Normalized defining polynomial
\( x^{12} - x^{11} + x^{9} - 2x^{8} + x^{7} + 3x^{4} - x^{3} - 2x^{2} + 1 \)
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: | \(44223092609\) \(\medspace = 89\cdot 22291^{2}\) | 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: | \(7.71\) | 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: | $89^{1/2}22291^{1/2}\approx 1408.509495885633$ | ||
Ramified primes: | \(89\), \(22291\) | 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: | \(\Q(\sqrt{89}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{22}a^{11}-\frac{2}{11}a^{10}+\frac{1}{22}a^{9}+\frac{9}{22}a^{8}+\frac{2}{11}a^{7}-\frac{1}{2}a^{4}+\frac{3}{22}a^{3}+\frac{1}{22}a^{2}+\frac{3}{11}a-\frac{7}{22}$
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: | $\frac{7}{22}a^{11}+\frac{5}{22}a^{10}+\frac{7}{22}a^{9}-\frac{3}{22}a^{8}+\frac{3}{11}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{22}a^{3}+\frac{29}{22}a^{2}+\frac{10}{11}a-\frac{8}{11}$, $\frac{6}{11}a^{11}+\frac{7}{22}a^{10}-\frac{5}{11}a^{9}+\frac{10}{11}a^{8}-\frac{9}{11}a^{7}-\frac{1}{2}a^{6}+\frac{1}{2}a^{5}-a^{4}+\frac{18}{11}a^{3}+\frac{17}{11}a^{2}-\frac{8}{11}a-\frac{7}{22}$, $\frac{21}{22}a^{11}-\frac{7}{22}a^{10}-\frac{6}{11}a^{9}+\frac{12}{11}a^{8}-\frac{37}{22}a^{7}+\frac{1}{2}a^{5}+\frac{37}{11}a^{3}+\frac{5}{11}a^{2}-\frac{39}{22}a-\frac{15}{22}$, $\frac{17}{22}a^{11}-\frac{13}{22}a^{10}+\frac{3}{11}a^{9}+\frac{5}{11}a^{8}-\frac{31}{22}a^{7}+a^{6}-\frac{1}{2}a^{5}+\frac{31}{11}a^{3}-\frac{8}{11}a^{2}-\frac{19}{22}a-\frac{9}{22}$, $\frac{6}{11}a^{11}-\frac{15}{22}a^{10}+\frac{1}{22}a^{9}+\frac{9}{22}a^{8}-\frac{29}{22}a^{7}+a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}+\frac{25}{22}a^{3}-\frac{21}{22}a^{2}-\frac{27}{22}a-\frac{9}{11}$ | 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: | \( 1.21605234252 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1.21605234252 \cdot 1}{2\cdot\sqrt{44223092609}}\cr\approx \mathstrut & 0.177900401388 \end{aligned}\]
Galois group
$C_2^6.S_6$ (as 12T293):
A non-solvable group of order 46080 |
The 65 conjugacy class representatives for $C_2^6.S_6$ |
Character table for $C_2^6.S_6$ |
Intermediate fields
6.0.22291.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 40 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(89\) | $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(22291\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |