Normalized defining polynomial
\( x^{19} - 5 x^{18} + 11 x^{17} - 10 x^{16} + 3 x^{15} + 3 x^{14} - 80 x^{13} + 481 x^{12} - 1180 x^{11} + \cdots - 1053 \)
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: | \(-101575284882268140616515967431\) \(\medspace = -\,3^{9}\cdot 557^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.63\) | 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: | $3^{1/2}557^{1/2}\approx 40.87786687193939$ | ||
Ramified primes: | \(3\), \(557\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1671}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}-\frac{1}{3}a$, $\frac{1}{9}a^{8}+\frac{1}{3}a^{4}-\frac{4}{9}a^{2}$, $\frac{1}{9}a^{9}+\frac{2}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}+\frac{2}{9}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{27}a^{11}-\frac{1}{27}a^{9}+\frac{1}{9}a^{7}+\frac{2}{27}a^{5}+\frac{1}{3}a^{4}+\frac{4}{27}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{10}+\frac{2}{27}a^{6}-\frac{5}{27}a^{4}+\frac{1}{3}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{81}a^{13}+\frac{1}{81}a^{11}+\frac{1}{27}a^{10}-\frac{2}{81}a^{9}-\frac{1}{27}a^{8}+\frac{8}{81}a^{7}-\frac{1}{9}a^{6}-\frac{1}{81}a^{5}-\frac{4}{27}a^{4}+\frac{11}{81}a^{3}+\frac{7}{27}a^{2}+\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{81}a^{14}+\frac{1}{81}a^{12}-\frac{2}{81}a^{10}-\frac{1}{81}a^{8}+\frac{1}{9}a^{7}-\frac{1}{81}a^{6}+\frac{1}{9}a^{5}+\frac{38}{81}a^{4}+\frac{4}{9}a^{3}-\frac{1}{9}a^{2}$, $\frac{1}{243}a^{15}+\frac{1}{243}a^{14}+\frac{1}{243}a^{13}+\frac{1}{243}a^{12}-\frac{2}{243}a^{11}-\frac{2}{243}a^{10}-\frac{10}{243}a^{9}+\frac{8}{243}a^{8}+\frac{35}{243}a^{7}+\frac{35}{243}a^{6}-\frac{34}{243}a^{5}-\frac{61}{243}a^{4}-\frac{8}{27}a^{3}+\frac{11}{27}a^{2}-\frac{1}{3}$, $\frac{1}{3159}a^{16}-\frac{2}{3159}a^{15}+\frac{16}{3159}a^{14}+\frac{10}{3159}a^{13}+\frac{1}{243}a^{12}+\frac{43}{3159}a^{11}+\frac{158}{3159}a^{10}-\frac{67}{3159}a^{9}-\frac{70}{3159}a^{8}-\frac{298}{3159}a^{7}+\frac{302}{3159}a^{6}+\frac{245}{3159}a^{5}+\frac{406}{1053}a^{4}+\frac{203}{1053}a^{3}+\frac{25}{351}a^{2}+\frac{10}{117}a-\frac{1}{3}$, $\frac{1}{161109}a^{17}+\frac{16}{161109}a^{16}-\frac{37}{53703}a^{15}-\frac{61}{53703}a^{14}-\frac{161}{53703}a^{13}-\frac{653}{53703}a^{12}-\frac{290}{161109}a^{11}-\frac{122}{161109}a^{10}+\frac{424}{53703}a^{9}+\frac{2878}{53703}a^{8}-\frac{331}{53703}a^{7}+\frac{34}{243}a^{6}-\frac{9530}{161109}a^{5}+\frac{70828}{161109}a^{4}+\frac{4753}{17901}a^{3}+\frac{3028}{17901}a^{2}-\frac{11}{117}a+\frac{52}{153}$, $\frac{1}{1302244047}a^{18}+\frac{428}{144693783}a^{17}-\frac{80209}{1302244047}a^{16}+\frac{793007}{434081349}a^{15}+\frac{1936471}{434081349}a^{14}-\frac{63280}{33390873}a^{13}+\frac{313963}{100172619}a^{12}-\frac{7269046}{434081349}a^{11}-\frac{67681507}{1302244047}a^{10}-\frac{23466806}{434081349}a^{9}+\frac{18566036}{434081349}a^{8}+\frac{1014310}{33390873}a^{7}-\frac{24995450}{1302244047}a^{6}-\frac{59695478}{434081349}a^{5}+\frac{630458807}{1302244047}a^{4}+\frac{2484602}{5359029}a^{3}-\frac{51382954}{144693783}a^{2}-\frac{6634339}{16077087}a+\frac{520313}{1236699}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | $\frac{2245027}{1302244047}a^{18}-\frac{1377262}{144693783}a^{17}+\frac{22557497}{1302244047}a^{16}-\frac{708839}{144693783}a^{15}-\frac{7222558}{434081349}a^{14}+\frac{276232}{48231261}a^{13}-\frac{199041563}{1302244047}a^{12}+\frac{132135868}{144693783}a^{11}-\frac{2557216489}{1302244047}a^{10}+\frac{126933494}{144693783}a^{9}+\frac{1748164948}{434081349}a^{8}-\frac{497693552}{48231261}a^{7}+\frac{1263225316}{100172619}a^{6}-\frac{382805542}{48231261}a^{5}-\frac{2114394868}{1302244047}a^{4}+\frac{3742811}{412233}a^{3}-\frac{1365238492}{144693783}a^{2}+\frac{94370168}{16077087}a-\frac{4057633}{1236699}$, $\frac{406838}{144693783}a^{18}-\frac{16183489}{1302244047}a^{17}+\frac{25830203}{1302244047}a^{16}-\frac{1763357}{434081349}a^{15}-\frac{3456923}{434081349}a^{14}-\frac{694141}{434081349}a^{13}-\frac{99833692}{434081349}a^{12}+\frac{1593348335}{1302244047}a^{11}-\frac{2996263279}{1302244047}a^{10}+\frac{342032915}{434081349}a^{9}+\frac{2037355205}{434081349}a^{8}-\frac{4824433418}{434081349}a^{7}+\frac{5858425957}{434081349}a^{6}-\frac{13216639594}{1302244047}a^{5}+\frac{6674182718}{1302244047}a^{4}-\frac{37093100}{16077087}a^{3}+\frac{437696276}{144693783}a^{2}-\frac{7291514}{1786343}a+\frac{1736531}{1236699}$, $\frac{253148}{76602591}a^{18}-\frac{2142689}{144693783}a^{17}+\frac{34669133}{1302244047}a^{16}-\frac{491263}{33390873}a^{15}-\frac{1517222}{434081349}a^{14}+\frac{131843}{33390873}a^{13}-\frac{349610864}{1302244047}a^{12}+\frac{622474220}{434081349}a^{11}-\frac{3929996398}{1302244047}a^{10}+\frac{826236733}{434081349}a^{9}+\frac{1951885688}{434081349}a^{8}-\frac{6175771847}{434081349}a^{7}+\frac{1497929795}{76602591}a^{6}-\frac{7545034148}{434081349}a^{5}+\frac{13080722648}{1302244047}a^{4}-\frac{770296021}{144693783}a^{3}+\frac{742806848}{144693783}a^{2}-\frac{1849121}{315237}a+\frac{5710457}{1236699}$, $\frac{194}{161109}a^{18}-\frac{776}{161109}a^{17}+\frac{1556}{161109}a^{16}-\frac{415}{53703}a^{15}+\frac{74}{17901}a^{14}+\frac{2}{243}a^{13}-\frac{15529}{161109}a^{12}+\frac{4613}{9477}a^{11}-\frac{167017}{161109}a^{10}+\frac{52042}{53703}a^{9}+\frac{2131}{1989}a^{8}-\frac{262618}{53703}a^{7}+\frac{1345769}{161109}a^{6}-\frac{1373501}{161109}a^{5}+\frac{909239}{161109}a^{4}-\frac{14555}{5967}a^{3}+\frac{34247}{17901}a^{2}-\frac{5060}{1989}a+\frac{353}{153}$, $\frac{290465}{434081349}a^{18}+\frac{23606}{144693783}a^{17}-\frac{3994465}{434081349}a^{16}+\frac{303121}{11130291}a^{15}-\frac{3611783}{144693783}a^{14}+\frac{1356740}{144693783}a^{13}-\frac{18545452}{434081349}a^{12}+\frac{347050}{11130291}a^{11}+\frac{361026104}{434081349}a^{10}-\frac{138542203}{48231261}a^{9}+\frac{571401761}{144693783}a^{8}+\frac{28897390}{144693783}a^{7}-\frac{4542944455}{434081349}a^{6}+\frac{1033309832}{48231261}a^{5}-\frac{10595745694}{434081349}a^{4}+\frac{2520860084}{144693783}a^{3}-\frac{28150219}{5359029}a^{2}-\frac{2531954}{1236699}a+\frac{1089784}{412233}$, $\frac{194}{161109}a^{18}-\frac{776}{161109}a^{17}+\frac{1556}{161109}a^{16}-\frac{415}{53703}a^{15}+\frac{74}{17901}a^{14}+\frac{2}{243}a^{13}-\frac{15529}{161109}a^{12}+\frac{4613}{9477}a^{11}-\frac{167017}{161109}a^{10}+\frac{52042}{53703}a^{9}+\frac{2131}{1989}a^{8}-\frac{262618}{53703}a^{7}+\frac{1345769}{161109}a^{6}-\frac{1373501}{161109}a^{5}+\frac{909239}{161109}a^{4}-\frac{8588}{5967}a^{3}-\frac{19456}{17901}a^{2}+\frac{2896}{1989}a+\frac{47}{153}$, $\frac{156175}{100172619}a^{18}-\frac{12834241}{1302244047}a^{17}+\frac{32766751}{1302244047}a^{16}-\frac{10663559}{434081349}a^{15}-\frac{28456}{11130291}a^{14}+\frac{4847834}{434081349}a^{13}-\frac{145765961}{1302244047}a^{12}+\frac{1199630408}{1302244047}a^{11}-\frac{266051672}{100172619}a^{10}+\frac{1317866165}{434081349}a^{9}+\frac{282965042}{144693783}a^{8}-\frac{5058850577}{434081349}a^{7}+\frac{25074823822}{1302244047}a^{6}-\frac{24764010793}{1302244047}a^{5}+\frac{16439622340}{1302244047}a^{4}-\frac{898602001}{144693783}a^{3}+\frac{747627193}{144693783}a^{2}-\frac{12141715}{1786343}a+\frac{6477520}{1236699}$, $\frac{76538}{100172619}a^{18}-\frac{609731}{144693783}a^{17}+\frac{786109}{100172619}a^{16}-\frac{1780010}{434081349}a^{15}-\frac{573553}{434081349}a^{14}-\frac{1039988}{434081349}a^{13}-\frac{83759752}{1302244047}a^{12}+\frac{169768072}{434081349}a^{11}-\frac{1151742266}{1302244047}a^{10}+\frac{15357535}{25534197}a^{9}+\frac{32303126}{25534197}a^{8}-\frac{1666019131}{434081349}a^{7}+\frac{7063602041}{1302244047}a^{6}-\frac{2124212620}{434081349}a^{5}+\frac{4426537708}{1302244047}a^{4}-\frac{27846064}{11130291}a^{3}+\frac{346637422}{144693783}a^{2}-\frac{41844076}{16077087}a+\frac{1474738}{1236699}$, $\frac{30167360}{1302244047}a^{18}-\frac{31363669}{434081349}a^{17}+\frac{24064966}{1302244047}a^{16}+\frac{124084066}{434081349}a^{15}-\frac{122024392}{434081349}a^{14}-\frac{96734486}{434081349}a^{13}-\frac{1859474611}{1302244047}a^{12}+\frac{1161346543}{144693783}a^{11}-\frac{6965635406}{1302244047}a^{10}-\frac{12267839923}{434081349}a^{9}+\frac{29746652116}{434081349}a^{8}-\frac{14844362749}{434081349}a^{7}-\frac{107417673898}{1302244047}a^{6}+\frac{64550291047}{434081349}a^{5}-\frac{94024211630}{1302244047}a^{4}-\frac{214282169}{11130291}a^{3}+\frac{24170956}{144693783}a^{2}+\frac{314037989}{5359029}a-\frac{41025089}{1236699}$ (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: | \( 191261859.157 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 191261859.157 \cdot 1}{2\cdot\sqrt{101575284882268140616515967431}}\cr\approx \mathstrut & 9.15910765549 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 38 |
The 11 conjugacy class representatives for $D_{19}$ |
Character table for $D_{19}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $19$ | R | $19$ | $19$ | $19$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{9}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/padicField/29.2.0.1}{2} }^{9}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $19$ | $19$ | $19$ | $19$ | ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(557\) | $\Q_{557}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |