Normalized defining polynomial
\( x^{9} - 4 x^{8} - 305 x^{7} - 4087 x^{6} - 29699 x^{5} - 131134 x^{4} - 357734 x^{3} - 587527 x^{2} + \cdots - 204251 \)
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: | \(-171707277277450704179928256\) \(\medspace = -\,2^{6}\cdot 31^{3}\cdot 709^{7}\) | 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: | \(822.20\) | 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: | $2^{2/3}31^{1/2}709^{5/6}\approx 2098.485825279839$ | ||
Ramified primes: | \(2\), \(31\), \(709\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-21979}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{31}a^{6}-\frac{13}{31}a^{5}+\frac{5}{31}a^{4}+\frac{5}{31}a^{3}-\frac{12}{31}a^{2}+\frac{13}{31}a-\frac{11}{31}$, $\frac{1}{31}a^{7}-\frac{9}{31}a^{5}+\frac{8}{31}a^{4}-\frac{9}{31}a^{3}+\frac{12}{31}a^{2}+\frac{3}{31}a+\frac{12}{31}$, $\frac{1}{31}a^{8}+\frac{15}{31}a^{5}+\frac{5}{31}a^{4}-\frac{5}{31}a^{3}-\frac{12}{31}a^{2}+\frac{5}{31}a-\frac{6}{31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{5652}$, which has order $90432$ (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: | $a+2$, $a^{8}-7a^{7}-284a^{6}-3235a^{5}-19994a^{4}-71152a^{3}-144278a^{2}-154693a-68084$, $\frac{10}{31}a^{8}-\frac{77}{31}a^{7}-\frac{2772}{31}a^{6}-\frac{30546}{31}a^{5}-\frac{182198}{31}a^{4}-\frac{619856}{31}a^{3}-\frac{1194760}{31}a^{2}-\frac{1215798}{31}a-\frac{508714}{31}$, $\frac{38}{31}a^{8}-\frac{233}{31}a^{7}-\frac{11091}{31}a^{6}-\frac{131685}{31}a^{5}-\frac{848497}{31}a^{4}-\frac{3180952}{31}a^{3}-\frac{6849066}{31}a^{2}-\frac{7831080}{31}a-\frac{3682150}{31}$, $\frac{98}{31}a^{8}-\frac{658}{31}a^{7}-\frac{28112}{31}a^{6}-\frac{324156}{31}a^{5}-\frac{2028460}{31}a^{4}-\frac{7322168}{31}a^{3}-\frac{15052688}{31}a^{2}-\frac{16316874}{31}a-\frac{7227819}{31}$ (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: | \( 21709.433536498364 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 21709.433536498364 \cdot 90432}{2\cdot\sqrt{171707277277450704179928256}}\cr\approx \mathstrut & 0.148653751216308 \end{aligned}\] (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
3.3.502681.1, 3.1.87916.3 |
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: | 6.0.85395746645213716144.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | R | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(31\) | 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
31.6.3.1 | $x^{6} + 961 x^{2} - 834148$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(709\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |