Normalized defining polynomial
\( x^{12} - 4x^{11} + 5x^{10} + 3x^{9} - 11x^{8} - 3x^{7} + 35x^{6} - 47x^{5} + 27x^{4} - 4x^{3} - x^{2} - x + 1 \)
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: | \(126548911552\) \(\medspace = 2^{6}\cdot 7^{11}\) | 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: | \(8.42\) | 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: | $2\cdot 7^{11/12}\approx 11.90428180002048$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(\Q(\sqrt{7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{43}a^{11}+\frac{11}{43}a^{10}-\frac{2}{43}a^{9}+\frac{16}{43}a^{8}+\frac{14}{43}a^{7}-\frac{8}{43}a^{6}+\frac{1}{43}a^{5}+\frac{11}{43}a^{4}+\frac{20}{43}a^{3}-\frac{5}{43}a^{2}+\frac{10}{43}a+\frac{20}{43}$
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: | \( -\frac{72}{43} a^{11} + \frac{283}{43} a^{10} - \frac{286}{43} a^{9} - \frac{378}{43} a^{8} + \frac{798}{43} a^{7} + \frac{576}{43} a^{6} - \frac{2652}{43} a^{5} + \frac{2562}{43} a^{4} - \frac{666}{43} a^{3} - \frac{285}{43} a^{2} + \frac{54}{43} a + \frac{108}{43} \) (order $14$) | 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: | $\frac{4}{43}a^{11}-\frac{42}{43}a^{10}+\frac{78}{43}a^{9}+\frac{21}{43}a^{8}-\frac{202}{43}a^{7}+\frac{11}{43}a^{6}+\frac{477}{43}a^{5}-\frac{601}{43}a^{4}+\frac{166}{43}a^{3}+\frac{66}{43}a^{2}-\frac{46}{43}a-\frac{49}{43}$, $\frac{57}{43}a^{11}-\frac{233}{43}a^{10}+\frac{273}{43}a^{9}+\frac{224}{43}a^{8}-\frac{664}{43}a^{7}-\frac{284}{43}a^{6}+\frac{2121}{43}a^{5}-\frac{2512}{43}a^{4}+\frac{1097}{43}a^{3}-\frac{27}{43}a^{2}+\frac{11}{43}a-\frac{107}{43}$, $\frac{67}{43}a^{11}-\frac{209}{43}a^{10}+\frac{124}{43}a^{9}+\frac{384}{43}a^{8}-\frac{438}{43}a^{7}-\frac{708}{43}a^{6}+\frac{1830}{43}a^{5}-\frac{1284}{43}a^{4}+\frac{93}{43}a^{3}+\frac{267}{43}a^{2}-\frac{18}{43}a-\frac{36}{43}$, $\frac{8}{43}a^{11}-\frac{41}{43}a^{10}+\frac{70}{43}a^{9}-\frac{1}{43}a^{8}-\frac{146}{43}a^{7}+\frac{65}{43}a^{6}+\frac{395}{43}a^{5}-\frac{686}{43}a^{4}+\frac{375}{43}a^{3}+\frac{3}{43}a^{2}-\frac{6}{43}a-\frac{12}{43}$, $\frac{44}{43}a^{11}-\frac{118}{43}a^{10}+\frac{41}{43}a^{9}+\frac{231}{43}a^{8}-\frac{158}{43}a^{7}-\frac{481}{43}a^{6}+\frac{904}{43}a^{5}-\frac{548}{43}a^{4}+\frac{149}{43}a^{3}-\frac{48}{43}a^{2}+\frac{53}{43}a+\frac{20}{43}$ | 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: | \( 17.4132666738 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 17.4132666738 \cdot 1}{14\cdot\sqrt{126548911552}}\cr\approx \mathstrut & 0.215130614841 \end{aligned}\]
Galois group
$C_3\times D_4$ (as 12T14):
A solvable group of order 24 |
The 15 conjugacy class representatives for $D_4 \times C_3$ |
Character table for $D_4 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 4.0.1372.1, \(\Q(\zeta_{7})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.6.8099130339328.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(7\) | 7.12.11.2 | $x^{12} + 7$ | $12$ | $1$ | $11$ | $D_4 \times C_3$ | $[\ ]_{12}^{2}$ |