Normalized defining polynomial
\( x^{18} - 7 x^{17} - 32 x^{16} + 294 x^{15} + 275 x^{14} - 4779 x^{13} + 1076 x^{12} + 37663 x^{11} - 29140 x^{10} - 146783 x^{9} + 161417 x^{8} + 251979 x^{7} - 350891 x^{6} - 118682 x^{5} + 269024 x^{4} - 41471 x^{3} - 43388 x^{2} + 12154 x - 191 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(34205654728777159191037355893457=17^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $56.48$ | 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: | $17, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(323=17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{323}(256,·)$, $\chi_{323}(1,·)$, $\chi_{323}(137,·)$, $\chi_{323}(271,·)$, $\chi_{323}(16,·)$, $\chi_{323}(273,·)$, $\chi_{323}(220,·)$, $\chi_{323}(290,·)$, $\chi_{323}(35,·)$, $\chi_{323}(101,·)$, $\chi_{323}(169,·)$, $\chi_{323}(237,·)$, $\chi_{323}(239,·)$, $\chi_{323}(305,·)$, $\chi_{323}(118,·)$, $\chi_{323}(120,·)$, $\chi_{323}(188,·)$, $\chi_{323}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{269705256860012329346369815362015263507} a^{17} - \frac{28590763442167704613012469214857821881}{269705256860012329346369815362015263507} a^{16} + \frac{71231950767800284092403172530807533808}{269705256860012329346369815362015263507} a^{15} + \frac{81961738872427785118660556317327184161}{269705256860012329346369815362015263507} a^{14} + \frac{12022051914415913023482402472646881123}{269705256860012329346369815362015263507} a^{13} - \frac{100729246868692134591493974624470119070}{269705256860012329346369815362015263507} a^{12} + \frac{102666630009180058782614132465763512351}{269705256860012329346369815362015263507} a^{11} - \frac{33805429253479703256034304206346512814}{269705256860012329346369815362015263507} a^{10} + \frac{127073179197050800207019306490450354772}{269705256860012329346369815362015263507} a^{9} - \frac{54976270187186864640794176490203682574}{269705256860012329346369815362015263507} a^{8} + \frac{35842865233268536189555421090361461851}{269705256860012329346369815362015263507} a^{7} + \frac{77265226650497830818160536283080664973}{269705256860012329346369815362015263507} a^{6} - \frac{124815737422564347309619506230600546859}{269705256860012329346369815362015263507} a^{5} - \frac{57489881253853958083136305761823179348}{269705256860012329346369815362015263507} a^{4} + \frac{30247942716775517275481898410465084322}{269705256860012329346369815362015263507} a^{3} + \frac{849293054802072599554422953383883098}{1786127528874253836730925929549769957} a^{2} - \frac{75988503471245088210278576440192005494}{269705256860012329346369815362015263507} a + \frac{66787555508311009042976650510072229808}{269705256860012329346369815362015263507}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 6552525312.42 \) (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
| \(\Q(\sqrt{17}) \), 3.3.361.1, 6.6.640267073.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | R | R | $18$ | $18$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 17 | Data not computed | ||||||
| $19$ | 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |