Normalized defining polynomial
\( x^{18} - 4 x^{16} - 6 x^{14} + 64 x^{12} - 145 x^{10} + 279 x^{8} - 446 x^{6} + 206 x^{4} + 341 x^{2} + 335 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-53137762492763568359375=-\,5^{9}\cdot 67^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.30$ | 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: | $5, 67$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{10} + \frac{3}{10} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} + \frac{3}{10} a^{7} - \frac{1}{2} a^{6} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{12} + \frac{3}{10} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{2}{5} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1150} a^{14} + \frac{3}{1150} a^{12} - \frac{8}{575} a^{10} - \frac{141}{1150} a^{8} - \frac{1}{2} a^{7} + \frac{257}{575} a^{6} - \frac{1}{2} a^{5} + \frac{99}{575} a^{4} - \frac{1}{2} a^{3} + \frac{531}{1150} a^{2} + \frac{17}{230}$, $\frac{1}{1150} a^{15} + \frac{3}{1150} a^{13} - \frac{8}{575} a^{11} - \frac{141}{1150} a^{9} - \frac{1}{2} a^{8} + \frac{257}{575} a^{7} - \frac{1}{2} a^{6} + \frac{99}{575} a^{5} - \frac{1}{2} a^{4} + \frac{531}{1150} a^{3} + \frac{17}{230} a$, $\frac{1}{145308250} a^{16} - \frac{5383}{145308250} a^{14} - \frac{5160699}{145308250} a^{12} + \frac{489246}{14530825} a^{10} + \frac{351989}{785450} a^{8} - \frac{46957531}{145308250} a^{6} - \frac{1}{2} a^{5} - \frac{1613297}{145308250} a^{4} - \frac{1}{2} a^{3} - \frac{4806628}{72654125} a^{2} - \frac{215877}{29061650}$, $\frac{1}{145308250} a^{17} - \frac{5383}{145308250} a^{15} - \frac{5160699}{145308250} a^{13} + \frac{489246}{14530825} a^{11} - \frac{20368}{392725} a^{9} - \frac{46957531}{145308250} a^{7} - \frac{1}{2} a^{6} + \frac{35520414}{72654125} a^{5} - \frac{4806628}{72654125} a^{3} + \frac{7157474}{14530825} a - \frac{1}{2}$
Class group and class number
$C_{2}$, which has order $2$
Unit group
| Rank: | $8$ | 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: | \( 14588.8980085 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 18 |
| The 6 conjugacy class representatives for $D_9$ |
| Character table for $D_9$ |
Intermediate fields
| \(\Q(\sqrt{-335}) \), 3.1.335.1 x3, 6.0.37595375.1, 9.1.12594450625.1 x9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 9 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $67$ | 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |