Normalized defining polynomial
\( x^{9} - 3x^{8} + 12x^{7} - 24x^{6} + 84x^{5} - 24x^{4} + 72x^{3} + 132x^{2} - 9x + 199 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(212576400000000\) \(\medspace = 2^{10}\cdot 3^{12}\cdot 5^{8}\) | 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: | \(39.08\) | 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: | $2^{17/10}3^{25/18}5^{13/10}\approx 121.08222909007846$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{3977812}a^{8}+\frac{82125}{1988906}a^{7}-\frac{238657}{994453}a^{6}+\frac{214580}{994453}a^{5}+\frac{5535}{994453}a^{4}-\frac{573839}{1988906}a^{3}+\frac{236281}{994453}a^{2}+\frac{340348}{994453}a+\frac{18587}{3977812}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{1252}{994453}a^{8}+\frac{151369}{3977812}a^{7}-\frac{441453}{3977812}a^{6}+\frac{1435147}{3977812}a^{5}-\frac{3484975}{3977812}a^{4}+\frac{9316993}{3977812}a^{3}-\frac{1409741}{3977812}a^{2}-\frac{5090897}{3977812}a+\frac{18499721}{3977812}$, $\frac{83073}{3977812}a^{8}-\frac{149363}{3977812}a^{7}+\frac{831547}{3977812}a^{6}-\frac{2046099}{3977812}a^{5}+\frac{4470435}{3977812}a^{4}-\frac{5828743}{3977812}a^{3}+\frac{7193967}{3977812}a^{2}-\frac{5216527}{3977812}a+\frac{1835077}{1988906}$, $\frac{105750}{994453}a^{8}-\frac{673051}{994453}a^{7}+\frac{1974201}{994453}a^{6}-\frac{5698509}{1988906}a^{5}+\frac{3708635}{1988906}a^{4}+\frac{73432}{994453}a^{3}-\frac{3619124}{994453}a^{2}+\frac{7447551}{1988906}a-\frac{9866739}{1988906}$, $\frac{77862887}{3977812}a^{8}-\frac{169375709}{1988906}a^{7}+\frac{583272943}{1988906}a^{6}-\frac{1364360827}{1988906}a^{5}+\frac{3849548903}{1988906}a^{4}-\frac{3683310541}{1988906}a^{3}-\frac{434405213}{1988906}a^{2}+\frac{6600109737}{1988906}a-\frac{12907914701}{3977812}$ (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: | \( 28802.3266185 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 28802.3266185 \cdot 1}{2\cdot\sqrt{212576400000000}}\cr\approx \mathstrut & 3.07885701572 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 181440 |
The 18 conjugacy class representatives for $A_9$ |
Character table for $A_9$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 36 sibling: | 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 | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
\(3\) | 3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.5.6.1 | $x^{5} + 10 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $[3/2]_{2}$ |