Normalized defining polynomial
\( x^{10} - x^{9} - x^{8} + 2x^{7} - 3x^{6} + 5x^{5} - x^{4} - 42x^{3} + 5x^{2} + 61x + 19 \)
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)
| |
Discriminant: | \(12602368800000\) \(\medspace = 2^{8}\cdot 3^{8}\cdot 5^{5}\cdot 7^{4}\) | 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: | \(20.42\) | 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: | $2^{4/5}3^{7/6}5^{1/2}7^{1/2}\approx 37.11072873249983$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{29733}a^{9}-\frac{3569}{29733}a^{8}-\frac{1444}{29733}a^{7}-\frac{11437}{29733}a^{6}-\frac{2095}{9911}a^{5}+\frac{6203}{29733}a^{4}-\frac{1042}{29733}a^{3}-\frac{8722}{29733}a^{2}-\frac{439}{29733}a+\frac{475}{29733}$
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)
| |
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)
| |
Fundamental units: | $\frac{1414}{29733}a^{9}-\frac{1867}{29733}a^{8}-\frac{50}{9911}a^{7}+\frac{2834}{29733}a^{6}-\frac{6734}{29733}a^{5}+\frac{9718}{29733}a^{4}-\frac{6560}{29733}a^{3}-\frac{21030}{9911}a^{2}+\frac{33380}{29733}a+\frac{37346}{29733}$, $\frac{1414}{29733}a^{9}-\frac{1867}{29733}a^{8}-\frac{50}{9911}a^{7}+\frac{2834}{29733}a^{6}-\frac{6734}{29733}a^{5}+\frac{9718}{29733}a^{4}-\frac{6560}{29733}a^{3}-\frac{21030}{9911}a^{2}+\frac{63113}{29733}a+\frac{37346}{29733}$, $\frac{1120}{29733}a^{9}-\frac{1049}{9911}a^{8}-\frac{1787}{29733}a^{7}+\frac{5483}{29733}a^{6}-\frac{12301}{29733}a^{5}+\frac{3220}{9911}a^{4}-\frac{5788}{9911}a^{3}-\frac{65771}{29733}a^{2}+\frac{102980}{29733}a+\frac{41883}{9911}$, $\frac{5171}{29733}a^{9}-\frac{10928}{29733}a^{8}+\frac{1990}{9911}a^{7}-\frac{1790}{29733}a^{6}-\frac{21388}{29733}a^{5}+\frac{43361}{29733}a^{4}-\frac{46153}{29733}a^{3}-\frac{45085}{9911}a^{2}+\frac{197770}{29733}a+\frac{87496}{29733}$, $\frac{1469}{29733}a^{9}+\frac{58}{29733}a^{8}-\frac{94}{9911}a^{7}-\frac{1808}{29733}a^{6}-\frac{5524}{29733}a^{5}+\frac{3998}{29733}a^{4}+\frac{5507}{29733}a^{3}-\frac{5839}{9911}a^{2}-\frac{20498}{29733}a-\frac{5906}{29733}$ | 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: | \( 732.657279074 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 732.657279074 \cdot 1}{2\cdot\sqrt{12602368800000}}\cr\approx \mathstrut & 0.643315840787 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.317520.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.317520.3 |
Degree 6 sibling: | data not computed |
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: | 5.1.317520.3 |
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 | R | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |