Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - 6x^{9} + 12x^{8} - 18x^{7} + 20x^{6} - 15x^{5} + 6x^{4} + x^{3} - x^{2} - x + 1 \)
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: | \(164648481361\) \(\medspace = 7^{8}\cdot 13^{4}\) | 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: | \(8.60\) | 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: | $7^{2/3}13^{1/2}\approx 13.193814370085985$ | ||
Ramified primes: | \(7\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{10}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{3}{7}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $a^{11}-\frac{11}{7}a^{10}+\frac{12}{7}a^{9}-\frac{32}{7}a^{8}+9a^{7}-\frac{79}{7}a^{6}+\frac{73}{7}a^{5}-\frac{30}{7}a^{4}-\frac{15}{7}a^{3}+\frac{25}{7}a^{2}+\frac{3}{7}a-\frac{10}{7}$, $\frac{1}{7}a^{11}-\frac{8}{7}a^{10}+\frac{6}{7}a^{9}-2a^{8}+\frac{31}{7}a^{7}-\frac{53}{7}a^{6}+8a^{5}-\frac{43}{7}a^{4}+\frac{9}{7}a^{3}+\frac{8}{7}a^{2}-\frac{5}{7}a-\frac{13}{7}$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{10}+\frac{1}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{4}{7}a^{3}-\frac{4}{7}a^{2}+\frac{6}{7}a-\frac{2}{7}$, $\frac{6}{7}a^{11}-\frac{11}{7}a^{10}+\frac{10}{7}a^{9}-\frac{30}{7}a^{8}+\frac{58}{7}a^{7}-11a^{6}+\frac{67}{7}a^{5}-\frac{34}{7}a^{4}-a^{3}+\frac{15}{7}a^{2}+\frac{2}{7}a-\frac{11}{7}$, $2a^{11}-\frac{20}{7}a^{10}+4a^{9}-\frac{67}{7}a^{8}+\frac{127}{7}a^{7}-\frac{172}{7}a^{6}+24a^{5}-\frac{104}{7}a^{4}+\frac{17}{7}a^{3}+\frac{18}{7}a^{2}+a-\frac{18}{7}$ | 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: | \( 2.70857178473 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2.70857178473 \cdot 1}{2\cdot\sqrt{164648481361}}\cr\approx \mathstrut & 0.205357562070 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
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
\(\Q(\zeta_{7})^+\), 6.2.31213.1 x2, 6.2.405769.1 |
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: | 6.2.31213.1 |
Degree 8 sibling: | 8.0.68574961.1 |
Degree 12 sibling: | 12.4.27825593350009.1 |
Minimal sibling: | 6.2.31213.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |