Normalized defining polynomial
\( x^{12} - 2x^{10} - 15x^{8} + 40x^{6} - 85x^{4} + 108x^{2} - 81 \)
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: | \(-64000000000000000000\) \(\medspace = -\,2^{24}\cdot 5^{18}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.72\) | 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: | $2^{247/96}5^{39/20}\approx 137.25134260974906$ | ||
Ramified primes: | \(2\), \(5\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{5}+\frac{2}{5}a^{3}+\frac{2}{5}a$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}+\frac{2}{5}$, $\frac{1}{15}a^{9}+\frac{1}{15}a^{7}+\frac{1}{15}a^{3}-\frac{1}{15}a$, $\frac{1}{225}a^{10}-\frac{4}{45}a^{8}-\frac{1}{15}a^{6}+\frac{8}{45}a^{4}+\frac{1}{45}a^{2}+\frac{7}{25}$, $\frac{1}{675}a^{11}-\frac{4}{135}a^{9}-\frac{4}{45}a^{7}+\frac{62}{135}a^{5}+\frac{28}{135}a^{3}-\frac{28}{75}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{2}{225}a^{10}+\frac{1}{45}a^{8}-\frac{2}{15}a^{6}-\frac{4}{9}a^{4}-\frac{7}{45}a^{2}-\frac{1}{25}$, $\frac{29}{675}a^{11}-\frac{8}{135}a^{9}-\frac{7}{9}a^{7}+\frac{178}{135}a^{5}-\frac{32}{27}a^{3}+\frac{103}{75}a$, $\frac{1}{27}a^{11}-\frac{23}{225}a^{10}+\frac{17}{135}a^{9}+\frac{2}{45}a^{8}-\frac{43}{45}a^{7}+\frac{32}{15}a^{6}-\frac{259}{135}a^{5}-\frac{76}{45}a^{4}+\frac{158}{27}a^{3}-\frac{149}{45}a^{2}-\frac{27}{5}a+\frac{144}{25}$, $\frac{1}{27}a^{11}+\frac{23}{225}a^{10}+\frac{17}{135}a^{9}-\frac{2}{45}a^{8}-\frac{43}{45}a^{7}-\frac{32}{15}a^{6}-\frac{259}{135}a^{5}+\frac{76}{45}a^{4}+\frac{158}{27}a^{3}+\frac{149}{45}a^{2}-\frac{27}{5}a-\frac{144}{25}$, $\frac{2}{225}a^{11}+\frac{4}{45}a^{10}-\frac{11}{45}a^{9}+\frac{2}{9}a^{8}-\frac{1}{5}a^{7}-\frac{23}{15}a^{6}+\frac{196}{45}a^{5}-\frac{146}{45}a^{4}-\frac{46}{45}a^{3}-\frac{43}{45}a^{2}+\frac{497}{75}a-\frac{24}{5}$, $\frac{80827}{675}a^{11}+\frac{55061}{225}a^{10}+\frac{35171}{135}a^{9}+\frac{23953}{45}a^{8}-\frac{31888}{45}a^{7}-\frac{21734}{15}a^{6}+\frac{246908}{135}a^{5}+\frac{168184}{45}a^{4}-\frac{343073}{135}a^{3}-\frac{233659}{45}a^{2}+\frac{174329}{75}a+\frac{118722}{25}$ (assuming GRH) | 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: | \( 294995.822327 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 294995.822327 \cdot 1}{2\cdot\sqrt{64000000000000000000}}\cr\approx \mathstrut & 0.722196228492 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr S_5$ (as 12T270):
A non-solvable group of order 7680 |
The 37 conjugacy class representatives for $C_2\wr S_5$ |
Character table for $C_2\wr S_5$ is not computed |
Intermediate fields
6.2.125000000.2 |
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 24 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 12.2.64000000000000000000.26 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.388 | $x^{12} + 4 x^{11} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6 x^{2} + 4 x + 14$ | $12$ | $1$ | $24$ | 12T149 | $[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$ |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.10.18.1 | $x^{10} + 200 x^{6} + 60 x^{5} - 1250 x^{4} + 10000 x^{2} + 6000 x + 900$ | $5$ | $2$ | $18$ | $F_{5}\times C_2$ | $[9/4]_{4}^{2}$ |