Normalized defining polynomial
\( x^{10} - 2x^{9} - 4x^{8} + 8x^{7} + 9x^{6} - 7x^{5} - 26x^{4} + 4x^{3} + 18x^{2} - 8x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-17067892507723\) \(\medspace = -\,25747^{3}\) | 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: | \(21.05\) | 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: | $25747^{1/2}\approx 160.45871743224174$ | ||
Ramified primes: | \(25747\) | 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: | \(\Q(\sqrt{-25747}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{541}a^{9}-\frac{210}{541}a^{8}-\frac{145}{541}a^{7}-\frac{128}{541}a^{6}+\frac{124}{541}a^{5}+\frac{169}{541}a^{4}-\frac{13}{541}a^{3}+\frac{3}{541}a^{2}-\frac{65}{541}a-\frac{13}{541}$
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: | $a$, $\frac{628}{541}a^{9}-\frac{958}{541}a^{8}-\frac{2877}{541}a^{7}+\frac{3471}{541}a^{6}+\frac{7001}{541}a^{5}-\frac{445}{541}a^{4}-\frac{15738}{541}a^{3}-\frac{5690}{541}a^{2}+\frac{6788}{541}a-\frac{1131}{541}$, $\frac{1042}{541}a^{9}-\frac{1879}{541}a^{8}-\frac{4479}{541}a^{7}+\frac{7284}{541}a^{6}+\frac{10729}{541}a^{5}-\frac{4596}{541}a^{4}-\frac{28153}{541}a^{3}-\frac{1743}{541}a^{2}+\frac{17748}{541}a-\frac{3808}{541}$, $\frac{805}{541}a^{9}-\frac{1340}{541}a^{8}-\frac{3656}{541}a^{7}+\frac{5160}{541}a^{6}+\frac{8932}{541}a^{5}-\frac{2451}{541}a^{4}-\frac{21826}{541}a^{3}-\frac{4077}{541}a^{2}+\frac{13136}{541}a-\frac{2350}{541}$, $\frac{540}{541}a^{9}-\frac{872}{541}a^{8}-\frac{2560}{541}a^{7}+\frac{3374}{541}a^{6}+\frac{6368}{541}a^{5}-\frac{1251}{541}a^{4}-\frac{14594}{541}a^{3}-\frac{3790}{541}a^{2}+\frac{8180}{541}a-\frac{1610}{541}$, $\frac{1450}{541}a^{9}-\frac{2622}{541}a^{8}-\frac{6293}{541}a^{7}+\frac{10242}{541}a^{6}+\frac{15336}{541}a^{5}-\frac{7056}{541}a^{4}-\frac{39408}{541}a^{3}-\frac{2683}{541}a^{2}+\frac{25852}{541}a-\frac{5325}{541}$ | 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: | \( 1882.92554161 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 1882.92554161 \cdot 1}{2\cdot\sqrt{17067892507723}}\cr\approx \mathstrut & 0.904425480254 \end{aligned}\]
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: | 6.0.25747.1, 6.4.17067892507723.1 |
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: | 6.0.25747.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(25747\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |