Normalized defining polynomial
\( x^{19} - x - 4 \)
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: | \(-518630842213267150872019351855\) \(\medspace = -\,5\cdot 103726168442653430174403870371\) | 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: | \(36.64\) | 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: | $5^{1/2}103726168442653430174403870371^{1/2}\approx 720160289250432.9$ | ||
Ramified primes: | \(5\), \(103726168442653430174403870371\) | 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: | $\Q(\sqrt{-51863\!\cdots\!51855}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{9}$
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)
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Fundamental units: | $\frac{1}{2}a^{10}-\frac{1}{2}a-1$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}+\frac{1}{2}a^{10}-\frac{1}{2}a^{7}+\frac{1}{2}a^{4}-\frac{1}{2}a-1$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{10}+a^{9}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{4}-a^{3}+a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{9}+\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-1$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+a^{15}-\frac{1}{2}a^{14}-a^{13}+a^{12}-a^{11}+a^{10}+a^{9}-\frac{3}{2}a^{8}+\frac{1}{2}a^{7}-a^{6}-\frac{1}{2}a^{5}+2a^{4}-3$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-a^{8}-\frac{3}{2}a^{7}-\frac{1}{2}a^{6}+a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-2a^{2}-a-1$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{11}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+a^{5}-a^{4}+a^{3}-\frac{3}{2}a^{2}+a-3$, $\frac{1}{2}a^{17}-a^{16}+a^{15}-a^{14}+\frac{3}{2}a^{13}-2a^{12}+\frac{5}{2}a^{11}-3a^{10}+4a^{9}-\frac{7}{2}a^{8}+3a^{7}-3a^{6}+3a^{5}-\frac{5}{2}a^{4}+3a^{3}-\frac{7}{2}a^{2}+3a-1$ (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: | \( 53039958.1922 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 53039958.1922 \cdot 1}{2\cdot\sqrt{518630842213267150872019351855}}\cr\approx \mathstrut & 1.12406860510 \end{aligned}\] (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 | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $19$ | $15{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | $19$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.17.0.1 | $x^{17} + 3 x^{2} + 2 x + 3$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(103\!\cdots\!371\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |