Normalized defining polynomial
\( x^{19} + 8x - 8 \)
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: | \(-601146240601341529876691943424\) \(\medspace = -\,2^{18}\cdot 46553428447\cdot 49259334849493\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(36.92\) | 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: | $2^{18/19}46553428447^{1/2}49259334849493^{1/2}\approx 2920158458207.675$ | ||
Ramified primes: | \(2\), \(46553428447\), \(49259334849493\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-22931\!\cdots\!27371}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{4}a^{13}$, $\frac{1}{4}a^{14}$, $\frac{1}{4}a^{15}$, $\frac{1}{4}a^{16}$, $\frac{1}{4}a^{17}$, $\frac{1}{4}a^{18}$
Monogenic: | Not computed | |
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: | $\frac{1}{4}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+3$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+3$, $\frac{1}{2}a^{8}+a^{4}+1$, $\frac{1}{4}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{4}a^{14}-1$, $\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{3}{4}a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+a^{7}+a^{2}+5$, $\frac{1}{4}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{12}+a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+a^{5}+a^{4}-a^{3}-2a^{2}+a+3$, $\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+a^{6}-a^{4}-a^{2}+a+1$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{9}+\frac{1}{2}a^{7}+a^{3}-a+1$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{2}a^{7}+a^{5}-a^{4}+a^{3}-a^{2}+a+1$ (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: | \( 28122221.8245 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 28122221.8245 \cdot 1}{2\cdot\sqrt{601146240601341529876691943424}}\cr\approx \mathstrut & 0.553577432468 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 121645100408832000 |
The 490 conjugacy class representatives for $S_{19}$ |
Character table for $S_{19}$ |
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 | $16{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | $15{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{19} + 2$ | $19$ | $1$ | $18$ | $F_{19}$ | $[\ ]_{19}^{18}$ |
\(46553428447\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(49259334849493\) | $\Q_{49259334849493}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |