Normalized defining polynomial
\( x^{19} + 19 x^{15} - 19 x^{14} - 19 x^{13} + 171 x^{12} + 19 x^{11} - 266 x^{10} + 19 x^{9} + 475 x^{8} + 741 x^{7} + 437 x^{6} + 38 x^{5} - 38 x^{4} - 19 x^{3} + 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(-714209495693373205673756419\)\(\medspace = -\,19^{21}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $25.90$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $19$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{15} - \frac{1}{10} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{3}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{3}{10} a^{6} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{370} a^{17} + \frac{6}{185} a^{16} - \frac{171}{370} a^{15} + \frac{51}{370} a^{14} + \frac{16}{185} a^{13} - \frac{123}{370} a^{12} - \frac{71}{185} a^{11} + \frac{69}{185} a^{10} - \frac{52}{185} a^{9} - \frac{14}{37} a^{8} + \frac{93}{370} a^{7} - \frac{41}{370} a^{6} + \frac{92}{185} a^{5} - \frac{91}{370} a^{4} - \frac{147}{370} a^{3} + \frac{15}{74} a^{2} - \frac{137}{370} a + \frac{15}{37}$, $\frac{1}{26521869359148268510} a^{18} - \frac{6826040825032609}{26521869359148268510} a^{17} + \frac{628721947265875046}{13260934679574134255} a^{16} + \frac{6413722722121603321}{13260934679574134255} a^{15} - \frac{1260785536523544687}{13260934679574134255} a^{14} - \frac{909665398604062088}{2652186935914826851} a^{13} - \frac{6616310951575836579}{26521869359148268510} a^{12} - \frac{396876292381548847}{5304373871829653702} a^{11} - \frac{1381901801247986266}{13260934679574134255} a^{10} + \frac{2427022988972449907}{13260934679574134255} a^{9} - \frac{4661541396092582077}{26521869359148268510} a^{8} - \frac{411399133396466912}{13260934679574134255} a^{7} - \frac{1276119457010125836}{2652186935914826851} a^{6} + \frac{608820512510773566}{2652186935914826851} a^{5} - \frac{3736187603432184098}{13260934679574134255} a^{4} + \frac{1977641008305862037}{26521869359148268510} a^{3} + \frac{480456212482391413}{26521869359148268510} a^{2} - \frac{5284893778945036104}{13260934679574134255} a + \frac{1201796264987315159}{5304373871829653702}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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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);
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Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 1437945.22979 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$D_{19}:C_3$ (as 19T4):
A solvable group of order 114 |
The 9 conjugacy class representatives for $C_{19}:C_{6}$ |
Character table for $C_{19}:C_{6}$ |
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 | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
19 | Data not computed |