Normalized defining polynomial
\( x^{19} - 6 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-200928214500803951615477553772817547264\) \(\medspace = -\,2^{18}\cdot 3^{18}\cdot 19^{19}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(103.74\) | 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: | $2^{18/19}3^{18/19}19^{359/342}\approx 120.09178438864606$ | ||
Ramified primes: | \(2\), \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | 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: | $a^{10}-a^{9}+a^{8}-a^{6}+a^{5}-a^{3}+a^{2}-a-1$, $3a^{18}-4a^{17}-3a^{16}+4a^{15}-8a^{14}-a^{13}+a^{12}-12a^{11}+5a^{10}-3a^{9}-12a^{8}+10a^{7}-16a^{6}-7a^{5}+10a^{4}-27a^{3}+9a^{2}-5a-31$, $4a^{18}-3a^{17}-a^{15}+3a^{14}+2a^{13}-5a^{12}+2a^{11}-5a^{10}+13a^{9}-9a^{8}+7a^{7}-17a^{6}+19a^{5}-11a^{4}+20a^{3}-28a^{2}+17a-11$, $11a^{18}+16a^{17}+12a^{16}-3a^{15}-16a^{14}-18a^{13}-14a^{12}-8a^{11}-2a^{10}+16a^{9}+34a^{8}+40a^{7}+10a^{6}-23a^{5}-48a^{4}-38a^{3}-29a^{2}-12a+11$, $2a^{18}-3a^{17}+8a^{16}+5a^{15}+a^{14}-2a^{13}+22a^{12}-11a^{11}+5a^{10}+24a^{9}-11a^{8}+9a^{7}+17a^{6}+11a^{5}-12a^{4}+31a^{3}+25a^{2}-36a+61$, $5a^{18}-5a^{16}-2a^{15}+6a^{14}+7a^{13}-2a^{12}-12a^{11}+2a^{10}+9a^{9}+7a^{8}-5a^{7}-21a^{6}+9a^{5}+15a^{4}+a^{3}-15a^{2}-26a+19$, $3a^{18}-a^{17}-a^{16}-2a^{15}+8a^{13}-9a^{12}+9a^{11}-18a^{10}+22a^{9}-14a^{8}+13a^{7}-18a^{6}+6a^{5}+5a^{4}+5a^{3}-5a^{2}-8a-1$, $185a^{18}-126a^{17}+22a^{16}+110a^{15}-242a^{14}+335a^{13}-352a^{12}+274a^{11}-109a^{10}-119a^{9}+365a^{8}-564a^{7}+644a^{6}-561a^{5}+313a^{4}+68a^{3}-513a^{2}+918a-1151$, $204a^{18}+141a^{17}+23a^{16}-68a^{15}-122a^{14}-111a^{13}-152a^{12}-165a^{11}-118a^{10}+76a^{9}+403a^{8}+571a^{7}+469a^{6}-80a^{5}-709a^{4}-1028a^{3}-772a^{2}+81a+841$ (assuming GRH) | 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: | \( 929704933317 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 929704933317 \cdot 1}{2\cdot\sqrt{200928214500803951615477553772817547264}}\cr\approx \mathstrut & 1.00102169086913 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 342 |
The 19 conjugacy class representatives for $F_{19}$ |
Character table for $F_{19}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | ${\href{/padicField/5.9.0.1}{9} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.3.0.1}{3} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.19.18.1 | $x^{19} + 2$ | $19$ | $1$ | $18$ | $F_{19}$ | $[\ ]_{19}^{18}$ |
\(3\) | 3.19.18.1 | $x^{19} + 3$ | $19$ | $1$ | $18$ | $F_{19}$ | $[\ ]_{19}^{18}$ |
\(19\) | 19.19.19.10 | $x^{19} + 19 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ |