Normalized defining polynomial
\( x^{9} - 3x^{8} + 6x^{7} - 99x^{6} + 372x^{5} - 279x^{4} + 2703x^{3} - 2610x^{2} + 2700x + 27000 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2146055133681216576\) \(\medspace = 2^{6}\cdot 3^{10}\cdot 7^{6}\cdot 13^{6}\) | 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: | \(108.86\) | 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: | $2^{2/3}3^{7/6}7^{2/3}13^{2/3}\approx 115.70595800507586$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a$, $\frac{1}{930}a^{7}-\frac{1}{930}a^{6}-\frac{46}{155}a^{5}+\frac{83}{310}a^{4}+\frac{1}{31}a^{3}-\frac{63}{310}a^{2}+\frac{29}{62}a+\frac{1}{31}$, $\frac{1}{10883036700}a^{8}-\frac{570721}{3627678900}a^{7}+\frac{55408481}{1813839450}a^{6}-\frac{502610051}{1209226300}a^{5}+\frac{382554451}{906919725}a^{4}+\frac{304060789}{1209226300}a^{3}+\frac{669261781}{3627678900}a^{2}-\frac{35381901}{120922630}a+\frac{1920668}{12092263}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{18}$, which has order $162$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{214393}{10883036700}a^{8}-\frac{488713}{3627678900}a^{7}+\frac{942553}{1813839450}a^{6}-\frac{1911883}{1209226300}a^{5}+\frac{9488713}{906919725}a^{4}-\frac{73745083}{1209226300}a^{3}+\frac{229826773}{3627678900}a^{2}+\frac{5388883}{120922630}a-\frac{1617707}{12092263}$, $\frac{1096657}{2720759175}a^{8}-\frac{2849703}{604613150}a^{7}+\frac{2385463}{1813839450}a^{6}-\frac{13342972}{302306575}a^{5}+\frac{845034341}{1813839450}a^{4}-\frac{72874132}{302306575}a^{3}-\frac{1217325781}{1813839450}a^{2}-\frac{1665447211}{120922630}a-\frac{288006364}{12092263}$, $\frac{2533949}{2720759175}a^{8}-\frac{2538158}{302306575}a^{7}-\frac{2218937}{906919725}a^{6}-\frac{23746219}{302306575}a^{5}+\frac{788087636}{906919725}a^{4}-\frac{5613699}{302306575}a^{3}-\frac{1713115726}{906919725}a^{2}-\frac{1566325456}{60461315}a-\frac{501806729}{12092263}$, $\frac{103320263}{10883036700}a^{8}+\frac{10737817}{3627678900}a^{7}-\frac{11066107}{1813839450}a^{6}-\frac{1120823133}{1209226300}a^{5}+\frac{507855998}{906919725}a^{4}+\frac{6758496467}{1209226300}a^{3}+\frac{125780807543}{3627678900}a^{2}+\frac{5892939941}{120922630}a+\frac{17828711}{12092263}$ (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)]
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Regulator: | \( 12087.707677111599 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 12087.707677111599 \cdot 162}{2\cdot\sqrt{2146055133681216576}}\cr\approx \mathstrut & 2.08332967079355 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 18 |
The 6 conjugacy class representatives for $C_3^2:C_2$ |
Character table for $C_3^2:C_2$ |
Intermediate fields
3.1.894348.1, 3.1.24843.1, 3.1.108.1, 3.1.894348.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 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 | ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.1.0.1}{1} }^{9}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
\(13\) | 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |