Normalized defining polynomial
\( x^{18} - x^{17} + 20 x^{16} - 20 x^{15} + 172 x^{14} - 172 x^{13} + 837 x^{12} - 837 x^{11} + 2566 x^{10} - 2566 x^{9} + 5283 x^{8} - 5283 x^{7} + 7791 x^{6} - 7791 x^{5} + 9045 x^{4} - 9045 x^{3} + 9330 x^{2} - 9330 x + 9349 \)
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: | \(-10703880581610941769412109375=-\,5^{9}\cdot 19^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.07$ | 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: | $5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(95=5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{95}(1,·)$, $\chi_{95}(66,·)$, $\chi_{95}(69,·)$, $\chi_{95}(6,·)$, $\chi_{95}(11,·)$, $\chi_{95}(14,·)$, $\chi_{95}(79,·)$, $\chi_{95}(16,·)$, $\chi_{95}(81,·)$, $\chi_{95}(84,·)$, $\chi_{95}(89,·)$, $\chi_{95}(26,·)$, $\chi_{95}(29,·)$, $\chi_{95}(94,·)$, $\chi_{95}(34,·)$, $\chi_{95}(36,·)$, $\chi_{95}(59,·)$, $\chi_{95}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4181} a^{10} + \frac{1597}{4181} a^{9} + \frac{10}{4181} a^{8} + \frac{1830}{4181} a^{7} + \frac{35}{4181} a^{6} + \frac{1309}{4181} a^{5} + \frac{50}{4181} a^{4} + \frac{1919}{4181} a^{3} + \frac{25}{4181} a^{2} + \frac{1830}{4181} a + \frac{2}{4181}$, $\frac{1}{4181} a^{11} + \frac{11}{4181} a^{9} - \frac{1597}{4181} a^{8} + \frac{44}{4181} a^{7} - \frac{233}{4181} a^{6} + \frac{77}{4181} a^{5} + \frac{1508}{4181} a^{4} + \frac{55}{4181} a^{3} - \frac{466}{4181} a^{2} + \frac{11}{4181} a + \frac{987}{4181}$, $\frac{1}{4181} a^{12} + \frac{1741}{4181} a^{9} - \frac{66}{4181} a^{8} + \frac{542}{4181} a^{7} - \frac{308}{4181} a^{6} - \frac{348}{4181} a^{5} - \frac{495}{4181} a^{4} - \frac{670}{4181} a^{3} - \frac{264}{4181} a^{2} + \frac{1762}{4181} a - \frac{22}{4181}$, $\frac{1}{4181} a^{13} - \frac{78}{4181} a^{9} - \frac{144}{4181} a^{8} - \frac{416}{4181} a^{7} + \frac{1432}{4181} a^{6} - \frac{819}{4181} a^{5} + \frac{81}{4181} a^{4} - \frac{624}{4181} a^{3} + \frac{47}{4181} a^{2} - \frac{130}{4181} a + \frac{699}{4181}$, $\frac{1}{4181} a^{14} - \frac{1008}{4181} a^{9} + \frac{364}{4181} a^{8} + \frac{2018}{4181} a^{7} + \frac{1911}{4181} a^{6} + \frac{1839}{4181} a^{5} - \frac{905}{4181} a^{4} - \frac{787}{4181} a^{3} + \frac{1820}{4181} a^{2} + \frac{1285}{4181} a + \frac{156}{4181}$, $\frac{1}{4181} a^{15} + \frac{455}{4181} a^{9} - \frac{445}{4181} a^{8} - \frac{1451}{4181} a^{7} - \frac{510}{4181} a^{6} + \frac{1552}{4181} a^{5} - \frac{559}{4181} a^{4} + \frac{369}{4181} a^{3} + \frac{1399}{4181} a^{2} + \frac{975}{4181} a + \frac{2016}{4181}$, $\frac{1}{4181} a^{16} + \frac{414}{4181} a^{9} - \frac{1820}{4181} a^{8} - \frac{1141}{4181} a^{7} - \frac{1830}{4181} a^{6} + \frac{1729}{4181} a^{5} - \frac{1476}{4181} a^{4} + \frac{2083}{4181} a^{3} - \frac{2038}{4181} a^{2} + \frac{1385}{4181} a - \frac{910}{4181}$, $\frac{1}{4181} a^{17} + \frac{1801}{4181} a^{9} - \frac{1100}{4181} a^{8} + \frac{1492}{4181} a^{7} - \frac{218}{4181} a^{6} + \frac{128}{4181} a^{5} - \frac{1893}{4181} a^{4} + \frac{2067}{4181} a^{3} - \frac{603}{4181} a^{2} - \frac{1769}{4181} a - \frac{828}{4181}$
Class group and class number
$C_{152}$, which has order $152$ (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: | \( 22305.8950792 \) (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
| \(\Q(\sqrt{-95}) \), 3.3.361.1, 6.0.309512375.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | $18$ | R | $18$ | $18$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | $18$ | $18$ | $18$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 19 | Data not computed | ||||||