Normalized defining polynomial
\( x^{19} + 4x - 8 \)
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)
| |
Discriminant: | \(-33192383974339170988381865771008\) \(\medspace = -\,2^{24}\cdot 311\cdot 13913\cdot 49356007\cdot 9263974663\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.60\) | 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: | not computed | ||
Ramified primes: | \(2\), \(311\), \(13913\), \(49356007\), \(9263974663\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-19784\!\cdots\!20463}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{17}-\frac{1}{2}a^{8}$, $\frac{1}{4}a^{18}$
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: | $\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+a^{3}+a+1$, $\frac{1}{2}a^{10}-1$, $\frac{1}{4}a^{18}+\frac{1}{2}a^{17}+\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}+a^{9}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}-2a^{4}-2a^{3}+a-1$, $\frac{1}{2}a^{17}+\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+a^{13}-a^{11}+\frac{1}{2}a^{9}-a^{8}-\frac{3}{2}a^{7}+\frac{1}{2}a^{6}+\frac{3}{2}a^{5}-a^{4}-2a^{3}+2a^{2}+2a-3$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{11}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-a^{6}+a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $\frac{1}{2}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-a^{13}+a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{3}{2}a^{8}-a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}-a^{4}+2a^{2}-a-1$, $\frac{1}{4}a^{17}+\frac{1}{2}a^{16}+\frac{1}{4}a^{14}+a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+a^{7}-\frac{1}{2}a^{5}+2a^{4}+2a^{3}-a^{2}+a+3$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{4}a^{14}-a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{9}+a^{8}-\frac{1}{2}a^{7}+\frac{5}{2}a^{5}+2a^{3}+2a^{2}-2a+3$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{3}{2}a^{8}+a^{7}-a^{6}-2a^{5}+3a^{4}-2a^{3}+4a-3$ (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: | \( 844213349.401 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 844213349.401 \cdot 1}{2\cdot\sqrt{33192383974339170988381865771008}}\cr\approx \mathstrut & 2.23641196756 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | $16{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.18.24.53 | $x^{18} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2$ | $18$ | $1$ | $24$ | 18T433 | $[16/9, 16/9, 16/9, 16/9, 16/9, 16/9]_{9}^{6}$ | |
\(311\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(13913\) | $\Q_{13913}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(49356007\) | $\Q_{49356007}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(9263974663\) | $\Q_{9263974663}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{9263974663}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |