Normalized defining polynomial
\( x^{12} + 4x^{10} - 12x^{9} - 6x^{8} - 4x^{7} - 8x^{6} - 4x^{5} + 4x^{4} + 4x^{3} + 2 \)
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: | \(-166168191041536\) \(\medspace = -\,2^{26}\cdot 19^{5}\) | 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: | \(15.31\) | 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: | $2^{13/6}19^{1/2}\approx 19.570794546160904$ | ||
Ramified primes: | \(2\), \(19\) | 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: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2132489}a^{11}+\frac{320877}{2132489}a^{10}-\frac{917254}{2132489}a^{9}+\frac{420010}{2132489}a^{8}+\frac{376453}{2132489}a^{7}+\frac{269872}{2132489}a^{6}-\frac{395576}{2132489}a^{5}+\frac{902591}{2132489}a^{4}+\frac{963754}{2132489}a^{3}-\frac{665051}{2132489}a^{2}+\frac{736992}{2132489}a-\frac{718160}{2132489}$
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: | $\frac{9550}{2132489}a^{11}-\frac{11343}{2132489}a^{10}+\frac{489112}{2132489}a^{9}-\frac{116309}{2132489}a^{8}+\frac{1882185}{2132489}a^{7}-\frac{5166579}{2132489}a^{6}-\frac{3245270}{2132489}a^{5}-\frac{1908977}{2132489}a^{4}-\frac{4236802}{2132489}a^{3}-\frac{2817297}{2132489}a^{2}+\frac{1059900}{2132489}a+\frac{1789113}{2132489}$, $\frac{33908}{2132489}a^{11}+\frac{338438}{2132489}a^{10}+\frac{103433}{2132489}a^{9}+\frac{937538}{2132489}a^{8}-\frac{4575808}{2132489}a^{7}-\frac{1823012}{2132489}a^{6}-\frac{1967687}{2132489}a^{5}-\frac{426500}{2132489}a^{4}+\frac{709196}{2132489}a^{3}+\frac{2654356}{2132489}a^{2}+\frac{1418634}{2132489}a-\frac{477389}{2132489}$, $\frac{11343}{2132489}a^{11}-\frac{450912}{2132489}a^{10}+\frac{1709}{2132489}a^{9}-\frac{1939485}{2132489}a^{8}+\frac{5128379}{2132489}a^{7}+\frac{3168870}{2132489}a^{6}+\frac{1870777}{2132489}a^{5}+\frac{4275002}{2132489}a^{4}+\frac{2855497}{2132489}a^{3}-\frac{1059900}{2132489}a^{2}-\frac{3921602}{2132489}a-\frac{2113389}{2132489}$, $\frac{720539}{2132489}a^{11}-\frac{64677}{2132489}a^{10}+\frac{2903375}{2132489}a^{9}-\frac{9253490}{2132489}a^{8}-\frac{3532633}{2132489}a^{7}-\frac{4105924}{2132489}a^{6}-\frac{1588213}{2132489}a^{5}+\frac{448752}{2132489}a^{4}+\frac{5022913}{2132489}a^{3}+\frac{2818168}{2132489}a^{2}+\frac{3332886}{2132489}a-\frac{3169945}{2132489}$, $\frac{76121}{2132489}a^{11}-\frac{50889}{2132489}a^{10}-\frac{336896}{2132489}a^{9}-\frac{826367}{2132489}a^{8}-\frac{2540858}{2132489}a^{7}+\frac{9189931}{2132489}a^{6}-\frac{896016}{2132489}a^{5}+\frac{3731398}{2132489}a^{4}+\frac{31656}{2132489}a^{3}+\frac{3074178}{2132489}a^{2}-\frac{952580}{2132489}a-\frac{701845}{2132489}$, $\frac{88709}{2132489}a^{11}+\frac{214621}{2132489}a^{10}+\frac{697687}{2132489}a^{9}-\frac{180718}{2132489}a^{8}-\frac{2141052}{2132489}a^{7}-\frac{5643733}{2132489}a^{6}-\frac{5309867}{2132489}a^{5}+\frac{1513025}{2132489}a^{4}+\frac{37087}{2132489}a^{3}-\frac{700974}{2132489}a^{2}+\frac{2108055}{2132489}a+\frac{853435}{2132489}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 313.364356743 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 313.364356743 \cdot 1}{2\cdot\sqrt{166168191041536}}\cr\approx \mathstrut & 0.476107461864 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
3.1.76.1, 4.2.4864.1 x2, 6.2.369664.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.2.4864.1 |
Degree 6 siblings: | 6.2.369664.1, 6.0.7023616.2 |
Degree 8 sibling: | 8.0.8540717056.5 |
Degree 12 sibling: | 12.0.49331181715456.7 |
Minimal sibling: | 4.2.4864.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.65 | $x^{12} + 4 x^{3} + 2$ | $12$ | $1$ | $26$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |