Normalized defining polynomial
\( x^{18} - 2 x^{17} + 174 x^{16} - 306 x^{15} + 14120 x^{14} - 21646 x^{13} + 698247 x^{12} - 919080 x^{11} + 23121289 x^{10} - 25526162 x^{9} + 530650253 x^{8} - 473795224 x^{7} + 8430968114 x^{6} - 5731976716 x^{5} + 89361791435 x^{4} - 41302878778 x^{3} + 573297458501 x^{2} - 135741604120 x + 1696770051481 \)
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: | \(-60058044833301692798474988224160989184=-\,2^{18}\cdot 3^{9}\cdot 7^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $125.55$ | 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: | $2, 3, 7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1596=2^{2}\cdot 3\cdot 7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1596}(1,·)$, $\chi_{1596}(841,·)$, $\chi_{1596}(587,·)$, $\chi_{1596}(83,·)$, $\chi_{1596}(1429,·)$, $\chi_{1596}(1175,·)$, $\chi_{1596}(923,·)$, $\chi_{1596}(671,·)$, $\chi_{1596}(419,·)$, $\chi_{1596}(169,·)$, $\chi_{1596}(1259,·)$, $\chi_{1596}(1261,·)$, $\chi_{1596}(757,·)$, $\chi_{1596}(503,·)$, $\chi_{1596}(505,·)$, $\chi_{1596}(251,·)$, $\chi_{1596}(253,·)$, $\chi_{1596}(85,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{17} + \frac{3680691950561589480264282898894991124753228683239080607733039260205611}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{16} - \frac{2433920577167324324265470432403901441399986449024383499776748948050488}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{15} + \frac{1656321242593602150670253213398691872597813605087704179587869371567217}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{14} - \frac{4140848308435769116115706782160679897047885583152810099043676318475738}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{13} - \frac{3419692888318670089849639393263394139971950547063646847797223474099150}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{12} + \frac{147267137973964197242832405498824143556155009992052479899463581252349}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{11} - \frac{2334383060024590495733255528186414097684749538198202320285087247492289}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{10} + \frac{2255286830937101487579571124566394096434847201304772350383554919156614}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{9} - \frac{4443085875463562451723434930210965807068039817326243639047864762780496}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{8} + \frac{566263145832087816880392627762388704491679752885968774220907167202488}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{7} - \frac{1414756626724472337177776721526079831276356816834553625599164138727268}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{6} + \frac{4372483252588004389532604846701489775267220150394167736965177060880875}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{5} - \frac{4626779535061220455828193367207913948672365583220654674265879875739769}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{4} + \frac{2059230495700352455516758482511556323643118642065177982027764483748487}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{3} - \frac{1724982198208093564427240138726323121235017089813450144135239734228694}{10548332041476250058015115506975484465401382180857877844857819408959833} a^{2} - \frac{2401710833197593667415252816259222856128862197782418757096068692895908}{10548332041476250058015115506975484465401382180857877844857819408959833} a - \frac{2833795865140859763633911808798399504111252744885324692602617004633162}{10548332041476250058015115506975484465401382180857877844857819408959833}$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{568332}$, which has order $9093312$ (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: | \( 22305.895079162343 \) (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
| \(\Q(\sqrt{-21}) \), 3.3.361.1, 6.0.77241777984.6, \(\Q(\zeta_{19})^+\) |
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 | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | $18$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | $18$ | $18$ | $18$ | $18$ |
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 | ||||||
| 3 | Data not computed | ||||||
| $7$ | 7.6.3.1 | $x^{6} - 14 x^{4} + 49 x^{2} - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 7.6.3.1 | $x^{6} - 14 x^{4} + 49 x^{2} - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.3.1 | $x^{6} - 14 x^{4} + 49 x^{2} - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $19$ | 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |