Normalized defining polynomial
\( x^{9} - 126x^{6} + 378x^{5} + 63x^{4} - 4242x^{3} + 6615x^{2} - 18963x - 58114 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3230456496696835875\) \(\medspace = -\,3^{22}\cdot 5^{3}\cdot 7^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(113.92\) | 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))
| |
Galois root discriminant: | $3^{22/9}5^{1/2}7^{5/6}\approx 165.97006698500886$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-35}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}$, $\frac{1}{7}a^{7}$, $\frac{1}{126857182119836}a^{8}+\frac{6377206380919}{126857182119836}a^{7}+\frac{2914545581025}{126857182119836}a^{6}+\frac{4645299180207}{18122454588548}a^{5}-\frac{8606130860121}{18122454588548}a^{4}+\frac{3541083732065}{9061227294274}a^{3}+\frac{2037870415879}{4530613647137}a^{2}-\frac{7605930925139}{18122454588548}a+\frac{840406032321}{9061227294274}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (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)
| |
Fundamental units: | $\frac{145894500}{1023041791289}a^{8}+\frac{296558592}{1023041791289}a^{7}+\frac{211098638}{1023041791289}a^{6}-\frac{2627059266}{146148827327}a^{5}+\frac{3429988194}{146148827327}a^{4}+\frac{13952999414}{146148827327}a^{3}-\frac{113861702808}{146148827327}a^{2}-\frac{149065672644}{146148827327}a+\frac{59090804125}{146148827327}$, $\frac{94999325294}{31714295529959}a^{8}-\frac{306814705839}{31714295529959}a^{7}+\frac{103144980308}{4530613647137}a^{6}-\frac{2112539110477}{4530613647137}a^{5}+\frac{11309350049482}{4530613647137}a^{4}-\frac{32699861760746}{4530613647137}a^{3}+\frac{31374253245863}{4530613647137}a^{2}-\frac{437249422993}{4530613647137}a-\frac{305422341035425}{4530613647137}$, $\frac{586527724641}{63428591059918}a^{8}-\frac{979752056933}{63428591059918}a^{7}+\frac{247540954383}{9061227294274}a^{6}-\frac{11018971251177}{9061227294274}a^{5}+\frac{50170165137305}{9061227294274}a^{4}-\frac{40452338297870}{4530613647137}a^{3}-\frac{103981317820533}{4530613647137}a^{2}+\frac{869275835484915}{9061227294274}a-\frac{14\!\cdots\!32}{4530613647137}$, $\frac{2148554728359}{63428591059918}a^{8}-\frac{11417305588239}{63428591059918}a^{7}+\frac{1691165327451}{9061227294274}a^{6}+\frac{3383517925207}{9061227294274}a^{5}-\frac{13926159602205}{9061227294274}a^{4}+\frac{1641414504478}{4530613647137}a^{3}-\frac{67855438035872}{4530613647137}a^{2}+\frac{23412226018215}{9061227294274}a-\frac{10\!\cdots\!00}{4530613647137}$, $\frac{223976433225}{63428591059918}a^{8}+\frac{313855920979}{63428591059918}a^{7}-\frac{237952031403}{9061227294274}a^{6}-\frac{5615377434959}{9061227294274}a^{5}+\frac{469009371761}{9061227294274}a^{4}+\frac{11155589636663}{4530613647137}a^{3}-\frac{14486281059938}{4530613647137}a^{2}+\frac{326024094766623}{9061227294274}a+\frac{311415878613606}{4530613647137}$ (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: | \( 506430.869081 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 506430.869081 \cdot 9}{2\cdot\sqrt{3230456496696835875}}\cr\approx \mathstrut & 2.51611407553 \end{aligned}\] (assuming GRH)
Galois group
$C_3\wr S_3$ (as 9T20):
A solvable group of order 162 |
The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
Character table for $C_3 \wr S_3 $ |
Intermediate fields
3.1.2835.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{3}$ | R | R | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.5.1 | $x^{6} + 21$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |