Normalized defining polynomial
\( x^{10} - 5x^{9} + 3x^{8} + 14x^{7} - 5x^{6} - 27x^{5} - 7x^{4} + 26x^{3} + 25x^{2} + 9x + 3 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-175784940288\) \(\medspace = -\,2^{8}\cdot 3^{5}\cdot 41^{4}\) | 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: | \(13.32\) | 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: | $2^{4/5}3^{1/2}41^{2/3}\approx 35.85699261035452$ | ||
Ramified primes: | \(2\), \(3\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2999}a^{9}+\frac{843}{2999}a^{8}+\frac{1105}{2999}a^{7}+\frac{1366}{2999}a^{6}+\frac{749}{2999}a^{5}-\frac{663}{2999}a^{4}-\frac{1418}{2999}a^{3}+\frac{161}{2999}a^{2}-\frac{1401}{2999}a-\frac{435}{2999}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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)
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Torsion generator: | \( -\frac{562}{2999} a^{9} + \frac{3075}{2999} a^{8} - \frac{3216}{2999} a^{7} - \frac{5946}{2999} a^{6} + \frac{4920}{2999} a^{5} + \frac{12726}{2999} a^{4} - \frac{818}{2999} a^{3} - \frac{12508}{2999} a^{2} - \frac{10372}{2999} a - \frac{1448}{2999} \) (order $6$) | 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: | $\frac{858}{2999}a^{9}-\frac{5463}{2999}a^{8}+\frac{9403}{2999}a^{7}+\frac{2418}{2999}a^{6}-\frac{11140}{2999}a^{5}-\frac{11040}{2999}a^{4}+\frac{9947}{2999}a^{3}+\frac{15179}{2999}a^{2}+\frac{3540}{2999}a+\frac{1645}{2999}$, $\frac{704}{2999}a^{9}-\frac{3329}{2999}a^{8}+\frac{1179}{2999}a^{7}+\frac{10981}{2999}a^{6}-\frac{3527}{2999}a^{5}-\frac{19901}{2999}a^{4}-\frac{2604}{2999}a^{3}+\frac{20375}{2999}a^{2}+\frac{15362}{2999}a+\frac{2657}{2999}$, $\frac{470}{2999}a^{9}-\frac{2657}{2999}a^{8}+\frac{3522}{2999}a^{7}+\frac{3233}{2999}a^{6}-\frac{4851}{2999}a^{5}-\frac{8711}{2999}a^{4}+\frac{5316}{2999}a^{3}+\frac{9692}{2999}a^{2}+\frac{1310}{2999}a-\frac{3517}{2999}$, $\frac{1078}{2999}a^{9}-\frac{5941}{2999}a^{8}+\frac{6585}{2999}a^{7}+\frac{12035}{2999}a^{6}-\frac{17303}{2999}a^{5}-\frac{15947}{2999}a^{4}+\frac{12882}{2999}a^{3}+\frac{14611}{2999}a^{2}+\frac{7216}{2999}a+\frac{1913}{2999}$ | 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: | \( 157.516418044 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 157.516418044 \cdot 1}{6\cdot\sqrt{175784940288}}\cr\approx \mathstrut & 0.613173088387 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.3.80688.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.3.80688.1 |
Degree 6 sibling: | 6.0.1220728752.3 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.3.80688.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(41\) | 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.6.4.1 | $x^{6} + 114 x^{5} + 4350 x^{4} + 56322 x^{3} + 30774 x^{2} + 180240 x + 2223605$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |