Normalized defining polynomial
\( x^{12} + 4x^{10} + 16x^{8} + 52x^{6} + 56x^{4} + 16x^{2} - 2 \)
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: | \(-60870372462952448\) \(\medspace = -\,2^{35}\cdot 11^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.04\) | 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: | $2^{331/96}11^{3/4}\approx 65.91296574252337$ | ||
Ramified primes: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{169}a^{10}-\frac{79}{169}a^{8}-\frac{18}{169}a^{6}+\frac{25}{169}a^{4}+\frac{9}{169}a^{2}-\frac{55}{169}$, $\frac{1}{169}a^{11}-\frac{79}{169}a^{9}-\frac{18}{169}a^{7}+\frac{25}{169}a^{5}+\frac{9}{169}a^{3}-\frac{55}{169}a$
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: | $\frac{66}{169}a^{10}+\frac{194}{169}a^{8}+\frac{840}{169}a^{6}+\frac{2495}{169}a^{4}+\frac{932}{169}a^{2}-\frac{419}{169}$, $a^{2}+1$, $\frac{147}{169}a^{11}+\frac{62}{169}a^{10}+\frac{555}{169}a^{9}+\frac{172}{169}a^{8}+\frac{2255}{169}a^{7}+\frac{743}{169}a^{6}+\frac{7224}{169}a^{5}+\frac{2226}{169}a^{4}+\frac{6900}{169}a^{3}+\frac{389}{169}a^{2}+\frac{1886}{169}a-\frac{875}{169}$, $\frac{84}{169}a^{11}+\frac{293}{169}a^{9}+\frac{1192}{169}a^{7}+\frac{3790}{169}a^{5}+\frac{2784}{169}a^{3}+\frac{281}{169}a-1$, $\frac{84}{169}a^{11}+\frac{293}{169}a^{9}+\frac{1192}{169}a^{7}+\frac{3790}{169}a^{5}+\frac{2784}{169}a^{3}+\frac{281}{169}a+1$, $\frac{250}{169}a^{11}+\frac{1252}{169}a^{10}+\frac{699}{169}a^{9}+\frac{4013}{169}a^{8}+\frac{3105}{169}a^{7}+\frac{16841}{169}a^{6}+\frac{9123}{169}a^{5}+\frac{51749}{169}a^{4}+\frac{2419}{169}a^{3}+\frac{29013}{169}a^{2}-\frac{737}{169}a-\frac{3119}{169}$ | 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: | \( 7570.43454126 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 7570.43454126 \cdot 1}{2\cdot\sqrt{60870372462952448}}\cr\approx \mathstrut & 0.600962214344 \end{aligned}\]
Galois group
$C_2^4:D_{12}$ (as 12T152):
A solvable group of order 384 |
The 22 conjugacy class representatives for $C_2^4:D_{12}$ |
Character table for $C_2^4:D_{12}$ |
Intermediate fields
3.1.44.1, 6.2.2725888.2 |
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 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.35.716 | $x^{12} + 8 x^{11} + 12 x^{10} + 12 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{3} + 10$ | $12$ | $1$ | $35$ | 12T152 | $[4/3, 4/3, 3, 19/6, 19/6, 4]_{3}^{2}$ |
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |