Normalized defining polynomial
\( x^{10} - 4x^{9} + 2x^{8} + 8x^{7} - 4x^{6} - 21x^{5} + 48x^{4} - 58x^{3} + 24x^{2} - 4x - 1 \)
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: | \(276654003125\) \(\medspace = 5^{5}\cdot 97^{4}\) | 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: | \(13.94\) | 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: | $5^{1/2}97^{2/3}\approx 47.206248873643986$ | ||
Ramified primes: | \(5\), \(97\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{6}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{4}-\frac{1}{5}a^{2}$, $\frac{1}{135}a^{9}+\frac{1}{15}a^{8}+\frac{11}{135}a^{7}-\frac{11}{135}a^{6}-\frac{4}{45}a^{5}+\frac{4}{45}a^{4}+\frac{14}{45}a^{3}-\frac{52}{135}a^{2}-\frac{31}{135}a-\frac{29}{135}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{22}{135}a^{9}-\frac{11}{15}a^{8}+\frac{16}{27}a^{7}+\frac{38}{27}a^{6}-\frac{61}{45}a^{5}-\frac{173}{45}a^{4}+\frac{434}{45}a^{3}-\frac{1549}{135}a^{2}+\frac{749}{135}a+\frac{37}{135}$, $\frac{11}{135}a^{9}-\frac{4}{15}a^{8}-\frac{14}{135}a^{7}+\frac{122}{135}a^{6}+\frac{2}{9}a^{5}-\frac{109}{45}a^{4}+\frac{20}{9}a^{3}-\frac{59}{135}a^{2}-\frac{233}{135}a+\frac{167}{135}$, $\frac{26}{135}a^{9}-\frac{7}{15}a^{8}-\frac{92}{135}a^{7}+\frac{227}{135}a^{6}+\frac{76}{45}a^{5}-\frac{193}{45}a^{4}+\frac{121}{45}a^{3}+\frac{79}{135}a^{2}-\frac{968}{135}a+\frac{218}{135}$, $\frac{11}{135}a^{9}-\frac{7}{15}a^{8}+\frac{94}{135}a^{7}+\frac{68}{135}a^{6}-\frac{71}{45}a^{5}-\frac{64}{45}a^{4}+\frac{334}{45}a^{3}-\frac{298}{27}a^{2}+\frac{1117}{135}a-\frac{53}{27}$, $\frac{22}{135}a^{9}-\frac{11}{15}a^{8}+\frac{16}{27}a^{7}+\frac{163}{135}a^{6}-\frac{52}{45}a^{5}-\frac{146}{45}a^{4}+\frac{85}{9}a^{3}-\frac{1711}{135}a^{2}+\frac{992}{135}a-\frac{287}{135}$ | 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: | \( 69.6672028353 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 69.6672028353 \cdot 1}{2\cdot\sqrt{276654003125}}\cr\approx \mathstrut & 0.412866231802 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.47045.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.47045.1 |
Degree 6 sibling: | 6.2.11066160125.1 |
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: | 5.1.47045.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(97\) | 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.6.4.3 | $x^{6} + 8536 x^{3} - 3415467$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |