Normalized defining polynomial
\( x^{9} - 3x^{8} + 12x^{7} - 20x^{6} + 69x^{5} - 99x^{4} + 132x^{3} - 72x^{2} - 6x + 10 \)
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: | \(2539579392000\) \(\medspace = 2^{14}\cdot 3^{11}\cdot 5^{3}\cdot 7\) | 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: | \(23.89\) | 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))
| |
Galois root discriminant: | $2^{17/6}3^{115/54}5^{3/4}7^{1/2}\approx 654.3142900594603$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{105}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{232652}a^{8}+\frac{18709}{116326}a^{7}+\frac{732}{58163}a^{6}+\frac{52951}{116326}a^{5}+\frac{11567}{33236}a^{4}-\frac{618}{58163}a^{3}+\frac{3247}{8309}a^{2}+\frac{24342}{58163}a+\frac{22475}{116326}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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)
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Fundamental units: | $\frac{992}{8309}a^{8}-\frac{3603}{16618}a^{7}+\frac{17779}{16618}a^{6}-\frac{16733}{16618}a^{5}+\frac{14901}{2374}a^{4}-\frac{34305}{8309}a^{3}+\frac{7520}{1187}a^{2}-\frac{3069}{8309}a-\frac{12312}{8309}$, $\frac{14629}{116326}a^{8}-\frac{21117}{58163}a^{7}+\frac{71035}{58163}a^{6}-\frac{110981}{58163}a^{5}+\frac{108875}{16618}a^{4}-\frac{574381}{58163}a^{3}+\frac{37165}{8309}a^{2}-\frac{7699}{58163}a-\frac{8664}{58163}$, $\frac{5837}{58163}a^{8}-\frac{44561}{116326}a^{7}+\frac{156117}{116326}a^{6}-\frac{303595}{116326}a^{5}+\frac{136543}{16618}a^{4}-\frac{760759}{58163}a^{3}+\frac{149202}{8309}a^{2}-\frac{729776}{58163}a+\frac{174346}{58163}$, $\frac{541}{58163}a^{8}+\frac{2414}{58163}a^{7}+\frac{13647}{58163}a^{6}-\frac{113899}{58163}a^{5}+\frac{42615}{8309}a^{4}-\frac{406744}{58163}a^{3}+\frac{46948}{8309}a^{2}-\frac{19590}{58163}a-\frac{52347}{58163}$ | 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: | \( 4771.43994028 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 4771.43994028 \cdot 1}{2\cdot\sqrt{2539579392000}}\cr\approx \mathstrut & 4.66646526943 \end{aligned}\]
Galois group
A non-solvable group of order 362880 |
The 30 conjugacy class representatives for $S_9$ |
Character table for $S_9$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 18 sibling: | data not computed |
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 | R | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.10.6 | $x^{4} + 4 x^{3} + 12 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(3\) | 3.3.5.1 | $x^{3} + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |