Normalized defining polynomial
\( x^{12} - 6 x^{11} + 15 x^{10} - 18 x^{9} - 3 x^{8} + 24 x^{7} + 3 x^{6} + 6 x^{5} + 33 x^{4} + 152 x^{3} + 273 x^{2} + 198 x + 51 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1093889542148001\) \(\medspace = 3^{16}\cdot 71^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(17.92\) | 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))
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Galois root discriminant: | $3^{4/3}71^{1/2}\approx 36.45783266912841$ | ||
Ramified primes: | \(3\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.166726039041.1$^{8}$, 12.0.1093889542148001.2$^{24}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{2}$, $\frac{1}{18}a^{8}+\frac{1}{18}a^{7}+\frac{1}{18}a^{6}-\frac{7}{18}a^{5}+\frac{5}{18}a^{4}+\frac{5}{18}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{54}a^{9}-\frac{1}{18}a^{7}+\frac{2}{27}a^{6}-\frac{1}{9}a^{5}+\frac{2}{9}a^{4}-\frac{4}{27}a^{3}+\frac{5}{18}a^{2}+\frac{1}{3}a+\frac{7}{18}$, $\frac{1}{162}a^{10}-\frac{1}{162}a^{9}-\frac{1}{54}a^{8}+\frac{7}{162}a^{7}-\frac{5}{81}a^{6}-\frac{2}{9}a^{5}-\frac{10}{81}a^{4}-\frac{31}{162}a^{3}+\frac{19}{54}a^{2}+\frac{19}{54}a-\frac{7}{54}$, $\frac{1}{486}a^{11}+\frac{1}{486}a^{10}+\frac{2}{243}a^{9}+\frac{1}{486}a^{8}-\frac{23}{486}a^{7}-\frac{10}{243}a^{6}-\frac{73}{243}a^{5}-\frac{125}{486}a^{4}-\frac{239}{486}a^{3}-\frac{10}{27}a^{2}-\frac{77}{162}a-\frac{5}{162}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
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)
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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)
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Fundamental units: | $\frac{1}{9}a^{11}-\frac{115}{162}a^{10}+\frac{313}{162}a^{9}-\frac{143}{54}a^{8}+\frac{25}{81}a^{7}+\frac{529}{162}a^{6}-\frac{29}{18}a^{5}+\frac{257}{162}a^{4}+\frac{212}{81}a^{3}+\frac{451}{27}a^{2}+\frac{667}{27}a+\frac{607}{54}$, $\frac{29}{486}a^{11}-\frac{95}{243}a^{10}+\frac{533}{486}a^{9}-\frac{386}{243}a^{8}+\frac{203}{486}a^{7}+\frac{863}{486}a^{6}-\frac{697}{486}a^{5}+\frac{310}{243}a^{4}+\frac{785}{486}a^{3}+\frac{194}{27}a^{2}+\frac{1841}{162}a+\frac{415}{81}$, $\frac{113}{486}a^{11}-\frac{392}{243}a^{10}+\frac{2411}{486}a^{9}-\frac{2081}{243}a^{8}+\frac{1582}{243}a^{7}+\frac{214}{243}a^{6}-\frac{311}{243}a^{5}+\frac{1331}{486}a^{4}+\frac{1174}{243}a^{3}+\frac{553}{18}a^{2}+\frac{2707}{81}a+\frac{823}{81}$, $\frac{23}{486}a^{11}-\frac{80}{243}a^{10}+\frac{473}{486}a^{9}-\frac{362}{243}a^{8}+\frac{175}{243}a^{7}+\frac{190}{243}a^{6}-\frac{32}{243}a^{5}-\frac{79}{486}a^{4}+\frac{187}{243}a^{3}+\frac{277}{54}a^{2}+\frac{562}{81}a+\frac{223}{81}$, $\frac{205}{486}a^{11}-\frac{658}{243}a^{10}+\frac{3601}{486}a^{9}-\frac{2503}{243}a^{8}+\frac{811}{486}a^{7}+\frac{5971}{486}a^{6}-\frac{3497}{486}a^{5}+\frac{1763}{243}a^{4}+\frac{5221}{486}a^{3}+\frac{1595}{27}a^{2}+\frac{14365}{162}a+\frac{3416}{81}$ | 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: | \( 1519.80421832 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1519.80421832 \cdot 1}{2\cdot\sqrt{1093889542148001}}\cr\approx \mathstrut & 1.41367678071 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4\times C_2$ |
Character table for $A_4\times C_2$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 6.0.465831.1 x2, 6.6.33074001.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.0.465831.1 |
Degree 8 sibling: | 8.0.166726039041.1 |
Degree 12 sibling: | 12.0.5514297181968073041.1 |
Minimal sibling: | 6.0.465831.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(71\) | 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |