Normalized defining polynomial
\( x^{10} - x^{9} + 3x^{8} + x^{7} + 8x^{6} + 11x^{5} + 11x^{4} - 8x^{3} - 11x^{2} - 8x - 3 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(24583787279533\) \(\medspace = 29077^{3}\) | 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: | \(21.83\) | 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: | $29077^{1/2}\approx 170.51979357247652$ | ||
Ramified primes: | \(29077\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29077}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4507}a^{9}+\frac{1584}{4507}a^{8}+\frac{244}{4507}a^{7}-\frac{861}{4507}a^{6}+\frac{944}{4507}a^{5}-\frac{73}{4507}a^{4}+\frac{1488}{4507}a^{3}+\frac{1311}{4507}a^{2}+\frac{197}{4507}a+\frac{1254}{4507}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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)
| |
Fundamental units: | $\frac{1436}{4507}a^{9}-\frac{1411}{4507}a^{8}+\frac{3345}{4507}a^{7}+\frac{3029}{4507}a^{6}+\frac{7991}{4507}a^{5}+\frac{16861}{4507}a^{4}+\frac{9464}{4507}a^{3}-\frac{19358}{4507}a^{2}-\frac{19077}{4507}a-\frac{6563}{4507}$, $\frac{94}{4507}a^{9}+\frac{165}{4507}a^{8}+\frac{401}{4507}a^{7}+\frac{192}{4507}a^{6}+\frac{3103}{4507}a^{5}+\frac{2152}{4507}a^{4}+\frac{9169}{4507}a^{3}+\frac{6052}{4507}a^{2}+\frac{4997}{4507}a+\frac{694}{4507}$, $\frac{1390}{4507}a^{9}-\frac{2163}{4507}a^{8}+\frac{5642}{4507}a^{7}-\frac{2435}{4507}a^{6}+\frac{14144}{4507}a^{5}+\frac{6698}{4507}a^{4}+\frac{13128}{4507}a^{3}-\frac{16566}{4507}a^{2}-\frac{10111}{4507}a-\frac{5656}{4507}$, $\frac{1970}{4507}a^{9}-\frac{2871}{4507}a^{8}+\frac{7445}{4507}a^{7}-\frac{1538}{4507}a^{6}+\frac{16317}{4507}a^{5}+\frac{13935}{4507}a^{4}+\frac{15331}{4507}a^{3}-\frac{22376}{4507}a^{2}-\frac{8526}{4507}a-\frac{8470}{4507}$, $\frac{1436}{4507}a^{9}-\frac{1411}{4507}a^{8}+\frac{3345}{4507}a^{7}+\frac{3029}{4507}a^{6}+\frac{7991}{4507}a^{5}+\frac{16861}{4507}a^{4}+\frac{9464}{4507}a^{3}-\frac{19358}{4507}a^{2}-\frac{14570}{4507}a-\frac{2056}{4507}$ | 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: | \( 843.535429333 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 843.535429333 \cdot 2}{2\cdot\sqrt{24583787279533}}\cr\approx \mathstrut & 1.06061654649 \end{aligned}\]
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: | 6.2.29077.1, 6.2.24583787279533.1 |
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: | 6.2.29077.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(29077\) | $\Q_{29077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29077}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |