Normalized defining polynomial
\( x^{12} - 6x^{10} + 10x^{8} + 6x^{6} - 39x^{4} + 50x^{2} - 23 \)
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: | \(-107983916761088\) \(\medspace = -\,2^{24}\cdot 23^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.77\) | 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: | $2^{9/4}23^{1/2}\approx 22.812947879511295$ | ||
Ramified primes: | \(2\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}+\frac{8}{19}a^{8}+\frac{8}{19}a^{6}+\frac{4}{19}a^{4}-\frac{2}{19}a^{2}+\frac{3}{19}$, $\frac{1}{19}a^{11}+\frac{8}{19}a^{9}+\frac{8}{19}a^{7}+\frac{4}{19}a^{5}-\frac{2}{19}a^{3}+\frac{3}{19}a$
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: | $\frac{2}{19}a^{10}-\frac{3}{19}a^{8}-\frac{3}{19}a^{6}+\frac{8}{19}a^{4}-\frac{23}{19}a^{2}+\frac{6}{19}$, $\frac{25}{19}a^{10}-\frac{104}{19}a^{8}+\frac{67}{19}a^{6}+\frac{252}{19}a^{4}-\frac{525}{19}a^{2}+\frac{341}{19}$, $\frac{2}{19}a^{10}-\frac{3}{19}a^{8}-\frac{3}{19}a^{6}+\frac{8}{19}a^{4}-\frac{4}{19}a^{2}-\frac{13}{19}$, $\frac{2}{19}a^{11}+\frac{10}{19}a^{10}-\frac{3}{19}a^{9}-\frac{34}{19}a^{8}-\frac{3}{19}a^{7}+\frac{4}{19}a^{6}+\frac{8}{19}a^{5}+\frac{97}{19}a^{4}-\frac{23}{19}a^{3}-\frac{134}{19}a^{2}+\frac{6}{19}a+\frac{49}{19}$, $\frac{4}{19}a^{11}-\frac{4}{19}a^{10}-\frac{25}{19}a^{9}+\frac{25}{19}a^{8}+\frac{32}{19}a^{7}-\frac{32}{19}a^{6}+\frac{54}{19}a^{5}-\frac{54}{19}a^{4}-\frac{160}{19}a^{3}+\frac{160}{19}a^{2}+\frac{126}{19}a-\frac{126}{19}$, $\frac{18}{19}a^{11}-\frac{13}{19}a^{10}-\frac{84}{19}a^{9}+\frac{67}{19}a^{8}+\frac{68}{19}a^{7}-\frac{66}{19}a^{6}+\frac{205}{19}a^{5}-\frac{147}{19}a^{4}-\frac{435}{19}a^{3}+\frac{368}{19}a^{2}+\frac{301}{19}a-\frac{286}{19}$ | 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: | \( 114.796116528 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 114.796116528 \cdot 1}{2\cdot\sqrt{107983916761088}}\cr\approx \mathstrut & 0.216360011511 \end{aligned}\]
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
3.1.23.1, 6.2.33856.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.270848.2, 6.0.6229504.2 |
Degree 8 siblings: | 8.4.138674176.1, 8.0.73358639104.2 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.270848.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.280 | $x^{12} + 12 x^{11} + 82 x^{10} + 260 x^{9} + 534 x^{8} + 968 x^{7} + 1280 x^{6} + 1376 x^{5} + 1740 x^{4} + 1376 x^{3} + 1096 x^{2} + 560 x - 1048$ | $4$ | $3$ | $24$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ |
\(23\) | 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |