Normalized defining polynomial
\( x^{10} - 9x^{8} - 2x^{7} + 35x^{6} + 8x^{5} - 63x^{4} - 9x^{3} + 30x^{2} + 8x - 7 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-585682963101963\) \(\medspace = -\,3^{3}\cdot 167^{6}\) | 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: | \(29.98\) | 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: | $3^{1/2}167^{2/3}\approx 52.52567088234156$ | ||
Ramified primes: | \(3\), \(167\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1929}a^{9}+\frac{374}{1929}a^{8}-\frac{950}{1929}a^{7}-\frac{122}{643}a^{6}+\frac{110}{1929}a^{5}+\frac{213}{643}a^{4}-\frac{91}{643}a^{3}+\frac{42}{643}a^{2}+\frac{286}{643}a+\frac{686}{1929}$
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)
| |
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: | $\frac{629}{1929}a^{9}-\frac{92}{1929}a^{8}-\frac{5347}{1929}a^{7}-\frac{221}{643}a^{6}+\frac{19036}{1929}a^{5}+\frac{876}{643}a^{4}-\frac{9014}{643}a^{3}+\frac{55}{643}a^{2}-\frac{789}{643}a+\frac{3256}{1929}$, $\frac{491}{643}a^{9}+\frac{379}{643}a^{8}-\frac{4133}{643}a^{7}-\frac{4167}{643}a^{6}+\frac{14144}{643}a^{5}+\frac{14754}{643}a^{4}-\frac{20232}{643}a^{3}-\frac{19795}{643}a^{2}+\frac{113}{643}a+\frac{4395}{643}$, $\frac{1498}{1929}a^{9}+\frac{842}{1929}a^{8}-\frac{13001}{1929}a^{7}-\frac{3359}{643}a^{6}+\frac{47111}{1929}a^{5}+\frac{12363}{643}a^{4}-\frac{25079}{643}a^{3}-\frac{17459}{643}a^{2}+\frac{6620}{643}a+\frac{22619}{1929}$, $\frac{2597}{1929}a^{9}+\frac{991}{1929}a^{8}-\frac{23107}{1929}a^{7}-\frac{4979}{643}a^{6}+\frac{85054}{1929}a^{5}+\frac{19471}{643}a^{4}-\frac{45999}{643}a^{3}-\frac{29814}{643}a^{2}+\frac{11651}{643}a+\frac{41584}{1929}$, $\frac{60}{643}a^{9}-\frac{65}{643}a^{8}-\frac{416}{643}a^{7}+\frac{545}{643}a^{6}+\frac{1456}{643}a^{5}-\frac{2169}{643}a^{4}-\frac{2234}{643}a^{3}+\frac{4345}{643}a^{2}+\frac{40}{643}a-\frac{635}{643}$, $\frac{82}{1929}a^{9}-\frac{196}{1929}a^{8}-\frac{740}{1929}a^{7}+\frac{284}{643}a^{6}+\frac{3233}{1929}a^{5}-\frac{538}{643}a^{4}-\frac{1675}{643}a^{3}-\frac{414}{643}a^{2}+\frac{304}{643}a+\frac{311}{1929}$ | 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: | \( 13527.5655241 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 13527.5655241 \cdot 1}{2\cdot\sqrt{585682963101963}}\cr\approx \mathstrut & 1.10922064148 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.0.83667.1, 6.4.21000500667.1 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.0.83667.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(167\) | $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
167.3.2.1 | $x^{3} + 167$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
167.3.2.1 | $x^{3} + 167$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
167.3.2.1 | $x^{3} + 167$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |