Normalized defining polynomial
\( x^{19} - 3 x^{18} + 2 x^{17} + 4 x^{16} - 8 x^{15} + 14 x^{13} - 9 x^{12} - 16 x^{11} + 21 x^{10} + \cdots - 1 \)
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: | \(-197294383575054168405587977516\) \(\medspace = -\,2^{2}\cdot 49323595893763542101396994379\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.82\) | 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: | $2^{2/3}49323595893763542101396994379^{1/2}\approx 352544569446363.8$ | ||
Ramified primes: | \(2\), \(49323595893763542101396994379\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-49323\!\cdots\!94379}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a$, $a^{18}-4a^{17}+6a^{16}-2a^{15}-6a^{14}+6a^{13}+8a^{12}-17a^{11}+a^{10}+20a^{9}-12a^{8}-13a^{7}+15a^{6}+a^{5}-5a^{4}-2a^{3}+3a^{2}-1$, $a^{18}-4a^{17}+6a^{16}-2a^{15}-7a^{14}+9a^{13}+5a^{12}-17a^{11}+4a^{10}+20a^{9}-20a^{8}-6a^{7}+19a^{6}-6a^{5}-5a^{4}+a^{3}+3a^{2}-a-1$, $a^{18}-2a^{17}-a^{16}+6a^{15}-4a^{14}-8a^{13}+14a^{12}+5a^{11}-25a^{10}+5a^{9}+29a^{8}-18a^{7}-21a^{6}+17a^{5}+11a^{4}-9a^{3}-7a^{2}+2a+2$, $a^{18}-4a^{17}+6a^{16}-3a^{15}-5a^{14}+11a^{13}-4a^{12}-10a^{11}+13a^{10}-21a^{8}+25a^{7}+5a^{6}-38a^{5}+24a^{4}+18a^{3}-21a^{2}-2a+6$, $a^{17}-2a^{16}+a^{15}+2a^{14}-5a^{13}+2a^{12}+4a^{11}-3a^{10}-3a^{9}+7a^{8}-5a^{7}-2a^{6}+7a^{5}-8a^{4}-3a^{3}+4a^{2}-2a-2$, $3a^{18}-11a^{17}+11a^{16}+12a^{15}-38a^{14}+17a^{13}+52a^{12}-67a^{11}-35a^{10}+116a^{9}-25a^{8}-116a^{7}+82a^{6}+53a^{5}-71a^{4}-3a^{3}+28a^{2}-5a-6$, $a^{18}+2a^{17}-13a^{16}+13a^{15}+10a^{14}-31a^{13}+11a^{12}+48a^{11}-44a^{10}-53a^{9}+74a^{8}+21a^{7}-66a^{6}+7a^{5}+31a^{4}-3a^{3}-16a^{2}+a+4$, $2a^{18}-7a^{17}+3a^{16}+16a^{15}-23a^{14}-7a^{13}+46a^{12}-20a^{11}-64a^{10}+60a^{9}+56a^{8}-85a^{7}-25a^{6}+69a^{5}+13a^{4}-33a^{3}-12a^{2}+12a+6$ (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: | \( 20856736.7017 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 20856736.7017 \cdot 1}{2\cdot\sqrt{197294383575054168405587977516}}\cr\approx \mathstrut & 0.716651167422 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 121645100408832000 |
The 490 conjugacy class representatives for $S_{19}$ are not computed |
Character table for $S_{19}$ is not computed |
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 | $15{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $19$ | $19$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.13.0.1 | $x^{13} + x^{4} + x^{3} + x + 1$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(493\!\cdots\!379\) | $\Q_{49\!\cdots\!79}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |