Normalized defining polynomial
\( x^{19} - 9x - 9 \)
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: | \(-629287868530805907924458261644707\) \(\medspace = -\,3^{18}\cdot 125471\cdot 142573\cdot 90800058249161\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(53.24\) | 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: | $3^{18/19}125471^{1/2}142573^{1/2}90800058249161^{1/2}\approx 3608635332174.5854$ | ||
Ramified primes: | \(3\), \(125471\), \(142573\), \(90800058249161\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-16243\!\cdots\!45163}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$, $\frac{1}{3}a^{16}$, $\frac{1}{3}a^{17}$, $\frac{1}{3}a^{18}$
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: | $\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-2$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-4$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-1$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{16}+\frac{2}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{2}{3}a^{10}+a^{9}+a^{8}+2a^{4}+3a^{3}+3a^{2}+2a+1$, $\frac{5}{3}a^{18}-2a^{17}+2a^{16}-a^{15}+\frac{2}{3}a^{13}-2a^{12}+\frac{10}{3}a^{11}-\frac{10}{3}a^{10}+3a^{9}-3a^{8}+2a^{7}+a^{6}-3a^{5}+3a^{4}-5a^{3}+8a^{2}-5a-14$, $\frac{2}{3}a^{18}-a^{17}+\frac{2}{3}a^{16}+a^{14}-\frac{5}{3}a^{13}+\frac{2}{3}a^{11}+a^{10}-a^{9}-a^{8}-a^{7}+2a^{6}+2a^{5}-3a^{4}-3a^{3}+3a^{2}+4a-7$, $\frac{2}{3}a^{18}-\frac{4}{3}a^{17}-\frac{2}{3}a^{16}-a^{15}+\frac{7}{3}a^{14}+a^{13}+a^{12}-\frac{10}{3}a^{11}-\frac{4}{3}a^{10}-a^{9}+4a^{8}+2a^{7}+2a^{6}-4a^{5}-4a^{4}-5a^{3}+4a^{2}+9a+4$, $\frac{10}{3}a^{18}-\frac{2}{3}a^{17}+2a^{16}-4a^{15}+\frac{1}{3}a^{14}-\frac{5}{3}a^{13}+\frac{16}{3}a^{12}+\frac{4}{3}a^{11}+\frac{8}{3}a^{10}-6a^{9}-3a^{8}-2a^{7}+9a^{6}+8a^{5}+4a^{4}-10a^{3}-11a^{2}-4a-14$, $\frac{1}{3}a^{18}+\frac{2}{3}a^{17}-\frac{5}{3}a^{16}+\frac{4}{3}a^{15}-\frac{8}{3}a^{14}+\frac{4}{3}a^{13}-\frac{5}{3}a^{12}+2a^{11}+\frac{5}{3}a^{10}+3a^{8}-3a^{7}+a^{6}-6a^{5}+7a-2$ (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: | \( 682842465.233 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 682842465.233 \cdot 1}{2\cdot\sqrt{629287868530805907924458261644707}}\cr\approx \mathstrut & 0.415446071966 \end{aligned}\] (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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | $16{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | $19$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19$ | $18{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.9.0.1}{9} }$ | $19$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.19.18.1 | $x^{19} + 3$ | $19$ | $1$ | $18$ | $F_{19}$ | $[\ ]_{19}^{18}$ |
\(125471\) | $\Q_{125471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(142573\) | $\Q_{142573}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{142573}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(90800058249161\) | $\Q_{90800058249161}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |