Normalized defining polynomial
\( x^{19} - 2x - 4 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-33988985718086113356299711291588608\)
\(\medspace = -\,2^{34}\cdot 1978419355471042995125737\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.68\) | 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: |
\(2\), \(1978419355471042995125737\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-19784\!\cdots\!25737}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}$
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: |
$a^{17}+a^{16}-a^{14}+3a^{13}+3a^{9}+a^{8}-2a^{7}+3a^{6}+3a^{5}-a^{3}+4a^{2}+5a-3$, $a^{18}+2a^{17}+2a^{16}+2a^{15}+2a^{14}+a^{13}+a^{12}-a^{11}-2a^{9}-a^{8}-4a^{7}-3a^{6}-6a^{5}-4a^{4}-6a^{3}-2a^{2}-a+3$, $a^{16}+a^{8}+2a^{7}-2a^{4}-2a^{3}-a^{2}-4a-1$, $a^{18}+a^{17}-a^{16}-2a^{15}-a^{12}+a^{11}+a^{10}-2a^{9}-2a^{8}-a^{7}+a^{6}+6a^{5}+5a^{4}-4a^{3}-5a^{2}+2a+3$, $a^{18}-3a^{17}+a^{16}-a^{15}+a^{14}-4a^{13}+a^{12}-2a^{11}+a^{10}-2a^{9}-2a^{8}-3a^{7}+a^{6}-5a^{4}-3a^{3}-2a^{2}+a-7$, $4a^{18}+a^{17}+a^{16}+7a^{15}-2a^{14}+10a^{13}-2a^{12}+5a^{11}+a^{10}-6a^{9}+3a^{8}-14a^{7}+a^{6}-15a^{5}-6a^{4}-7a^{3}-13a^{2}+8a-17$, $a^{18}-a^{17}+a^{16}-a^{14}+2a^{12}-2a^{11}+2a^{9}-2a^{8}-a^{7}+4a^{6}-a^{5}-5a^{4}+4a^{3}+2a^{2}-3a-3$, $3a^{18}-3a^{17}+3a^{15}-4a^{14}+a^{13}+3a^{12}-2a^{11}-a^{10}+5a^{9}-3a^{8}-6a^{7}+7a^{6}-5a^{5}-6a^{4}+8a^{3}+a^{2}-5a+1$, $4a^{18}+a^{17}-8a^{16}+a^{15}+11a^{14}-7a^{13}-7a^{12}+9a^{11}+4a^{10}-7a^{9}-6a^{8}+12a^{7}+9a^{6}-21a^{5}-a^{4}+23a^{3}-9a^{2}-17a+1$
| 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: | \( 32498884517.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 32498884517.5 \cdot 1}{2\cdot\sqrt{33988985718086113356299711291588608}}\cr\approx \mathstrut & 2.69040813487 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $[\ ]$ |
2.18.34.21 | $x^{18} + 2 x^{17} + 6 x^{16} + 2 x^{14} + 6 x^{12} + 4 x^{10} + 4 x^{8} + 4 x^{5} + 6$ | $18$ | $1$ | $34$ | 18T656 | $[2, 8/3, 8/3, 26/9, 26/9, 26/9, 26/9, 26/9, 26/9]_{9}^{6}$ | |
\(197\!\cdots\!737\)
| 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 $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |