Normalized defining polynomial
\( x^{12} + 2x^{10} + 12x^{8} + 26x^{6} + 115x^{4} + 36x^{2} + 64 \)
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: | \(209824841467085056\) \(\medspace = 2^{8}\cdot 31^{10}\) | 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: | \(27.76\) | 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: | $2\cdot 31^{5/6}\approx 34.980934492937045$ | ||
Ramified primes: | \(2\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{7}+\frac{3}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{16}a^{8}+\frac{3}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{10}-\frac{3}{256}a^{8}-\frac{5}{256}a^{6}-\frac{45}{256}a^{4}-\frac{1}{4}a^{3}-\frac{3}{64}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{256}a^{11}-\frac{3}{256}a^{9}-\frac{5}{256}a^{7}+\frac{19}{256}a^{5}+\frac{13}{64}a^{3}+\frac{1}{4}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{18}$, which has order $18$
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: | $\frac{1}{128}a^{11}+\frac{5}{128}a^{9}+\frac{19}{128}a^{7}+\frac{43}{128}a^{5}+\frac{55}{32}a^{3}+\frac{7}{4}a$, $\frac{1}{64}a^{11}+\frac{1}{64}a^{10}+\frac{1}{64}a^{9}-\frac{3}{64}a^{8}+\frac{7}{64}a^{7}+\frac{11}{64}a^{6}+\frac{31}{64}a^{5}-\frac{13}{64}a^{4}+\frac{5}{8}a^{3}+\frac{1}{16}a^{2}+\frac{3}{4}a-1$, $\frac{9}{256}a^{11}+\frac{9}{256}a^{10}+\frac{21}{256}a^{9}+\frac{21}{256}a^{8}+\frac{115}{256}a^{7}+\frac{115}{256}a^{6}+\frac{315}{256}a^{5}+\frac{315}{256}a^{4}+\frac{285}{64}a^{3}+\frac{285}{64}a^{2}+\frac{13}{4}a+\frac{1}{4}$, $\frac{1}{64}a^{11}+\frac{1}{64}a^{9}+\frac{7}{64}a^{7}+\frac{31}{64}a^{5}+\frac{1}{8}a^{3}+\frac{1}{4}a$, $\frac{47}{256}a^{11}+\frac{5}{256}a^{10}+\frac{67}{256}a^{9}-\frac{15}{256}a^{8}+\frac{453}{256}a^{7}+\frac{71}{256}a^{6}+\frac{749}{256}a^{5}-\frac{33}{256}a^{4}+\frac{1047}{64}a^{3}+\frac{9}{64}a^{2}-7a-\frac{55}{4}$ | 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: | \( 31367.7469501 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 31367.7469501 \cdot 18}{2\cdot\sqrt{209824841467085056}}\cr\approx \mathstrut & 37.9207216451 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-31}) \), 3.3.961.1, 6.4.458066416.1, 6.0.28629151.1, 6.2.14776336.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.458066416.1 |
Degree 8 sibling: | 8.0.227200942336.3 |
Degree 12 sibling: | 12.4.3357197463473360896.26 |
Minimal sibling: | 6.4.458066416.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
\(31\) | 31.6.5.5 | $x^{6} + 31$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
31.6.5.5 | $x^{6} + 31$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |