5}{16}a^{6}-\frac{7}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{11}-\frac{1}{16}a^{9}+\frac{5}{32}a^{7}+\frac{9}{32}a^{5}+\frac{3}{8}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{14}$, which has order $14$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{1}{16}a^{10}+\frac{3}{8}a^{8}+\frac{21}{16}a^{6}+\frac{49}{16}a^{4}+\frac{17}{4}a^{2}+3$, $\frac{1}{16}a^{10}+\frac{3}{8}a^{8}+\frac{21}{16}a^{6}+\frac{49}{16}a^{4}+\frac{17}{4}a^{2}+4$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{3}{8}a^{8}-\frac{1}{4}a^{7}+\frac{21}{16}a^{6}-\frac{5}{8}a^{5}+\frac{49}{16}a^{4}-\frac{5}{8}a^{3}+\frac{17}{4}a^{2}-a+4$, $\frac{1}{8}a^{9}+\frac{1}{4}a^{7}+\frac{5}{8}a^{5}+\frac{5}{8}a^{3}+a-1$, $\frac{1}{32}a^{11}+\frac{3}{16}a^{9}+\frac{21}{32}a^{7}+\frac{49}{32}a^{5}+\frac{21}{8}a^{3}+2a-1$ | 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: | \( 325.675402795 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 325.675402795 \cdot 14}{2\cdot\sqrt{61585005599133696}}\cr\approx \mathstrut & 0.565229365139 \end{aligned}\]
Galois group
$C_2^2\times A_4$ (as 12T25):
A solvable group of order 48 |
The 16 conjugacy class representatives for $C_2^2 \times A_4$ |
Character table for $C_2^2 \times A_4$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), 6.0.3877551.1, \(\Q(\zeta_{36})^+\), 6.0.82721088.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
\(3\) | 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
\(197\) | 197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |