Normalized defining polynomial
\( x^{9} - 3x^{8} + 39x^{7} + 198x^{6} + 444x^{5} + 603x^{4} + 515x^{3} + 273x^{2} + 72x + 1 \)
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: | \(-821824919051661504\) \(\medspace = -\,2^{6}\cdot 3^{12}\cdot 7^{6}\cdot 59^{3}\) | 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: | \(97.84\) | 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: | $2^{2/3}3^{4/3}7^{2/3}59^{1/2}\approx 193.05143575878094$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(59\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-59}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2897}a^{8}-\frac{737}{2897}a^{7}-\frac{742}{2897}a^{6}+\frac{190}{2897}a^{5}+\frac{40}{2897}a^{4}+\frac{213}{2897}a^{3}+\frac{611}{2897}a^{2}+\frac{834}{2897}a-\frac{817}{2897}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{9}$, which has order $243$ (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: | $\frac{2267}{2897}a^{8}-\frac{7901}{2897}a^{7}+\frac{90850}{2897}a^{6}+\frac{410451}{2897}a^{5}+\frac{748299}{2897}a^{4}+\frac{784159}{2897}a^{3}+\frac{458097}{2897}a^{2}+\frac{114817}{2897}a+\frac{1941}{2897}$, $a+1$, $\frac{254}{2897}a^{8}-\frac{1790}{2897}a^{7}+\frac{14322}{2897}a^{6}+\frac{4805}{2897}a^{5}-\frac{33295}{2897}a^{4}-\frac{99439}{2897}a^{3}-\frac{117124}{2897}a^{2}-\frac{74967}{2897}a-\frac{25007}{2897}$, $\frac{1941}{2897}a^{8}-\frac{8090}{2897}a^{7}+\frac{83600}{2897}a^{6}+\frac{293468}{2897}a^{5}+\frac{451353}{2897}a^{4}+\frac{422124}{2897}a^{3}+\frac{215456}{2897}a^{2}+\frac{71796}{2897}a+\frac{24935}{2897}$, $\frac{72}{2897}a^{8}-\frac{918}{2897}a^{7}+\frac{4516}{2897}a^{6}-\frac{9496}{2897}a^{5}-\frac{133279}{2897}a^{4}-\frac{239600}{2897}a^{3}-\frac{222532}{2897}a^{2}-\frac{90596}{2897}a+\frac{2013}{2897}$ (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: | \( 970.684878684 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 970.684878684 \cdot 243}{2\cdot\sqrt{821824919051661504}}\cr\approx \mathstrut & 0.258163286673 \end{aligned}\] (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
3.3.3969.2, 3.1.936684.2 |
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: | 6.0.13042388016.14 |
Minimal sibling: | 6.0.13042388016.14 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | R |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ |
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
\(59\) | 59.3.0.1 | $x^{3} + 5 x + 57$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
59.6.3.2 | $x^{6} + 187 x^{4} + 114 x^{3} + 10468 x^{2} - 19608 x + 175293$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |