Normalized defining polynomial
\( x^{19} - 3 x^{18} - 8 x^{17} + 30 x^{16} + 16 x^{15} - 115 x^{14} + 31 x^{13} + 208 x^{12} - 175 x^{11} + \cdots - 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1598194952468587114325789491156\) \(\medspace = -\,2^{2}\cdot 937\cdot 971\cdot 4676220259\cdot 93910895860973\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.87\) | 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: | \(2\), \(937\), \(971\), \(4676220259\), \(93910895860973\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-39954\!\cdots\!72789}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | $a$, $a^{2}-a-1$, $a^{18}-3a^{17}-8a^{16}+30a^{15}+16a^{14}-115a^{13}+31a^{12}+208a^{11}-175a^{10}-165a^{9}+280a^{8}+5a^{7}-206a^{6}+82a^{5}+75a^{4}-53a^{3}-13a^{2}+10a+1$, $a^{18}-3a^{17}-8a^{16}+30a^{15}+16a^{14}-115a^{13}+31a^{12}+208a^{11}-175a^{10}-165a^{9}+280a^{8}+5a^{7}-206a^{6}+82a^{5}+75a^{4}-53a^{3}-14a^{2}+10a+2$, $6a^{18}-20a^{17}-40a^{16}+190a^{15}+21a^{14}-664a^{13}+438a^{12}+977a^{11}-1370a^{10}-311a^{9}+1629a^{8}-682a^{7}-754a^{6}+739a^{5}+46a^{4}-263a^{3}+48a^{2}+33a-6$, $a^{3}-a^{2}-2a+1$, $3a^{18}-11a^{17}-18a^{16}+105a^{15}-9a^{14}-369a^{13}+294a^{12}+545a^{11}-836a^{10}-166a^{9}+996a^{8}-411a^{7}-506a^{6}+463a^{5}+59a^{4}-193a^{3}+23a^{2}+27a-3$, $3a^{18}-9a^{17}-24a^{16}+91a^{15}+44a^{14}-349a^{13}+126a^{12}+609a^{11}-617a^{10}-388a^{9}+920a^{8}-192a^{7}-561a^{6}+398a^{5}+95a^{4}-179a^{3}+26a^{2}+22a-6$, $a^{18}-2a^{17}-10a^{16}+20a^{15}+36a^{14}-79a^{13}-48a^{12}+160a^{11}-15a^{10}-180a^{9}+100a^{8}+105a^{7}-101a^{6}-19a^{5}+56a^{4}+2a^{3}-11a^{2}+2a+2$, $4a^{18}-12a^{17}-31a^{16}+117a^{15}+57a^{14}-433a^{13}+134a^{12}+741a^{11}-665a^{10}-519a^{9}+984a^{8}-71a^{7}-641a^{6}+315a^{5}+182a^{4}-160a^{3}-17a^{2}+23a+1$, $a^{18}-a^{17}-13a^{16}+12a^{15}+65a^{14}-61a^{13}-154a^{12}+173a^{11}+165a^{10}-293a^{9}-36a^{8}+280a^{7}-80a^{6}-131a^{5}+89a^{4}+37a^{3}-32a^{2}-8a+2$, $8a^{18}-26a^{17}-57a^{16}+252a^{15}+63a^{14}-915a^{13}+467a^{12}+1477a^{11}-1694a^{10}-798a^{9}+2257a^{8}-546a^{7}-1302a^{6}+895a^{5}+264a^{4}-403a^{3}+16a^{2}+58a-5$, $a^{18}+5a^{17}-30a^{16}-36a^{15}+229a^{14}+40a^{13}-751a^{12}+305a^{11}+1163a^{10}-1113a^{9}-722a^{8}+1478a^{7}-133a^{6}-864a^{5}+417a^{4}+249a^{3}-176a^{2}-30a+19$ (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: | \( 222480440.47 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{5}\cdot 222480440.47 \cdot 1}{2\cdot\sqrt{1598194952468587114325789491156}}\cr\approx \mathstrut & 0.44118045619 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }$ | $19$ | $17{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.17.0.1 | $x^{17} + x^{3} + 1$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(937\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(971\) | $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(4676220259\) | $\Q_{4676220259}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(93910895860973\) | $\Q_{93910895860973}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |