Normalized defining polynomial
\( x^{12} - 6x^{11} + 9x^{10} + 4x^{9} - 9x^{8} - 12x^{7} + 10x^{6} + 6x^{4} - 6x^{2} + 2 \)
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: | \(-180551034077184\) \(\medspace = -\,2^{22}\cdot 3^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.42\) | 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: | $2^{9/4}3^{4/3}\approx 20.58160140743025$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{873}a^{11}-\frac{136}{873}a^{10}-\frac{62}{873}a^{9}-\frac{28}{291}a^{8}+\frac{145}{291}a^{7}+\frac{61}{291}a^{6}-\frac{209}{873}a^{5}+\frac{107}{873}a^{4}-\frac{227}{873}a^{3}+\frac{410}{873}a^{2}-\frac{53}{873}a+\frac{197}{873}$
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)
| |
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: | $\frac{10}{97}a^{11}-\frac{103}{291}a^{10}-\frac{38}{97}a^{9}+\frac{130}{97}a^{8}+\frac{179}{97}a^{7}-\frac{207}{97}a^{6}-\frac{344}{97}a^{5}-\frac{88}{291}a^{4}+\frac{155}{97}a^{3}+\frac{123}{97}a^{2}-\frac{38}{291}a-\frac{67}{97}$, $\frac{454}{873}a^{11}-\frac{2380}{873}a^{10}+\frac{2407}{873}a^{9}+\frac{965}{291}a^{8}-\frac{227}{291}a^{7}-\frac{1988}{291}a^{6}-\frac{602}{873}a^{5}-\frac{1183}{873}a^{4}+\frac{2575}{873}a^{3}+\frac{191}{873}a^{2}-\frac{1364}{873}a-\frac{481}{873}$, $\frac{1022}{873}a^{11}-\frac{5423}{873}a^{10}+\frac{5312}{873}a^{9}+\frac{2812}{291}a^{8}-\frac{1384}{291}a^{7}-\frac{5170}{291}a^{6}-\frac{586}{873}a^{5}+\frac{1102}{873}a^{4}+\frac{6917}{873}a^{3}+\frac{5218}{873}a^{2}-\frac{3532}{873}a-\frac{2657}{873}$, $\frac{818}{873}a^{11}-\frac{4451}{873}a^{10}+\frac{5156}{873}a^{9}+\frac{1540}{291}a^{8}-\frac{991}{291}a^{7}-\frac{3355}{291}a^{6}+\frac{1892}{873}a^{5}-\frac{2102}{873}a^{4}+\frac{2882}{873}a^{3}+\frac{1021}{873}a^{2}-\frac{1741}{873}a-\frac{359}{873}$, $\frac{715}{873}a^{11}-\frac{3829}{873}a^{10}+\frac{3976}{873}a^{9}+\frac{1805}{291}a^{8}-\frac{1085}{291}a^{7}-\frac{3236}{291}a^{6}+\frac{721}{873}a^{5}-\frac{1192}{873}a^{4}+\frac{2983}{873}a^{3}+\frac{5060}{873}a^{2}-\frac{1229}{873}a-\frac{1735}{873}$, $\frac{28}{873}a^{11}-\frac{25}{873}a^{10}-\frac{572}{873}a^{9}+\frac{380}{291}a^{8}+\frac{277}{291}a^{7}-\frac{620}{291}a^{6}-\frac{614}{873}a^{5}+\frac{668}{873}a^{4}-\frac{1700}{873}a^{3}-\frac{742}{873}a^{2}+\frac{2590}{873}a-\frac{13}{873}$ | 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: | \( 699.749160748 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 699.749160748 \cdot 1}{2\cdot\sqrt{180551034077184}}\cr\approx \mathstrut & 1.01993321140 \end{aligned}\]
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
3.1.324.1, 6.2.1679616.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.3359232.1, 6.0.3359232.2 |
Degree 8 siblings: | 8.4.429981696.1, 8.0.429981696.2 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.3359232.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.8.18.55 | $x^{8} + 2 x^{6} + 4 x^{3} + 10$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
\(3\) | 3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ |
3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ | |
3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ |