Normalized defining polynomial
\( x^{10} - 3x^{8} - 2x^{7} + 3x^{6} - 40x^{5} - 5x^{4} + 14x^{3} - 52x^{2} + 4x - 8 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(178869124096000\) \(\medspace = 2^{13}\cdot 5^{3}\cdot 13^{3}\cdot 43^{3}\) | 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: | \(26.62\) | 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: | $2^{11/6}5^{1/2}13^{1/2}43^{1/2}\approx 188.39927686169324$ | ||
Ramified primes: | \(2\), \(5\), \(13\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5590}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{34024}a^{9}-\frac{3461}{34024}a^{8}+\frac{1035}{17012}a^{7}+\frac{3143}{17012}a^{6}+\frac{2505}{34024}a^{5}+\frac{6275}{34024}a^{4}+\frac{818}{4253}a^{3}-\frac{1788}{4253}a^{2}+\frac{293}{8506}a-\frac{929}{4253}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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)
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Fundamental units: | $\frac{1921}{17012}a^{9}-\frac{571}{8506}a^{8}-\frac{2169}{8506}a^{7}-\frac{1557}{8506}a^{6}+\frac{6215}{17012}a^{5}-\frac{39767}{8506}a^{4}+\frac{20843}{8506}a^{3}-\frac{14561}{8506}a^{2}+\frac{1457}{4253}a-\frac{951}{4253}$, $\frac{195}{4253}a^{9}+\frac{1075}{17012}a^{8}-\frac{385}{4253}a^{7}-\frac{2441}{8506}a^{6}-\frac{620}{4253}a^{5}-\frac{26221}{17012}a^{4}-\frac{8326}{4253}a^{3}-\frac{3565}{4253}a^{2}-\frac{5375}{4253}a+\frac{1033}{4253}$, $\frac{15675}{34024}a^{9}+\frac{93}{34024}a^{8}-\frac{6772}{4253}a^{7}-\frac{17239}{17012}a^{6}+\frac{70227}{34024}a^{5}-\frac{598179}{34024}a^{4}-\frac{49363}{17012}a^{3}+\frac{59912}{4253}a^{2}-\frac{170585}{8506}a-\frac{12562}{4253}$, $\frac{8257}{34024}a^{9}+\frac{2683}{34024}a^{8}-\frac{7647}{8506}a^{7}-\frac{17067}{17012}a^{6}+\frac{31217}{34024}a^{5}-\frac{278069}{34024}a^{4}-\frac{61947}{17012}a^{3}+\frac{78101}{8506}a^{2}-\frac{4909}{8506}a+\frac{5912}{4253}$, $\frac{4917}{34024}a^{9}+\frac{11275}{34024}a^{8}-\frac{873}{8506}a^{7}-\frac{18283}{17012}a^{6}-\frac{33627}{34024}a^{5}-\frac{158701}{34024}a^{4}-\frac{187807}{17012}a^{3}-\frac{56579}{8506}a^{2}-\frac{13845}{8506}a-\frac{4424}{4253}$ (assuming GRH) | 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: | \( 12104.8481357 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 12104.8481357 \cdot 1}{2\cdot\sqrt{178869124096000}}\cr\approx \mathstrut & 2.82124709054 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.89440.1 |
Degree 6 sibling: | 6.2.44717281024000.1 |
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: | 5.1.89440.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | R | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.11.9 | $x^{6} + 4 x^{5} + 4 x^{2} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(43\) | $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
43.6.3.1 | $x^{6} + 1849 x^{2} - 3180280$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |