Normalized defining polynomial
\( x^{19} + 4x - 4 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-22395541656295175507680165888\) \(\medspace = -\,2^{18}\cdot 1699471\cdot 109607929\cdot 458633653\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(31.05\) | 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))
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Galois root discriminant: | $2^{18/19}1699471^{1/2}109607929^{1/2}458633653^{1/2}\approx 563633938950.1552$ | ||
Ramified primes: | \(2\), \(1699471\), \(109607929\), \(458633653\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{-85432\!\cdots\!77027}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{10}a^{18}-\frac{1}{5}a^{17}-\frac{1}{10}a^{16}+\frac{1}{5}a^{15}+\frac{1}{10}a^{14}-\frac{1}{5}a^{13}-\frac{1}{10}a^{12}+\frac{1}{5}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$
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)
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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)
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Fundamental units: | $a-1$, $\frac{1}{5}a^{18}+\frac{1}{10}a^{17}-\frac{1}{5}a^{16}-\frac{1}{10}a^{15}-\frac{3}{10}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{10}a^{11}+\frac{1}{5}a^{10}+\frac{3}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{3}{5}a^{3}-\frac{4}{5}a^{2}+\frac{3}{5}a+\frac{3}{5}$, $\frac{1}{10}a^{18}-\frac{1}{5}a^{17}-\frac{3}{5}a^{16}-\frac{3}{10}a^{15}+\frac{1}{10}a^{14}-\frac{1}{5}a^{13}-\frac{3}{5}a^{12}-\frac{3}{10}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}-\frac{3}{5}a^{8}+\frac{1}{5}a^{7}+\frac{3}{5}a^{6}-\frac{1}{5}a^{5}-\frac{3}{5}a^{4}+\frac{1}{5}a^{3}+\frac{3}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{10}a^{18}+\frac{3}{10}a^{17}+\frac{2}{5}a^{16}+\frac{1}{5}a^{15}+\frac{1}{10}a^{14}-\frac{1}{5}a^{13}-\frac{1}{10}a^{12}+\frac{1}{5}a^{11}+\frac{3}{5}a^{10}+\frac{4}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{6}{5}a^{3}+\frac{3}{5}a^{2}+\frac{4}{5}a-\frac{1}{5}$, $\frac{3}{10}a^{18}+\frac{2}{5}a^{17}+\frac{1}{5}a^{16}+\frac{1}{10}a^{15}+\frac{3}{10}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{1}{10}a^{11}+\frac{3}{10}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{3}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{3}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{7}{5}$, $\frac{1}{10}a^{18}+\frac{3}{10}a^{17}+\frac{2}{5}a^{16}+\frac{1}{5}a^{15}+\frac{1}{10}a^{14}-\frac{1}{5}a^{13}-\frac{1}{10}a^{12}-\frac{3}{10}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{3}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{3}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{10}a^{18}-\frac{1}{5}a^{17}+\frac{2}{5}a^{16}-\frac{3}{10}a^{15}+\frac{1}{10}a^{14}+\frac{3}{10}a^{13}-\frac{1}{10}a^{12}+\frac{1}{5}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{4}{5}a^{5}-\frac{3}{5}a^{4}+\frac{1}{5}a^{3}+\frac{3}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{3}{10}a^{18}-\frac{1}{10}a^{17}+\frac{1}{5}a^{16}+\frac{3}{5}a^{15}+\frac{3}{10}a^{14}-\frac{1}{10}a^{13}+\frac{1}{5}a^{12}+\frac{3}{5}a^{11}+\frac{3}{10}a^{10}-\frac{3}{5}a^{9}+\frac{1}{5}a^{8}+\frac{3}{5}a^{7}-\frac{1}{5}a^{6}-\frac{3}{5}a^{5}+\frac{1}{5}a^{4}+\frac{8}{5}a^{3}-\frac{1}{5}a^{2}-\frac{3}{5}a+\frac{7}{5}$, $\frac{3}{10}a^{18}+\frac{2}{5}a^{17}+\frac{1}{5}a^{16}+\frac{1}{10}a^{15}+\frac{3}{10}a^{14}-\frac{1}{10}a^{13}-\frac{3}{10}a^{12}+\frac{1}{10}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{4}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{3}{5}a^{3}-\frac{6}{5}a^{2}-\frac{3}{5}a+\frac{7}{5}$ (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)]
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Regulator: | \( 8836400.64766 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 8836400.64766 \cdot 1}{2\cdot\sqrt{22395541656295175507680165888}}\cr\approx \mathstrut & 0.901183909135 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $18{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.19.18.1 | $x^{19} + 2$ | $19$ | $1$ | $18$ | $F_{19}$ | $[\ ]_{19}^{18}$ |
\(1699471\) | $\Q_{1699471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(109607929\) | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(458633653\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |