Normalized defining polynomial
\( x^{18} + 17x^{16} + 120x^{14} + 455x^{12} + 1001x^{10} + 1287x^{8} + 924x^{6} + 330x^{4} + 45x^{2} + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-75613185918270483380568064\) \(\medspace = -\,2^{18}\cdot 19^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.40\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2\cdot 19^{8/9}\approx 27.39680150407798$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(76=2^{2}\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{76}(1,·)$, $\chi_{76}(5,·)$, $\chi_{76}(7,·)$, $\chi_{76}(9,·)$, $\chi_{76}(11,·)$, $\chi_{76}(17,·)$, $\chi_{76}(23,·)$, $\chi_{76}(25,·)$, $\chi_{76}(35,·)$, $\chi_{76}(39,·)$, $\chi_{76}(43,·)$, $\chi_{76}(45,·)$, $\chi_{76}(47,·)$, $\chi_{76}(49,·)$, $\chi_{76}(73,·)$, $\chi_{76}(55,·)$, $\chi_{76}(61,·)$, $\chi_{76}(63,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{256}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{19}$, which has order $19$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( a^{17} + 16 a^{15} + 105 a^{13} + 364 a^{11} + 715 a^{9} + 792 a^{7} + 462 a^{5} + 120 a^{3} + 9 a \) (order $4$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a^{15}+15a^{13}+90a^{11}+275a^{9}+450a^{7}+378a^{5}+140a^{3}+15a$, $a^{12}+12a^{10}+54a^{8}+112a^{6}+105a^{4}+36a^{2}+2$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+451a^{7}+385a^{5}+154a^{3}+21a$, $a^{8}+8a^{6}+20a^{4}+16a^{2}+2$, $a^{13}+13a^{11}+65a^{9}+156a^{7}+182a^{5}+91a^{3}+13a$, $a^{17}+16a^{15}+104a^{13}+352a^{11}+661a^{9}+680a^{7}+356a^{5}+80a^{3}+5a$, $a^{3}+2a$, $a^{5}+5a^{3}+5a$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 22305.8950792 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 19}{4\cdot\sqrt{75613185918270483380568064}}\cr\approx \mathstrut & 0.185965897882 \end{aligned}\]
Galois group
A cyclic group of order 18 |
The 18 conjugacy class representatives for $C_{18}$ |
Character table for $C_{18}$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.3.361.1, 6.0.8340544.1, \(\Q(\zeta_{19})^+\) |
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 | R | $18$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | R | $18$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.1.0.1}{1} }^{18}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.18.115 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4320 x^{14} + 16128 x^{13} + 53696 x^{12} + 165120 x^{11} + 449824 x^{10} + 1006400 x^{9} + 1826368 x^{8} + 2905088 x^{7} + 3317760 x^{6} - 418816 x^{5} - 6684672 x^{4} - 4984832 x^{3} + 2483456 x^{2} + 3566080 x + 1829376$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
\(19\) | 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |