Normalized defining polynomial
\( x^{10} - 3 x^{9} + 14 x^{8} - 28 x^{7} + 130 x^{6} - 192 x^{5} + 706 x^{4} - 677 x^{3} + 2494 x^{2} - 1125 x + 4531 \)
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-162778775259375=-\,3^{5}\cdot 5^{5}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $26.37$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(165=3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(91,·)$, $\chi_{165}(136,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(104,·)$, $\chi_{165}(119,·)$, $\chi_{165}(89,·)$, $\chi_{165}(59,·)$, $\chi_{165}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6220541566471} a^{9} - \frac{896799671201}{6220541566471} a^{8} - \frac{1618887395821}{6220541566471} a^{7} + \frac{1175028657309}{6220541566471} a^{6} - \frac{673304891324}{6220541566471} a^{5} - \frac{1932629225050}{6220541566471} a^{4} - \frac{458038938915}{6220541566471} a^{3} + \frac{1192056148340}{6220541566471} a^{2} - \frac{450043259933}{6220541566471} a + \frac{117256218066}{270458328977}$
Class group and class number
$C_{22}$, which has order $22$
Unit group
| Rank: | $4$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 26.1711060094 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 10 |
| The 10 conjugacy class representatives for $C_{10}$ |
| Character table for $C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ | R | R | ${\href{/LocalNumberField/7.10.0.1}{10} }$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.10.5.1 | $x^{10} - 18 x^{6} + 81 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| $5$ | 5.10.5.2 | $x^{10} - 625 x^{2} + 6250$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |