Normalized defining polynomial
\( x^{9} - 3 x^{8} - 1221 x^{7} - 33360 x^{6} - 468492 x^{5} - 4028151 x^{4} - 21971391 x^{3} + \cdots - 116934011 \)
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: | \(-1618930577917087333772795971776\) \(\medspace = -\,2^{6}\cdot 3^{15}\cdot 19^{3}\cdot 307^{7}\) | 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: | \(2272.90\) | 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: | $2^{2/3}3^{31/18}19^{1/2}307^{5/6}\approx 5424.848382525166$ | ||
Ramified primes: | \(2\), \(3\), \(19\), \(307\) | 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: | \(\Q(\sqrt{-17499}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{57}a^{6}-\frac{2}{57}a^{5}+\frac{2}{57}a^{4}-\frac{7}{19}a^{3}+\frac{25}{57}a^{2}-\frac{10}{57}a+\frac{5}{57}$, $\frac{1}{57}a^{7}-\frac{2}{57}a^{5}-\frac{17}{57}a^{4}-\frac{17}{57}a^{3}-\frac{17}{57}a^{2}-\frac{5}{19}a+\frac{10}{57}$, $\frac{1}{57}a^{8}-\frac{2}{57}a^{5}+\frac{25}{57}a^{4}+\frac{17}{57}a^{3}-\frac{1}{19}a^{2}+\frac{28}{57}a-\frac{28}{57}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{42}\times C_{34398}$, which has order $13002444$ (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: | $a^{8}-8a^{7}-1181a^{6}-27455a^{5}-331217a^{4}-2372066a^{3}-10111061a^{2}-23661254a-23386801$, $a^{8}-9a^{7}-1167a^{6}-26358a^{5}-310344a^{4}-2166087a^{3}-8974869a^{2}-20367345a-19489002$, $\frac{5}{57}a^{8}-\frac{43}{57}a^{7}-\frac{5864}{57}a^{6}-\frac{133964}{57}a^{5}-\frac{1592489}{57}a^{4}-\frac{11227381}{57}a^{3}-\frac{47030854}{57}a^{2}-\frac{107984284}{57}a-\frac{104592134}{57}$, $\frac{52}{57}a^{8}-\frac{138}{19}a^{7}-\frac{20480}{19}a^{6}-\frac{1429855}{57}a^{5}-\frac{5755002}{19}a^{4}-\frac{41252836}{19}a^{3}-\frac{528036104}{57}a^{2}-\frac{412301195}{19}a-\frac{407917732}{19}$, $\frac{63}{19}a^{8}-\frac{549}{19}a^{7}-\frac{73791}{19}a^{6}-\frac{1679949}{19}a^{5}-\frac{19909575}{19}a^{4}-\frac{139893237}{19}a^{3}-\frac{583685019}{19}a^{2}-\frac{1333940436}{19}a-\frac{1285102781}{19}$ (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: | \( 55601.09550130088 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 55601.09550130088 \cdot 13002444}{2\cdot\sqrt{1618930577917087333772795971776}}\cr\approx \mathstrut & 0.563759435889165 \end{aligned}\] (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
3.3.7634169.1, 3.1.629964.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | data not computed |
Minimal sibling: | 6.0.5890659982904518319664.2 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.1.0.1}{1} }^{9}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.9.15.27 | $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 3$ | $9$ | $1$ | $15$ | $S_3\times C_3$ | $[3/2, 2]_{2}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(307\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |