Normalized defining polynomial
\( x^{9} - 7833 x^{7} - 154049 x^{6} + 20679120 x^{5} + 817921860 x^{4} - 10296478500 x^{3} + \cdots - 146020198499000 \)
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)
| |
Discriminant: | \(-28137346150667129747744198503104000\) \(\medspace = -\,2^{9}\cdot 3^{12}\cdot 5^{3}\cdot 7^{7}\cdot 373^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(6725.09\) | 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: | $2^{3/2}3^{4/3}5^{1/2}7^{5/6}373^{5/6}\approx 19254.432748637053$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\), \(373\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-26110}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{10}a^{5}-\frac{3}{10}a^{3}+\frac{1}{10}a^{2}$, $\frac{1}{26110}a^{6}-\frac{3}{10}a^{4}+\frac{1}{10}a^{3}$, $\frac{1}{261100}a^{7}-\frac{3}{100}a^{5}+\frac{41}{100}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{261100}a^{8}-\frac{3}{261100}a^{6}+\frac{1}{100}a^{5}+\frac{3}{10}a^{4}+\frac{1}{10}a^{3}-\frac{2}{5}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{21}\times C_{267414}$, which has order $50541246$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{1604048}{65275}a^{8}-\frac{19034709}{26110}a^{7}-\frac{11152766619}{65275}a^{6}+\frac{32081013}{25}a^{5}+\frac{4700929257}{10}a^{4}+\frac{12306536955}{2}a^{3}-\frac{4355695550169}{10}a^{2}-14066662227246a-120952409112747$, $\frac{11904}{65275}a^{8}-\frac{201797}{37300}a^{7}-\frac{165535239}{130550}a^{6}+\frac{952263}{100}a^{5}+\frac{348868831}{100}a^{4}+\frac{228331266}{5}a^{3}-\frac{32324746587}{10}a^{2}-104393401306a-897633674219$, $\frac{11747}{65275}a^{8}-\frac{1401929}{261100}a^{7}-\frac{163115277}{130550}a^{6}+\frac{38223}{4}a^{5}+\frac{343637221}{100}a^{4}+\frac{446663001}{10}a^{3}-\frac{15928491648}{5}a^{2}-102597748810a-880723791209$, $\frac{4717}{52220}a^{8}-\frac{699659}{261100}a^{7}-\frac{6559425}{10444}a^{6}+\frac{235821}{50}a^{5}+\frac{172801741}{100}a^{4}+\frac{113100018}{5}a^{3}-\frac{8005532298}{5}a^{2}-51708580465a-444622456258$, $\frac{43\!\cdots\!37}{13055}a^{8}-\frac{13\!\cdots\!64}{13055}a^{7}-\frac{60\!\cdots\!02}{2611}a^{6}+\frac{87\!\cdots\!39}{5}a^{5}+\frac{32\!\cdots\!23}{5}a^{4}+\frac{42\!\cdots\!91}{5}a^{3}-59\!\cdots\!62a^{2}-19\!\cdots\!74a-16\!\cdots\!93$ (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: | \( 1578972.4007431671 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1578972.4007431671 \cdot 50541246}{2\cdot\sqrt{28137346150667129747744198503104000}}\cr\approx \mathstrut & 0.472039686303261 \end{aligned}\] (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
3.3.552203001.1, 3.1.104440.1 |
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: | 6.0.50954714298387396391104000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.1.0.1}{1} }^{9}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(3\) | 3.3.4.1 | $x^{3} + 6 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.6.8.6 | $x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.5.3 | $x^{6} + 35$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(373\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |