Normalized defining polynomial
\( x^{12} - 2x^{10} - 8x^{9} - 18x^{8} - 20x^{7} - 10x^{6} + 8x^{5} + 11x^{4} + 4x^{3} - 8x^{2} - 8x - 2 \)
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: | \(-199336411529216\) \(\medspace = -\,2^{29}\cdot 13^{5}\) | 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: | \(15.55\) | 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: | $2^{11/4}13^{1/2}\approx 24.255161140409015$ | ||
Ramified primes: | \(2\), \(13\) | 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: | \(\Q(\sqrt{-26}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6004}a^{11}+\frac{485}{6004}a^{10}+\frac{1067}{6004}a^{9}+\frac{1143}{6004}a^{8}-\frac{1033}{6004}a^{7}-\frac{2693}{6004}a^{6}+\frac{2757}{6004}a^{5}-\frac{1739}{6004}a^{4}+\frac{1}{38}a^{3}-\frac{709}{3002}a^{2}+\frac{1361}{3002}a-\frac{359}{3002}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{5607}{6004}a^{11}-\frac{3419}{6004}a^{10}-\frac{9323}{6004}a^{9}-\frac{39495}{6004}a^{8}-\frac{76223}{6004}a^{7}-\frac{62633}{6004}a^{6}-\frac{13809}{6004}a^{5}+\frac{59963}{6004}a^{4}+\frac{173}{38}a^{3}-\frac{715}{3002}a^{2}-\frac{23973}{3002}a-\frac{7577}{3002}$, $\frac{3392}{1501}a^{11}-\frac{1477}{1501}a^{10}-\frac{12803}{3002}a^{9}-\frac{24043}{1501}a^{8}-\frac{50135}{1501}a^{7}-\frac{44600}{1501}a^{6}-\frac{24489}{3002}a^{5}+\frac{33264}{1501}a^{4}+\frac{267}{19}a^{3}+\frac{2350}{1501}a^{2}-\frac{26645}{1501}a-\frac{14343}{1501}$, $\frac{1151}{6004}a^{11}-\frac{137}{6004}a^{10}-\frac{2703}{6004}a^{9}-\frac{8289}{6004}a^{8}-\frac{18203}{6004}a^{7}-\frac{19591}{6004}a^{6}-\frac{8813}{6004}a^{5}+\frac{6749}{6004}a^{4}+\frac{11}{38}a^{3}-\frac{2517}{3002}a^{2}-\frac{3535}{3002}a-\frac{1935}{3002}$, $\frac{16987}{6004}a^{11}-\frac{7799}{6004}a^{10}-\frac{30967}{6004}a^{9}-\frac{120875}{6004}a^{8}-\frac{250047}{6004}a^{7}-\frac{220661}{6004}a^{6}-\frac{64085}{6004}a^{5}+\frac{167399}{6004}a^{4}+\frac{685}{38}a^{3}+\frac{6245}{3002}a^{2}-\frac{71143}{3002}a-\frac{34293}{3002}$, $\frac{125}{1501}a^{11}-\frac{331}{3002}a^{10}-\frac{214}{1501}a^{9}-\frac{941}{3002}a^{8}-\frac{1540}{1501}a^{7}+\frac{699}{3002}a^{6}+\frac{896}{1501}a^{5}+\frac{2041}{3002}a^{4}+\frac{22}{19}a^{3}-\frac{1633}{1501}a^{2}-\frac{477}{1501}a+\frac{310}{1501}$, $\frac{1199}{6004}a^{11}-\frac{873}{6004}a^{10}-\frac{2521}{6004}a^{9}-\frac{7461}{6004}a^{8}-\frac{13751}{6004}a^{7}-\frac{10763}{6004}a^{6}+\frac{6445}{6004}a^{5}+\frac{13337}{6004}a^{4}+\frac{21}{38}a^{3}+\frac{2477}{3002}a^{2}-\frac{1249}{3002}a-\frac{1155}{3002}$ | 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: | \( 1091.06965821 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 1091.06965821 \cdot 1}{2\cdot\sqrt{199336411529216}}\cr\approx \mathstrut & 1.51352116370 \end{aligned}\]
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
3.1.104.1, 6.2.692224.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.1384448.1, 6.0.2249728.1 |
Degree 8 siblings: | 8.4.708837376.1, 8.0.7487094784.4 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.1384448.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.10.6 | $x^{4} + 4 x^{3} + 12 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ |
2.4.9.7 | $x^{4} + 2 x^{2} + 6$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
2.4.10.6 | $x^{4} + 4 x^{3} + 12 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |