Normalized defining polynomial
\( x^{12} - 14x^{8} + 21x^{4} - 7 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-33174037869887488\) \(\medspace = -\,2^{24}\cdot 7^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.81\) | 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: | $2^{2}7^{11/12}\approx 23.80856360004096$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}+\frac{5}{13}a^{4}-\frac{1}{13}$, $\frac{1}{13}a^{9}+\frac{5}{13}a^{5}-\frac{1}{13}a$, $\frac{1}{13}a^{10}+\frac{5}{13}a^{6}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{11}+\frac{5}{13}a^{7}-\frac{1}{13}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $8$ | 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: | $\frac{2}{13}a^{8}-\frac{29}{13}a^{4}+\frac{37}{13}$, $\frac{3}{13}a^{8}-\frac{37}{13}a^{4}+\frac{23}{13}$, $\frac{5}{13}a^{10}-\frac{66}{13}a^{6}+\frac{60}{13}a^{2}-1$, $\frac{2}{13}a^{10}-\frac{29}{13}a^{6}+\frac{50}{13}a^{2}-1$, $\frac{5}{13}a^{10}+\frac{5}{13}a^{8}-\frac{66}{13}a^{6}-\frac{66}{13}a^{4}+\frac{47}{13}a^{2}+\frac{47}{13}$, $\frac{5}{13}a^{9}+\frac{5}{13}a^{8}-\frac{66}{13}a^{5}-\frac{66}{13}a^{4}+\frac{47}{13}a+\frac{47}{13}$, $\frac{3}{13}a^{8}-\frac{37}{13}a^{4}-a+\frac{23}{13}$, $\frac{5}{13}a^{9}+\frac{2}{13}a^{8}-\frac{66}{13}a^{5}-\frac{29}{13}a^{4}+\frac{60}{13}a+\frac{37}{13}$ | 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: | \( 3278.78874467 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 3278.78874467 \cdot 2}{2\cdot\sqrt{33174037869887488}}\cr\approx \mathstrut & 0.285781325742 \end{aligned}\]
Galois group
$C_3\times D_4$ (as 12T14):
A solvable group of order 24 |
The 15 conjugacy class representatives for $D_4 \times C_3$ |
Character table for $D_4 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), 4.2.87808.1, \(\Q(\zeta_{28})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.0.518344341716992.1 |
Minimal sibling: | 12.0.518344341716992.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.315 | $x^{12} - 8 x^{11} + 78 x^{10} + 76 x^{9} - 30 x^{8} + 720 x^{7} + 1200 x^{6} + 1248 x^{5} + 2676 x^{4} + 3712 x^{3} + 3448 x^{2} + 2160 x + 4072$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
\(7\) | 7.12.11.4 | $x^{12} + 42$ | $12$ | $1$ | $11$ | $D_4 \times C_3$ | $[\ ]_{12}^{2}$ |