Normalized defining polynomial
\( x^{18} + 221 x^{16} + 18798 x^{14} + 773383 x^{12} + 15825836 x^{10} + 146193112 x^{8} + 451182511 x^{6} + 175757803 x^{4} + 11193715 x^{2} + 107653 \)
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: | \(-3870328444867146736146206444709275111784448=-\,2^{18}\cdot 13^{15}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $232.27$ | 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: | $2, 13, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(988=2^{2}\cdot 13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{988}(1,·)$, $\chi_{988}(199,·)$, $\chi_{988}(9,·)$, $\chi_{988}(311,·)$, $\chi_{988}(81,·)$, $\chi_{988}(467,·)$, $\chi_{988}(251,·)$, $\chi_{988}(729,·)$, $\chi_{988}(283,·)$, $\chi_{988}(803,·)$, $\chi_{988}(549,·)$, $\chi_{988}(491,·)$, $\chi_{988}(885,·)$, $\chi_{988}(823,·)$, $\chi_{988}(633,·)$, $\chi_{988}(571,·)$, $\chi_{988}(61,·)$, $\chi_{988}(757,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{91} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{2}$, $\frac{1}{91} a^{7} + \frac{1}{7} a$, $\frac{1}{91} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{91} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7007} a^{10} - \frac{3}{637} a^{8} - \frac{1}{637} a^{6} + \frac{3}{49} a^{4} + \frac{19}{49} a^{2} + \frac{1}{11}$, $\frac{1}{7007} a^{11} - \frac{3}{637} a^{9} - \frac{1}{637} a^{7} + \frac{3}{49} a^{5} + \frac{19}{49} a^{3} + \frac{1}{11} a$, $\frac{1}{637637} a^{12} + \frac{2}{49049} a^{10} + \frac{3}{4459} a^{8} + \frac{20}{4459} a^{6} - \frac{23}{49} a^{4} + \frac{360}{3773} a^{2} - \frac{31}{77}$, $\frac{1}{637637} a^{13} + \frac{2}{49049} a^{11} + \frac{3}{4459} a^{9} + \frac{20}{4459} a^{7} - \frac{2}{49} a^{5} - \frac{1796}{3773} a^{3} + \frac{2}{77} a$, $\frac{1}{49098049} a^{14} - \frac{4}{49098049} a^{12} + \frac{29}{3776773} a^{10} + \frac{232}{49049} a^{8} + \frac{1024}{343343} a^{6} - \frac{69171}{290521} a^{4} + \frac{12321}{290521} a^{2} - \frac{2864}{5929}$, $\frac{1}{49098049} a^{15} - \frac{4}{49098049} a^{13} + \frac{29}{3776773} a^{11} + \frac{232}{49049} a^{9} + \frac{1024}{343343} a^{7} + \frac{13835}{290521} a^{5} + \frac{95327}{290521} a^{3} - \frac{1170}{5929} a$, $\frac{1}{456292767479549} a^{16} + \frac{2616564}{456292767479549} a^{14} - \frac{1348807}{41481160679959} a^{12} - \frac{1518385432}{35099443652273} a^{10} - \frac{12262518952}{3190858513843} a^{8} + \frac{15116022357}{5014206236039} a^{6} - \frac{234306125519}{2699957204021} a^{4} - \frac{37541932345}{245450654911} a^{2} - \frac{3844424433}{55101167429}$, $\frac{1}{456292767479549} a^{17} + \frac{2616564}{456292767479549} a^{15} - \frac{1348807}{41481160679959} a^{13} - \frac{1518385432}{35099443652273} a^{11} - \frac{12262518952}{3190858513843} a^{9} + \frac{15116022357}{5014206236039} a^{7} + \frac{151402046484}{2699957204021} a^{5} - \frac{2477553072}{245450654911} a^{3} + \frac{4027170914}{55101167429} a$
Class group and class number
$C_{3}\times C_{9}\times C_{909594}$, which has order $24559038$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 107879432.94074221 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-13}) \), 3.3.361.1, 6.0.18324175168.4, 9.9.81976414938366169.1 |
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 | R | $18$ | $18$ | ${\href{/LocalNumberField/7.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | $18$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | $18$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 13 | Data not computed | ||||||
| $19$ | 19.9.8.3 | $x^{9} - 77824$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.3 | $x^{9} - 77824$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |