Normalized defining polynomial
\( x^{19} - 2 x^{18} - 3 x^{17} + 9 x^{16} + x^{15} - 16 x^{14} + 5 x^{13} + 17 x^{12} - 10 x^{11} - 12 x^{10} + 10 x^{9} + x^{8} + 3 x^{6} - 9 x^{5} + 2 x^{4} + 6 x^{3} - 5 x^{2} - x + 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-8969938904536782790298366089328\) \(\medspace = -\,2^{4}\cdot 3947\cdot 1431584761\cdot 99216821342546749\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(42.57\) | 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: | not computed | ||
Ramified primes: | \(2\), \(3947\), \(1431584761\), \(99216821342546749\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-56062\!\cdots\!80583}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $11$ | 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: | $a$, $a^{3}-a+1$, $a^{18}-3a^{17}+9a^{15}-8a^{14}-8a^{13}+13a^{12}+4a^{11}-14a^{10}+2a^{9}+8a^{8}-7a^{7}+7a^{6}-4a^{5}-5a^{4}+6a^{3}+a^{2}-5a+2$, $2a^{18}-3a^{17}-8a^{16}+15a^{15}+11a^{14}-31a^{13}-6a^{12}+39a^{11}-3a^{10}-34a^{9}+8a^{8}+12a^{7}+a^{6}+6a^{5}-15a^{4}-4a^{3}+13a^{2}-5a-5$, $a^{18}-2a^{17}-3a^{16}+9a^{15}+a^{14}-16a^{13}+6a^{12}+15a^{11}-11a^{10}-7a^{9}+7a^{8}-2a^{7}+4a^{6}+2a^{5}-9a^{4}+3a^{3}+3a^{2}-4a$, $a^{18}-2a^{16}-2a^{15}+a^{14}+8a^{13}-4a^{12}-12a^{11}+6a^{10}+11a^{9}-7a^{8}-12a^{7}+2a^{6}-2a^{5}+5a^{4}+2a^{3}-6a^{2}+5a+3$, $2a^{18}-4a^{17}-5a^{16}+14a^{15}+a^{14}-17a^{13}+2a^{12}+14a^{11}-2a^{10}-7a^{9}+a^{8}-8a^{7}+15a^{6}+5a^{5}-8a^{4}+3a^{3}-a^{2}+2a+2$, $a^{9}-a^{8}-3a^{7}+3a^{6}+2a^{5}-a^{4}-3a^{2}+1$, $4a^{18}-6a^{17}-16a^{16}+28a^{15}+25a^{14}-53a^{13}-25a^{12}+61a^{11}+18a^{10}-48a^{9}-13a^{8}+7a^{7}+25a^{6}+24a^{5}-33a^{4}-22a^{3}+12a^{2}+a$, $2a^{18}-6a^{17}-a^{16}+22a^{15}-20a^{14}-22a^{13}+42a^{12}+3a^{11}-42a^{10}+10a^{9}+31a^{8}-26a^{7}+10a^{6}-21a^{4}+33a^{3}-13a^{2}-10a+5$, $5a^{18}-3a^{17}-20a^{16}+17a^{15}+31a^{14}-37a^{13}-30a^{12}+46a^{11}+17a^{10}-41a^{9}-5a^{8}+a^{7}-a^{6}+18a^{5}-16a^{4}-13a^{3}+17a^{2}-5$ (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: | \( 363184844.905 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{7}\cdot 363184844.905 \cdot 1}{2\cdot\sqrt{8969938904536782790298366089328}}\cr\approx \mathstrut & 0.750087687179 \end{aligned}\] (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 | $19$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $19$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $15{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
2.13.0.1 | $x^{13} + x^{4} + x^{3} + x + 1$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(3947\) | $\Q_{3947}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(1431584761\) | 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(99216821342546749\) | $\Q_{99216821342546749}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{99216821342546749}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |