Normalized defining polynomial
\( x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} - 19 x^{9} - 20 x^{8} + 32 x^{7} - 9 x^{6} - 17 x^{5} + 17 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(146525077331272123613812972\) \(\medspace = 2^{2}\cdot 139\cdot 126311\cdot 1021043\cdot 2043393285469\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.83\) | 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: | \(2\), \(139\), \(126311\), \(1021043\), \(2043393285469\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{36631\!\cdots\!53243}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $a$, $a^{18}-2a^{17}-a^{16}+8a^{15}-8a^{14}-7a^{13}+21a^{12}-10a^{11}-21a^{10}+31a^{9}-32a^{7}+20a^{6}+11a^{5}-17a^{4}+6a^{3}+3a^{2}-5a+1$, $5a^{18}-13a^{17}+44a^{15}-64a^{14}-16a^{13}+133a^{12}-113a^{11}-89a^{10}+233a^{9}-85a^{8}-185a^{7}+210a^{6}+14a^{5}-145a^{4}+84a^{3}+20a^{2}-40a+12$, $3a^{17}-3a^{16}-5a^{15}+18a^{14}-8a^{13}-24a^{12}+38a^{11}+a^{10}-56a^{9}+41a^{8}+30a^{7}-65a^{6}+7a^{5}+34a^{4}-26a^{3}-4a^{2}+10a-2$, $a^{18}+a^{17}-3a^{16}+3a^{15}+8a^{14}-11a^{13}-a^{12}+21a^{11}-16a^{10}-18a^{9}+30a^{8}-5a^{7}-33a^{6}+12a^{5}+6a^{4}-18a^{3}+3a-3$, $5a^{18}-6a^{17}-8a^{16}+33a^{15}-20a^{14}-41a^{13}+78a^{12}-14a^{11}-101a^{10}+99a^{9}+35a^{8}-130a^{7}+45a^{6}+58a^{5}-67a^{4}+8a^{3}+22a^{2}-13a+2$, $7a^{18}-6a^{17}-13a^{16}+41a^{15}-13a^{14}-61a^{13}+84a^{12}+14a^{11}-135a^{10}+83a^{9}+84a^{8}-147a^{7}-a^{6}+85a^{5}-54a^{4}-19a^{3}+24a^{2}-3a-1$, $9a^{18}-9a^{17}-17a^{16}+57a^{15}-23a^{14}-84a^{13}+126a^{12}+9a^{11}-193a^{10}+138a^{9}+112a^{8}-226a^{7}+23a^{6}+133a^{5}-93a^{4}-21a^{3}+43a^{2}-9a-2$, $a^{18}-5a^{16}+8a^{15}+4a^{14}-24a^{13}+20a^{12}+17a^{11}-47a^{10}+17a^{9}+43a^{8}-53a^{7}-4a^{6}+42a^{5}-24a^{4}-6a^{3}+16a^{2}-6a+2$, $2a^{18}+a^{17}-6a^{16}+6a^{15}+14a^{14}-23a^{13}-4a^{12}+45a^{11}-35a^{10}-39a^{9}+70a^{8}-7a^{7}-76a^{6}+37a^{5}+26a^{4}-42a^{3}+4a^{2}+12a-6$ (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: | \( 737873.586636 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{8}\cdot 737873.586636 \cdot 1}{2\cdot\sqrt{146525077331272123613812972}}\cr\approx \mathstrut & 0.592277074992 \end{aligned}\] (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 | R | ${\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | $17{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $19$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.13.0.1 | $x^{13} + x^{4} + x^{3} + x + 1$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(139\) | $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.4.0.1 | $x^{4} + 7 x^{2} + 96 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
139.11.0.1 | $x^{11} + 7 x + 137$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(126311\) | $\Q_{126311}$ | $x$ | $1$ | $1$ | $0$ | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(1021043\) | $\Q_{1021043}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(2043393285469\) | $\Q_{2043393285469}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |