Normalized defining polynomial
\( x^{19} - 6 x^{18} + 18 x^{17} - 39 x^{16} + 73 x^{15} - 200 x^{14} + 265 x^{13} + 305 x^{12} - 931 x^{11} + 1905 x^{10} - 5214 x^{9} + 10284 x^{8} - 13343 x^{7} + 12719 x^{6} - 8662 x^{5} + 4443 x^{4} - 1732 x^{3} + 614 x^{2} - 152 x + 39 \)
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: | \(-231310367559550740879663744871\)\(\medspace = -\,1831^{9}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $35.11$ | 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: | $1831$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{5} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{27} a^{6} - \frac{1}{27} a^{5} + \frac{11}{27} a^{4} - \frac{4}{27} a^{3} - \frac{4}{27} a^{2} - \frac{7}{27} a - \frac{4}{9}$, $\frac{1}{189} a^{16} - \frac{1}{63} a^{15} + \frac{2}{63} a^{14} + \frac{4}{189} a^{13} - \frac{1}{21} a^{12} - \frac{1}{63} a^{11} + \frac{25}{189} a^{10} + \frac{1}{9} a^{9} + \frac{8}{63} a^{8} - \frac{10}{189} a^{7} - \frac{8}{63} a^{6} + \frac{31}{63} a^{4} + \frac{1}{9} a^{3} - \frac{31}{63} a^{2} - \frac{74}{189} a + \frac{25}{63}$, $\frac{1}{5677371} a^{17} + \frac{4}{9639} a^{16} - \frac{39061}{5677371} a^{15} + \frac{23431}{630819} a^{14} + \frac{44839}{1892457} a^{13} - \frac{108944}{5677371} a^{12} - \frac{115825}{5677371} a^{11} - \frac{83947}{811053} a^{10} - \frac{7723}{630819} a^{9} + \frac{683246}{5677371} a^{8} - \frac{322213}{5677371} a^{7} - \frac{128144}{811053} a^{6} + \frac{2283073}{5677371} a^{5} + \frac{195559}{811053} a^{4} - \frac{2635103}{5677371} a^{3} + \frac{177047}{5677371} a^{2} + \frac{1613687}{5677371} a - \frac{90832}{270351}$, $\frac{1}{6376620657827511} a^{18} - \frac{286725058}{6376620657827511} a^{17} + \frac{5114476798316}{2125540219275837} a^{16} - \frac{91393300682065}{6376620657827511} a^{15} + \frac{29099678211083}{2125540219275837} a^{14} - \frac{353851745036513}{6376620657827511} a^{13} + \frac{56230169410196}{2125540219275837} a^{12} - \frac{5525634038423}{205697440575081} a^{11} + \frac{10920697707662}{6376620657827511} a^{10} - \frac{408800903654779}{6376620657827511} a^{9} + \frac{827255474969296}{6376620657827511} a^{8} + \frac{4028471867507}{54501031263483} a^{7} - \frac{98232255326917}{910945808261073} a^{6} - \frac{11019316777724}{375095332813383} a^{5} - \frac{1126468650470110}{6376620657827511} a^{4} - \frac{15674151521204}{125031777604461} a^{3} + \frac{1758133891772611}{6376620657827511} a^{2} + \frac{638318059312385}{6376620657827511} a - \frac{10834185621670}{163503093790449}$
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);
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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 (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 34462435.5384 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
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 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $19$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | $19$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $19$ | $19$ | $19$ |
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 |
---|---|---|---|---|---|---|---|
1831 | Data not computed |