Normalized defining polynomial
\( x^{19} - 2 x^{18} - 14 x^{17} + 23 x^{16} + 83 x^{15} - 104 x^{14} - 274 x^{13} + 234 x^{12} + 555 x^{11} + \cdots + 1 \)
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: | \(-1388142379667992699741943954318476\) \(\medspace = -\,2^{2}\cdot 34\!\cdots\!19\) | 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: | \(55.51\) | 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: | $2^{2/3}347035594916998174935485988579619^{1/2}\approx 2.9571521810644656e+16$ | ||
Ramified primes: | \(2\), \(34703\!\cdots\!79619\) | 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: | $\Q(\sqrt{-34703\!\cdots\!79619}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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Fundamental units: | $a$, $a^{18}-2a^{17}-14a^{16}+23a^{15}+83a^{14}-104a^{13}-274a^{12}+234a^{11}+555a^{10}-274a^{9}-708a^{8}+162a^{7}+554a^{6}-50a^{5}-246a^{4}+17a^{3}+55a^{2}-6a-3$, $2a^{18}-3a^{17}-30a^{16}+32a^{15}+189a^{14}-125a^{13}-652a^{12}+194a^{11}+1344a^{10}+7a^{9}-1690a^{8}-384a^{7}+1270a^{6}+454a^{5}-542a^{4}-212a^{3}+129a^{2}+41a-17$, $a^{18}-3a^{17}-11a^{16}+34a^{15}+49a^{14}-153a^{13}-121a^{12}+355a^{11}+200a^{10}-474a^{9}-234a^{8}+396a^{7}+158a^{6}-208a^{5}-38a^{4}+55a^{3}+a^{2}-8a+2$, $3a^{18}-8a^{17}-37a^{16}+94a^{15}+191a^{14}-442a^{13}-553a^{12}+1077a^{11}+1019a^{10}-1501a^{9}-1236a^{8}+1289a^{7}+905a^{6}-726a^{5}-309a^{4}+246a^{3}+20a^{2}-35a+6$, $34a^{18}-55a^{17}-498a^{16}+594a^{15}+3061a^{14}-2393a^{13}-10291a^{12}+4145a^{11}+20622a^{10}-1720a^{9}-25023a^{8}-3675a^{7}+17773a^{6}+4782a^{5}-6780a^{4}-1853a^{3}+1277a^{2}+208a-97$, $19a^{18}-31a^{17}-277a^{16}+335a^{15}+1694a^{14}-1357a^{13}-5666a^{12}+2408a^{11}+11305a^{10}-1235a^{9}-13697a^{8}-1589a^{7}+9783a^{6}+2263a^{5}-3802a^{4}-869a^{3}+732a^{2}+90a-53$, $72a^{18}-120a^{17}-1047a^{16}+1304a^{15}+6399a^{14}-5320a^{13}-21443a^{12}+9537a^{11}+42966a^{10}-5012a^{9}-52299a^{8}-6304a^{7}+37323a^{6}+9262a^{5}-14268a^{4}-3726a^{3}+2663a^{2}+438a-204$, $18a^{18}-33a^{17}-257a^{16}+370a^{15}+1549a^{14}-1600a^{13}-5161a^{12}+3286a^{11}+10418a^{10}-3033a^{9}-13011a^{8}+518a^{7}+9750a^{6}+923a^{5}-4024a^{4}-477a^{3}+814a^{2}+52a-59$, $33a^{18}-54a^{17}-482a^{16}+585a^{15}+2955a^{14}-2371a^{13}-9916a^{12}+4176a^{11}+19859a^{10}-1965a^{9}-24125a^{8}-3216a^{7}+17186a^{6}+4410a^{5}-6586a^{4}-1736a^{3}+1248a^{2}+199a-97$, $113a^{18}-177a^{17}-1660a^{16}+1881a^{15}+10214a^{14}-7345a^{13}-34268a^{12}+11685a^{11}+68182a^{10}-1620a^{9}-81479a^{8}-16807a^{7}+56219a^{6}+18664a^{5}-20327a^{4}-6903a^{3}+3547a^{2}+741a-267$, $25a^{18}-51a^{17}-343a^{16}+576a^{15}+1992a^{14}-2538a^{13}-6442a^{12}+5485a^{11}+12761a^{10}-5968a^{9}-15778a^{8}+2942a^{7}+11674a^{6}-428a^{5}-4639a^{4}+20a^{3}+842a^{2}-34a-41$, $100a^{18}-155a^{17}-1472a^{16}+1642a^{15}+9070a^{14}-6368a^{13}-30446a^{12}+9916a^{11}+60538a^{10}-620a^{9}-72188a^{8}-15797a^{7}+49602a^{6}+17098a^{5}-17812a^{4}-6286a^{3}+3091a^{2}+679a-239$ (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)]
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Regulator: | \( 8259829376.29 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{5}\cdot 8259829376.29 \cdot 1}{2\cdot\sqrt{1388142379667992699741943954318476}}\cr\approx \mathstrut & 0.555767794281 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $19$ | $18{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }^{2}$ | $17{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.14.0.1 | $x^{14} + x^{7} + x^{5} + x^{3} + 1$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(347\!\cdots\!619\) | $\Q_{34\!\cdots\!19}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |