Normalized defining polynomial
\( x^{10} - 9x^{7} + 7x^{6} + 6x^{5} - 18x^{4} + 14x^{3} - 2x^{2} - x + 1 \)
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: | \(-929428432479\) \(\medspace = -\,3^{3}\cdot 3253^{3}\) | 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: | \(15.73\) | 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: | $3^{1/2}3253^{1/2}\approx 98.78765105011861$ | ||
Ramified primes: | \(3\), \(3253\) | 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: | \(\Q(\sqrt{-9759}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1789}a^{9}+\frac{172}{1789}a^{8}-\frac{829}{1789}a^{7}+\frac{523}{1789}a^{6}+\frac{513}{1789}a^{5}+\frac{581}{1789}a^{4}-\frac{270}{1789}a^{3}+\frac{88}{1789}a^{2}+\frac{822}{1789}a+\frac{52}{1789}$
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: | $a^{9}-9a^{6}+7a^{5}+6a^{4}-18a^{3}+14a^{2}-2a-1$, $\frac{810}{1789}a^{9}-\frac{222}{1789}a^{8}-\frac{615}{1789}a^{7}-\frac{7519}{1789}a^{6}+\frac{7638}{1789}a^{5}+\frac{9048}{1789}a^{4}-\frac{18332}{1789}a^{3}+\frac{10454}{1789}a^{2}+\frac{2101}{1789}a-\frac{2605}{1789}$, $\frac{1568}{1789}a^{9}+\frac{1346}{1789}a^{8}+\frac{731}{1789}a^{7}-\frac{13610}{1789}a^{6}-\frac{666}{1789}a^{5}+\frac{12930}{1789}a^{4}-\frac{19046}{1789}a^{3}+\frac{2020}{1789}a^{2}+\frac{4394}{1789}a-\frac{758}{1789}$, $\frac{958}{1789}a^{9}+\frac{188}{1789}a^{8}+\frac{134}{1789}a^{7}-\frac{8831}{1789}a^{6}+\frac{4846}{1789}a^{5}+\frac{5586}{1789}a^{4}-\frac{13567}{1789}a^{3}+\frac{10955}{1789}a^{2}-\frac{3262}{1789}a+\frac{3302}{1789}$, $\frac{1480}{1789}a^{9}+\frac{522}{1789}a^{8}+\frac{334}{1789}a^{7}-\frac{13120}{1789}a^{6}+\frac{6071}{1789}a^{5}+\frac{10105}{1789}a^{4}-\frac{22121}{1789}a^{3}+\frac{12166}{1789}a^{2}+\frac{40}{1789}a+\frac{33}{1789}$, $\frac{4855}{1789}a^{9}-\frac{403}{1789}a^{8}-\frac{1334}{1789}a^{7}-\frac{44151}{1789}a^{6}+\frac{37896}{1789}a^{5}+\frac{38860}{1789}a^{4}-\frac{94330}{1789}a^{3}+\frac{62284}{1789}a^{2}+\frac{4918}{1789}a-\frac{12312}{1789}$ | 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: | \( 178.062928684 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 178.062928684 \cdot 1}{2\cdot\sqrt{929428432479}}\cr\approx \mathstrut & 0.366517847371 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.3.9759.1 |
Degree 6 sibling: | 6.0.929428432479.1 |
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: | 5.3.9759.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(3253\) | $\Q_{3253}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |