Normalized defining polynomial
\( x^{9} - 18x^{7} - 18x^{6} + 108x^{4} + 504x^{3} - 648x^{2} - 1296x + 1368 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(74310349154112\) \(\medspace = 2^{6}\cdot 3^{22}\cdot 37\) | 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: | \(34.77\) | 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: | $2^{2/3}3^{22/9}37^{1/2}\approx 141.6067404717884$ | ||
Ramified primes: | \(2\), \(3\), \(37\) | 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: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{6}a^{5}$, $\frac{1}{12}a^{6}$, $\frac{1}{60}a^{7}-\frac{1}{60}a^{6}+\frac{1}{15}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{1581900}a^{8}-\frac{113}{1581900}a^{7}+\frac{12751}{1581900}a^{6}+\frac{4597}{790950}a^{5}+\frac{2613}{263650}a^{4}-\frac{31601}{263650}a^{3}+\frac{5861}{131825}a^{2}-\frac{3222}{131825}a-\frac{31497}{131825}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{1389}{527300}a^{8}+\frac{1233}{527300}a^{7}-\frac{71153}{1581900}a^{6}-\frac{10731}{131825}a^{5}-\frac{26739}{263650}a^{4}+\frac{38173}{263650}a^{3}+\frac{219717}{131825}a^{2}-\frac{59019}{131825}a-\frac{398504}{131825}$, $\frac{231}{263650}a^{8}+\frac{131}{131825}a^{7}-\frac{18029}{1581900}a^{6}-\frac{11751}{263650}a^{5}-\frac{8376}{131825}a^{4}+\frac{9822}{131825}a^{3}+\frac{55656}{131825}a^{2}+\frac{121818}{131825}a-\frac{126227}{131825}$, $\frac{403}{527300}a^{8}-\frac{1198}{395475}a^{7}-\frac{1261}{263650}a^{6}+\frac{7041}{263650}a^{5}-\frac{4683}{263650}a^{4}+\frac{23641}{263650}a^{3}-\frac{32601}{131825}a^{2}-\frac{72473}{131825}a+\frac{149377}{131825}$, $\frac{203}{263650}a^{8}-\frac{5809}{1581900}a^{7}-\frac{6158}{395475}a^{6}+\frac{10416}{131825}a^{5}+\frac{9417}{131825}a^{4}-\frac{128943}{263650}a^{3}+\frac{20148}{131825}a^{2}+\frac{162179}{131825}a-\frac{134096}{131825}$, $\frac{9313}{395475}a^{8}+\frac{30827}{790950}a^{7}-\frac{181751}{527300}a^{6}-\frac{135091}{131825}a^{5}-\frac{501039}{263650}a^{4}-\frac{80696}{131825}a^{3}+\frac{1429117}{131825}a^{2}+\frac{936676}{131825}a-\frac{1876564}{131825}$, $\frac{7939}{790950}a^{8}-\frac{697}{790950}a^{7}-\frac{38957}{263650}a^{6}-\frac{24224}{131825}a^{5}-\frac{141121}{263650}a^{4}+\frac{61731}{131825}a^{3}+\frac{440713}{131825}a^{2}-\frac{617211}{131825}a+\frac{113779}{131825}$ | 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: | \( 5160.6310219 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 5160.6310219 \cdot 1}{2\cdot\sqrt{74310349154112}}\cr\approx \mathstrut & 0.37814457754 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
A solvable group of order 648 |
The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
Character table for $S_3 \wr C_3 $ |
Intermediate fields
\(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | R | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |