Normalized defining polynomial
\( x^{19} + 7x - 4 \)
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: | \(-1711431646479514505060267844358279\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.12\) | 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: | $1711431646479514505060267844358279^{1/2}\approx 4.1369453059951304e+16$ | ||
Ramified primes: | \(17114\!\cdots\!58279\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-17114\!\cdots\!58279}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{9}$
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)
| |
Fundamental units: | $\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}-a^{9}-a^{8}-a^{7}+a^{5}+\frac{3}{2}a^{4}+\frac{3}{2}a^{3}+\frac{1}{2}a^{2}-1$, $\frac{3}{2}a^{18}+a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{14}+\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{7}+\frac{1}{2}a^{5}-a^{4}+2a^{3}-\frac{3}{2}a^{2}-\frac{1}{2}a+11$, $\frac{3}{2}a^{18}+a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+a^{5}+\frac{3}{2}a^{4}-a^{3}-a^{2}-\frac{1}{2}a+13$, $a^{16}-\frac{3}{2}a^{15}-a^{14}-\frac{5}{2}a^{13}+\frac{1}{2}a^{12}+a^{11}+3a^{10}+a^{9}-a^{8}-3a^{7}-\frac{7}{2}a^{6}+a^{5}+\frac{3}{2}a^{4}+\frac{15}{2}a^{3}-a^{2}+2a-11$, $\frac{1}{2}a^{17}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+a^{11}+a^{9}-\frac{1}{2}a^{8}-\frac{3}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+a^{2}-2a+1$, $\frac{3}{2}a^{18}-a^{16}-\frac{3}{2}a^{15}-a^{14}+\frac{1}{2}a^{13}+\frac{3}{2}a^{12}+2a^{11}-a^{10}-\frac{5}{2}a^{9}-3a^{8}-3a^{7}+\frac{3}{2}a^{6}+3a^{5}+\frac{7}{2}a^{4}-\frac{3}{2}a^{3}-4a^{2}-5a+5$, $\frac{1}{2}a^{17}-\frac{3}{2}a^{13}+a^{12}+\frac{1}{2}a^{10}+a^{9}-\frac{1}{2}a^{8}-3a^{6}+2a^{5}-\frac{1}{2}a^{4}+4a^{2}-\frac{7}{2}a+1$, $a^{18}-a^{16}-2a^{15}-2a^{14}-\frac{3}{2}a^{13}-\frac{1}{2}a^{12}-a^{11}-\frac{5}{2}a^{10}-5a^{9}-6a^{8}-5a^{7}-2a^{6}-\frac{1}{2}a^{4}-\frac{5}{2}a^{3}-5a^{2}-\frac{9}{2}a+5$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{14}+\frac{3}{2}a^{13}+\frac{3}{2}a^{11}-\frac{5}{2}a^{10}+a^{9}-\frac{3}{2}a^{8}+\frac{7}{2}a^{7}-2a^{6}-\frac{7}{2}a^{4}+2a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a-1$ (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: | \( 3321587861.16 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 3321587861.16 \cdot 1}{2\cdot\sqrt{1711431646479514505060267844358279}}\cr\approx \mathstrut & 1.22541948248 \end{aligned}\] (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 | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $19$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(171\!\cdots\!279\) | $\Q_{17\!\cdots\!79}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17\!\cdots\!79}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17\!\cdots\!79}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |