Normalized defining polynomial
\( x^{10} - 2x^{9} + 6x^{8} + 14x^{6} - 26x^{5} - 4x^{4} - 4x^{3} + 32x^{2} - 12x + 4 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3903069328128\) \(\medspace = -\,2^{8}\cdot 3^{5}\cdot 89^{4}\) | 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: | \(18.16\) | 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: | $2^{4/5}3^{1/2}89^{1/2}\approx 28.44982682753188$ | ||
Ramified primes: | \(2\), \(3\), \(89\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{18}a^{8}+\frac{1}{6}a^{7}-\frac{2}{9}a^{6}+\frac{2}{9}a^{5}+\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{10998}a^{9}-\frac{269}{10998}a^{8}+\frac{953}{10998}a^{7}+\frac{56}{1833}a^{6}+\frac{149}{1222}a^{5}+\frac{301}{1833}a^{4}-\frac{734}{1833}a^{3}+\frac{761}{5499}a^{2}-\frac{310}{1833}a+\frac{1460}{5499}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{119}{1833} a^{9} + \frac{239}{1833} a^{8} - \frac{1355}{3666} a^{7} + \frac{73}{3666} a^{6} - \frac{3271}{3666} a^{5} + \frac{3824}{1833} a^{4} + \frac{1060}{1833} a^{3} + \frac{320}{611} a^{2} - \frac{5341}{1833} a + \frac{2012}{1833} \) (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)
| |
Fundamental units: | $\frac{113}{1222}a^{9}-\frac{1067}{10998}a^{8}+\frac{1681}{3666}a^{7}+\frac{5051}{10998}a^{6}+\frac{8882}{5499}a^{5}-\frac{4259}{5499}a^{4}-\frac{6229}{5499}a^{3}-\frac{13031}{5499}a^{2}+\frac{3073}{5499}a-\frac{2965}{5499}$, $\frac{10}{5499}a^{9}-\frac{82}{1833}a^{8}+\frac{365}{5499}a^{7}-\frac{917}{5499}a^{6}-\frac{3119}{10998}a^{5}-\frac{881}{5499}a^{4}-\frac{1270}{5499}a^{3}+\frac{10943}{5499}a^{2}-\frac{8213}{5499}a+\frac{772}{1833}$, $\frac{593}{10998}a^{9}+\frac{565}{10998}a^{8}+\frac{565}{10998}a^{7}+\frac{4919}{5499}a^{6}+\frac{5650}{5499}a^{5}+\frac{4520}{5499}a^{4}-\frac{17798}{5499}a^{3}-\frac{3955}{1833}a^{2}+\frac{4520}{5499}a+\frac{7325}{5499}$, $\frac{3215}{10998}a^{9}-\frac{3190}{5499}a^{8}+\frac{9641}{5499}a^{7}-\frac{4}{5499}a^{6}+\frac{46535}{10998}a^{5}-\frac{41875}{5499}a^{4}-\frac{6494}{5499}a^{3}-\frac{554}{1833}a^{2}+\frac{51608}{5499}a-\frac{19354}{5499}$ | 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: | \( 291.926530312 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 291.926530312 \cdot 3}{6\cdot\sqrt{3903069328128}}\cr\approx \mathstrut & 0.723502071464 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.3.380208.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.3.380208.1 |
Degree 6 sibling: | 6.0.3421872.4 |
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: | 5.3.380208.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(89\) | 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
89.4.2.2 | $x^{4} - 7298 x^{2} + 23763$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
89.4.2.2 | $x^{4} - 7298 x^{2} + 23763$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |