Normalized defining polynomial
\( x^{9} - 32x^{7} - 122x^{6} + 124x^{5} + 1516x^{4} - 7288x^{3} + 21472x^{2} - 21480x - 88504 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1597657402133118656\) \(\medspace = -\,2^{6}\cdot 11^{3}\cdot 163^{6}\) | 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: | \(105.34\) | 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: | $2^{2/3}11^{1/2}163^{2/3}\approx 157.09947822705817$ | ||
Ramified primes: | \(2\), \(11\), \(163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{92}a^{6}-\frac{5}{46}a^{5}+\frac{3}{23}a^{4}-\frac{3}{23}a^{3}+\frac{5}{23}a^{2}-\frac{10}{23}a$, $\frac{1}{276}a^{7}+\frac{1}{276}a^{6}+\frac{10}{69}a^{5}+\frac{7}{69}a^{4}-\frac{5}{69}a^{3}+\frac{22}{69}a^{2}+\frac{5}{69}a+\frac{1}{3}$, $\frac{1}{1166143468068}a^{8}-\frac{788657785}{1166143468068}a^{7}+\frac{100354957}{50701889916}a^{6}+\frac{49803211706}{291535867017}a^{5}+\frac{28519653200}{291535867017}a^{4}+\frac{2738569430}{291535867017}a^{3}-\frac{100263953}{32392874113}a^{2}+\frac{107423668993}{291535867017}a+\frac{4487157472}{12675472479}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{162}$, which has order $324$ (assuming GRH)
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: | $\frac{47855}{291535867017}a^{8}+\frac{2010095}{583071734034}a^{7}-\frac{189214883}{1166143468068}a^{6}+\frac{16677439}{25350944958}a^{5}+\frac{2169764585}{583071734034}a^{4}+\frac{10120449076}{291535867017}a^{3}-\frac{1860550054}{32392874113}a^{2}+\frac{25331077688}{291535867017}a-\frac{9033776716}{12675472479}$, $\frac{29040761}{1166143468068}a^{8}+\frac{124511681}{583071734034}a^{7}+\frac{213685243}{1166143468068}a^{6}-\frac{3918839461}{583071734034}a^{5}-\frac{13255204013}{291535867017}a^{4}-\frac{78662397907}{583071734034}a^{3}-\frac{12236104720}{32392874113}a^{2}-\frac{222968850220}{291535867017}a-\frac{5967037846}{12675472479}$, $\frac{16459039}{194357244678}a^{8}+\frac{61241069}{388714489356}a^{7}-\frac{1178600107}{388714489356}a^{6}-\frac{3270871979}{194357244678}a^{5}+\frac{23443955}{8450314986}a^{4}+\frac{24109112465}{97178622339}a^{3}-\frac{25202359469}{97178622339}a^{2}-\frac{79960998518}{97178622339}a-\frac{264577277}{1408385831}$, $\frac{250405}{1166143468068}a^{8}+\frac{8569895}{1166143468068}a^{7}-\frac{384922987}{1166143468068}a^{6}+\frac{966961753}{583071734034}a^{5}+\frac{1161097568}{291535867017}a^{4}-\frac{4258467229}{291535867017}a^{3}-\frac{709327559}{32392874113}a^{2}+\frac{17807212567}{291535867017}a-\frac{10591124795}{12675472479}$, $\frac{80357095}{1166143468068}a^{8}-\frac{24451729}{291535867017}a^{7}-\frac{3222178645}{1166143468068}a^{6}-\frac{3216276347}{583071734034}a^{5}+\frac{26023148365}{583071734034}a^{4}+\frac{107070976285}{583071734034}a^{3}-\frac{86017228366}{97178622339}a^{2}+\frac{26758298932}{291535867017}a+\frac{24155788798}{12675472479}$ (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: | \( 2367.83568108 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 2367.83568108 \cdot 324}{2\cdot\sqrt{1597657402133118656}}\cr\approx \mathstrut & 0.602218841891 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
3.3.26569.1, 3.1.1169036.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | data not computed |
Minimal sibling: | 6.0.565813424.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.1.0.1}{1} }^{9}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.1.0.1}{1} }^{9}$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(11\) | 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(163\) | 163.9.6.1 | $x^{9} + 21 x^{7} + 972 x^{6} + 147 x^{5} + 3339 x^{4} - 393290 x^{3} + 47628 x^{2} - 1122660 x + 34068133$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |