Normalized defining polynomial
\( x^{12} - 2x^{11} + 6x^{9} - 23x^{8} + 42x^{7} - 46x^{6} + 42x^{5} - 23x^{4} + 6x^{3} - 2x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-102241426653184\) \(\medspace = -\,2^{14}\cdot 7^{5}\cdot 13^{5}\) | 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: | \(14.71\) | 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: | $2^{7/6}7^{1/2}13^{1/2}\approx 21.415210999701447$ | ||
Ramified primes: | \(2\), \(7\), \(13\) | 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: | \(\Q(\sqrt{-91}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}-\frac{1}{7}a^{5}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{9}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}-\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{10}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{4}{7}a^{11}-\frac{6}{7}a^{10}-\frac{1}{7}a^{9}+\frac{20}{7}a^{8}-12a^{7}+20a^{6}-\frac{157}{7}a^{5}+22a^{4}-\frac{61}{7}a^{3}+\frac{16}{7}a^{2}+\frac{15}{7}a-\frac{13}{7}$, $a$, $\frac{8}{7}a^{11}-\frac{13}{7}a^{10}-\frac{4}{7}a^{9}+\frac{45}{7}a^{8}-\frac{169}{7}a^{7}+40a^{6}-\frac{279}{7}a^{5}+\frac{253}{7}a^{4}-\frac{92}{7}a^{3}+\frac{6}{7}a^{2}+\frac{10}{7}a-\frac{12}{7}$, $\frac{19}{7}a^{11}-\frac{30}{7}a^{10}-\frac{13}{7}a^{9}+\frac{110}{7}a^{8}-56a^{7}+\frac{629}{7}a^{6}-\frac{594}{7}a^{5}+\frac{519}{7}a^{4}-\frac{184}{7}a^{3}+\frac{22}{7}a^{2}+\frac{6}{7}a-5$, $\frac{29}{7}a^{11}-7a^{10}-\frac{17}{7}a^{9}+\frac{176}{7}a^{8}-88a^{7}+\frac{1009}{7}a^{6}-137a^{5}+\frac{785}{7}a^{4}-\frac{269}{7}a^{3}-2a^{2}+\frac{57}{7}a-5$, $\frac{17}{7}a^{11}-\frac{24}{7}a^{10}-\frac{17}{7}a^{9}+\frac{97}{7}a^{8}-\frac{332}{7}a^{7}+\frac{500}{7}a^{6}-\frac{426}{7}a^{5}+52a^{4}-13a^{3}-\frac{16}{7}a^{2}+a-\frac{20}{7}$ | 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: | \( 303.966776362 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 303.966776362 \cdot 1}{2\cdot\sqrt{102241426653184}}\cr\approx \mathstrut & 0.588765026974 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
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
3.1.364.1, 6.2.529984.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.6889792.1, 6.0.3709888.1 |
Degree 8 siblings: | 8.4.358269184.1, 8.0.103876864.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.3709888.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.14.1 | $x^{12} + 2 x^{3} + 2$ | $12$ | $1$ | $14$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |