Normalized defining polynomial
\( x^{12} + 9x^{8} + 28x^{6} + 52x^{4} + 24x^{2} + 4 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(47464824438784\)
\(\medspace = 2^{26}\cdot 29^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.79\) | 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: | $2^{9/4}29^{1/2}\approx 25.616305216426433$ | ||
Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{904}a^{10}-\frac{3}{113}a^{8}-\frac{1}{2}a^{7}-\frac{93}{904}a^{6}-\frac{1}{2}a^{3}-\frac{87}{452}a^{2}+\frac{33}{226}$, $\frac{1}{904}a^{11}-\frac{3}{113}a^{9}-\frac{93}{904}a^{7}-\frac{87}{452}a^{3}+\frac{33}{226}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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: |
$\frac{103}{452}a^{11}+\frac{71}{452}a^{10}-\frac{85}{904}a^{9}-\frac{9}{452}a^{8}+\frac{465}{226}a^{7}+\frac{629}{452}a^{6}+\frac{45}{8}a^{5}+\frac{17}{4}a^{4}+\frac{4113}{452}a^{3}+\frac{1733}{226}a^{2}+\frac{601}{452}a+\frac{505}{226}$, $\frac{21}{226}a^{10}+\frac{9}{452}a^{8}+\frac{97}{113}a^{6}+\frac{11}{4}a^{4}+\frac{1205}{226}a^{2}+\frac{625}{226}$, $\frac{299}{904}a^{11}-\frac{179}{904}a^{10}-\frac{57}{904}a^{9}+\frac{115}{904}a^{8}+\frac{2703}{904}a^{7}-\frac{1659}{904}a^{6}+\frac{69}{8}a^{5}-\frac{35}{8}a^{4}+\frac{1774}{113}a^{3}-\frac{1649}{226}a^{2}+\frac{1993}{452}a+\frac{51}{452}$, $\frac{25}{226}a^{11}+\frac{33}{904}a^{10}-\frac{27}{904}a^{9}-\frac{1}{904}a^{8}+\frac{435}{452}a^{7}+\frac{321}{904}a^{6}+\frac{23}{8}a^{5}+\frac{9}{8}a^{4}+\frac{2035}{452}a^{3}+\frac{429}{226}a^{2}+\frac{385}{452}a+\frac{483}{452}$, $\frac{163}{452}a^{11}-\frac{87}{226}a^{10}-\frac{27}{904}a^{9}+\frac{103}{904}a^{8}+\frac{363}{113}a^{7}-\frac{1559}{452}a^{6}+\frac{79}{8}a^{5}-\frac{79}{8}a^{4}+\frac{7911}{452}a^{3}-\frac{7579}{452}a^{2}+\frac{3097}{452}a-\frac{1611}{452}$
| 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: | \( 98.5165623303 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 98.5165623303 \cdot 2}{2\cdot\sqrt{47464824438784}}\cr\approx \mathstrut & 0.879837604159 \end{aligned}\]
Galois group
$C_2^2\times S_4$ (as 12T48):
A solvable group of order 96 |
The 20 conjugacy class representatives for $C_2^2\times S_4$ |
Character table for $C_2^2\times S_4$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), 3.1.116.1, 6.2.430592.1, 6.0.861184.2, 6.0.1722368.1 |
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 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Minimal sibling: | 12.0.741637881856.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.4.8.4 | $x^{4} + 6 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.8.18.54 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |