Normalized defining polynomial
\( x^{12} - 4x^{11} + 2x^{10} + 4x^{9} + 3x^{8} - 8x^{6} - 12x^{5} + 11x^{4} + 8x^{3} - 10x^{2} + 4x - 1 \)
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: | \(-10807949656064\) \(\medspace = -\,2^{26}\cdot 11^{5}\) | 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: | \(12.19\) | 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: | $2^{7/3}11^{1/2}\approx 16.714741551887755$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{22991}a^{11}+\frac{3617}{22991}a^{10}-\frac{7711}{22991}a^{9}-\frac{10453}{22991}a^{8}-\frac{7124}{22991}a^{7}-\frac{102}{22991}a^{6}-\frac{18}{277}a^{5}-\frac{6901}{22991}a^{4}+\frac{2707}{22991}a^{3}+\frac{7889}{22991}a^{2}+\frac{11237}{22991}a-\frac{4889}{22991}$
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: | $\frac{11627}{22991}a^{11}-\frac{41662}{22991}a^{10}+\frac{9103}{22991}a^{9}+\frac{39377}{22991}a^{8}+\frac{51807}{22991}a^{7}+\frac{32569}{22991}a^{6}-\frac{705}{277}a^{5}-\frac{160274}{22991}a^{4}+\frac{45592}{22991}a^{3}+\frac{60286}{22991}a^{2}-\frac{97218}{22991}a+\frac{35331}{22991}$, $\frac{32673}{22991}a^{11}-\frac{110454}{22991}a^{10}-\frac{6125}{22991}a^{9}+\frac{138382}{22991}a^{8}+\frac{182360}{22991}a^{7}+\frac{93013}{22991}a^{6}-\frac{2536}{277}a^{5}-\frac{532429}{22991}a^{4}+\frac{68407}{22991}a^{3}+\frac{350061}{22991}a^{2}-\frac{134724}{22991}a+\frac{3171}{22991}$, $\frac{21661}{22991}a^{11}-\frac{74464}{22991}a^{10}+\frac{1644}{22991}a^{9}+\frac{84899}{22991}a^{8}+\frac{117583}{22991}a^{7}+\frac{66687}{22991}a^{6}-\frac{1544}{277}a^{5}-\frac{339944}{22991}a^{4}+\frac{32268}{22991}a^{3}+\frac{175454}{22991}a^{2}-\frac{116015}{22991}a+\frac{18908}{22991}$, $\frac{23704}{22991}a^{11}-\frac{88035}{22991}a^{10}+\frac{19897}{22991}a^{9}+\frac{111050}{22991}a^{8}+\frac{93563}{22991}a^{7}+\frac{19238}{22991}a^{6}-\frac{2308}{277}a^{5}-\frac{322213}{22991}a^{4}+\frac{182775}{22991}a^{3}+\frac{267954}{22991}a^{2}-\frac{195806}{22991}a+\frac{31766}{22991}$, $\frac{2526}{22991}a^{11}-\frac{13876}{22991}a^{10}+\frac{18382}{22991}a^{9}+\frac{12381}{22991}a^{8}-\frac{16262}{22991}a^{7}-\frac{27742}{22991}a^{6}-\frac{317}{277}a^{5}+\frac{18243}{22991}a^{4}+\frac{124510}{22991}a^{3}+\frac{17408}{22991}a^{2}-\frac{78196}{22991}a+\frac{19544}{22991}$, $\frac{22077}{22991}a^{11}-\frac{87198}{22991}a^{10}+\frac{35599}{22991}a^{9}+\frac{104741}{22991}a^{8}+\frac{73856}{22991}a^{7}-\frac{21727}{22991}a^{6}-\frac{2384}{277}a^{5}-\frac{290903}{22991}a^{4}+\frac{261731}{22991}a^{3}+\frac{261529}{22991}a^{2}-\frac{223551}{22991}a+\frac{31283}{22991}$ | 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: | \( 70.8187286134 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 70.8187286134 \cdot 1}{2\cdot\sqrt{10807949656064}}\cr\approx \mathstrut & 0.421896387429 \end{aligned}\]
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
3.1.44.1, 6.2.123904.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.123904.1, 6.2.1362944.2 |
Degree 8 siblings: | 8.4.3838050304.1, 8.0.31719424.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.123904.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.15 | $x^{12} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $C_2 \times S_4$ | $[2, 8/3, 8/3]_{3}^{2}$ |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |