Normalized defining polynomial
\( x^{12} - 114x^{8} + 10108x^{4} - 521284 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-72943847830649307136\)
\(\medspace = -\,2^{32}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.21\) | 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))
| |
Galois root discriminant: | $2^{41/12}19^{3/4}\approx 97.18166881943587$ | ||
Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{19}a^{4}$, $\frac{1}{19}a^{5}$, $\frac{1}{722}a^{6}+\frac{8}{19}a^{2}$, $\frac{1}{722}a^{7}+\frac{8}{19}a^{3}$, $\frac{1}{9386}a^{8}-\frac{6}{247}a^{4}-\frac{2}{13}$, $\frac{1}{9386}a^{9}-\frac{6}{247}a^{5}-\frac{2}{13}a$, $\frac{1}{178334}a^{10}+\frac{1}{9386}a^{6}+\frac{89}{247}a^{2}$, $\frac{1}{356668}a^{11}-\frac{3}{4693}a^{7}-\frac{1}{38}a^{5}+\frac{116}{247}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: |
$\frac{3}{178334}a^{10}+\frac{3}{9386}a^{6}+\frac{20}{247}a^{2}$, $\frac{1}{178334}a^{10}+\frac{1}{9386}a^{8}-\frac{6}{4693}a^{6}-\frac{6}{247}a^{4}-\frac{15}{247}a^{2}+\frac{11}{13}$, $\frac{3}{178334}a^{10}+\frac{3}{9386}a^{8}+\frac{3}{9386}a^{6}-\frac{5}{247}a^{4}+\frac{20}{247}a^{2}+\frac{20}{13}$, $\frac{46}{89167}a^{11}-\frac{77}{89167}a^{10}+\frac{2}{4693}a^{9}+\frac{24}{4693}a^{8}-\frac{435}{4693}a^{7}+\frac{1184}{4693}a^{6}-\frac{154}{247}a^{5}+\frac{310}{247}a^{4}+\frac{986}{247}a^{3}-\frac{3410}{247}a^{2}+\frac{564}{13}a-\frac{1539}{13}$, $\frac{423}{356668}a^{11}+\frac{309}{89167}a^{10}+\frac{48}{4693}a^{9}+\frac{141}{4693}a^{8}-\frac{445}{9386}a^{7}-\frac{1293}{9386}a^{6}-\frac{203}{494}a^{5}-\frac{301}{247}a^{4}+\frac{2086}{247}a^{3}+\frac{6122}{247}a^{2}+\frac{952}{13}a+\frac{2777}{13}$, $\frac{245}{356668}a^{11}-\frac{14}{4693}a^{9}+\frac{77}{4693}a^{8}-\frac{495}{9386}a^{7}-\frac{85}{722}a^{6}+\frac{271}{494}a^{5}-\frac{183}{247}a^{4}+\frac{158}{247}a^{3}+\frac{194}{19}a^{2}-\frac{256}{13}a-\frac{295}{13}$
| 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: | \( 52316.7688349 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 52316.7688349 \cdot 4}{2\cdot\sqrt{72943847830649307136}}\cr\approx \mathstrut & 0.479883599554 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4.S_4$ (as 12T140):
A solvable group of order 384 |
The 28 conjugacy class representatives for $C_2^4.S_4$ |
Character table for $C_2^4.S_4$ |
Intermediate fields
3.1.76.1, 6.2.369664.1 |
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 32 siblings: | data not computed |
Minimal sibling: | 12.0.202060520306507776.55 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.12.32.381 | $x^{12} + 2 x^{10} + 4 x^{9} + 8 x^{8} + 12 x^{6} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 6$ | $12$ | $1$ | $32$ | 12T140 | $[2, 8/3, 8/3, 3, 11/3, 11/3]_{3}^{2}$ |
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.4.3.1 | $x^{4} + 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
19.4.3.1 | $x^{4} + 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |