Normalized defining polynomial
\( x^{12} - 2x^{11} - x^{10} + 2x^{9} - 6x^{8} + 10x^{7} - 9x^{6} + 10x^{5} - 6x^{4} + 2x^{3} - x^{2} - 2x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(151457444000000\) \(\medspace = 2^{8}\cdot 5^{6}\cdot 31^{4}\cdot 41\) | 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: | \(15.19\) | 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: | $2^{2/3}5^{1/2}31^{2/3}41^{1/2}\approx 224.2873209948301$ | ||
Ramified primes: | \(2\), \(5\), \(31\), \(41\) | 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: | \(\Q(\sqrt{41}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{3}{4}a^{11}-\frac{5}{4}a^{10}-\frac{5}{4}a^{9}+\frac{3}{2}a^{8}-\frac{17}{4}a^{7}+\frac{21}{4}a^{6}-\frac{17}{4}a^{5}+\frac{9}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{3}{4}a-2$, $a^{11}-2a^{10}-a^{9}+2a^{8}-6a^{7}+10a^{6}-9a^{5}+10a^{4}-6a^{3}+2a^{2}-a-2$, $\frac{1}{4}a^{11}-a^{9}-\frac{3}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+\frac{5}{4}a^{4}-\frac{5}{4}a^{3}+2a^{2}-2a-\frac{1}{4}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{5}{4}a^{8}-\frac{3}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{7}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{3}{4}$, $\frac{5}{4}a^{11}+\frac{7}{4}a^{10}+\frac{9}{4}a^{9}-a^{8}+\frac{27}{4}a^{7}-\frac{33}{4}a^{6}+\frac{27}{4}a^{5}-10a^{4}+\frac{13}{4}a^{3}-\frac{13}{4}a^{2}+\frac{3}{4}a+1$, $\frac{1}{4}a^{11}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-4a^{6}+2a^{5}-\frac{9}{4}a^{4}+\frac{13}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{3}{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-a^{9}-\frac{1}{2}a^{8}-\frac{5}{2}a^{7}+3a^{6}-\frac{5}{2}a^{5}+\frac{11}{2}a^{4}-3a^{3}+\frac{5}{2}a^{2}-\frac{3}{2}a$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 367.3117987857954 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 367.3117987857954 \cdot 1}{2\cdot\sqrt{151457444000000}}\cr\approx \mathstrut & 0.372133722872273 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), 6.6.1922000.1 |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(31\) | 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
31.6.4.2 | $x^{6} - 899 x^{3} + 2883$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.6.0.1 | $x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |