Normalized defining polynomial
\( x^{12} - 12 x^{10} - 12 x^{9} + 54 x^{8} + 108 x^{7} - 40 x^{6} - 324 x^{5} - 327 x^{4} + 180 x^{3} + 612 x^{2} + 432 x + 112 \)
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: | \(1624959306694656\) \(\medspace = 2^{22}\cdot 3^{18}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.52\) | 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: | $2^{2}3^{11/6}\approx 29.976594395601754$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{40}a^{9}+\frac{1}{40}a^{7}+\frac{1}{10}a^{6}+\frac{7}{40}a^{5}-\frac{1}{10}a^{4}-\frac{1}{8}a^{3}+\frac{2}{5}a^{2}+\frac{1}{10}a+\frac{1}{5}$, $\frac{1}{40}a^{10}+\frac{1}{40}a^{8}+\frac{1}{10}a^{7}-\frac{3}{40}a^{6}-\frac{1}{10}a^{5}-\frac{1}{8}a^{4}-\frac{1}{10}a^{3}+\frac{7}{20}a^{2}-\frac{3}{10}a$, $\frac{1}{560}a^{11}+\frac{1}{560}a^{10}+\frac{3}{560}a^{9}+\frac{1}{112}a^{8}-\frac{67}{560}a^{7}-\frac{29}{560}a^{6}-\frac{11}{112}a^{5}-\frac{57}{560}a^{4}-\frac{9}{56}a^{3}+\frac{13}{70}a^{2}-\frac{33}{70}a-\frac{2}{5}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1}{20} a^{9} - \frac{9}{20} a^{7} - \frac{11}{20} a^{6} + \frac{27}{20} a^{5} + \frac{33}{10} a^{4} + \frac{5}{4} a^{3} - \frac{99}{20} a^{2} - \frac{39}{5} a - \frac{13}{5} \) (order $6$) | 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: | $\frac{1}{40}a^{9}-\frac{9}{40}a^{7}-\frac{3}{20}a^{6}+\frac{27}{40}a^{5}+\frac{9}{10}a^{4}+\frac{1}{8}a^{3}-\frac{27}{20}a^{2}-\frac{12}{5}a-\frac{4}{5}$, $\frac{13}{280}a^{11}+\frac{13}{280}a^{10}-\frac{27}{70}a^{9}-\frac{25}{28}a^{8}+\frac{137}{280}a^{7}+\frac{213}{56}a^{6}+\frac{167}{35}a^{5}-\frac{199}{140}a^{4}-\frac{299}{28}a^{3}-\frac{345}{28}a^{2}-\frac{48}{7}a-\frac{8}{5}$, $\frac{1}{80}a^{11}-\frac{1}{80}a^{10}-\frac{3}{16}a^{9}+\frac{3}{80}a^{8}+\frac{87}{80}a^{7}+\frac{13}{16}a^{6}-\frac{213}{80}a^{5}-\frac{75}{16}a^{4}-\frac{13}{20}a^{3}+\frac{147}{20}a^{2}+\frac{81}{10}a+\frac{12}{5}$, $\frac{1}{112}a^{11}-\frac{23}{560}a^{10}-\frac{27}{560}a^{9}+\frac{137}{560}a^{8}+\frac{211}{560}a^{7}-\frac{369}{560}a^{6}-\frac{737}{560}a^{5}-\frac{117}{560}a^{4}+\frac{353}{140}a^{3}+\frac{319}{140}a^{2}-\frac{109}{70}a-\frac{8}{5}$, $\frac{9}{560}a^{11}-\frac{1}{112}a^{10}-\frac{127}{560}a^{9}+\frac{31}{560}a^{8}+\frac{587}{560}a^{7}+\frac{57}{112}a^{6}-\frac{1237}{560}a^{5}-\frac{1227}{560}a^{4}-\frac{34}{35}a^{3}+\frac{619}{140}a^{2}+\frac{207}{70}a+\frac{6}{5}$ | 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: | \( 2457.79965637 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2457.79965637 \cdot 1}{6\cdot\sqrt{1624959306694656}}\cr\approx \mathstrut & 0.625249271158 \end{aligned}\]
Galois group
$S_3\times D_6$ (as 12T37):
A solvable group of order 72 |
The 18 conjugacy class representatives for $S_3\times D_6$ |
Character table for $S_3\times D_6$ |
Intermediate fields
\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.2.13436928.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 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 | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 3.12.18.87 | $x^{12} - 6 x^{11} + 45 x^{10} + 30 x^{9} + 99 x^{8} + 36 x^{7} + 51 x^{6} + 198 x^{5} + 108 x^{4} + 198 x^{3} + 333$ | $6$ | $2$ | $18$ | $C_6\times S_3$ | $[2, 2]_{2}^{2}$ |