Normalized defining polynomial
\(x^{19} - 5 x^{18} + 10 x^{17} - 5 x^{16} + 9 x^{15} - 42 x^{14} + 51 x^{13} - 6 x^{12} - 23 x^{11} + 4 x^{10} - 29 x^{9} + 112 x^{8} - 63 x^{7} - 147 x^{6} + 183 x^{5} + 15 x^{4} - 77 x^{3} - 17 x^{2} + 37 x + 1\)
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: | \(-467562425055097089773569879\)\(\medspace = -\,919^{9}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $25.33$ | 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: | $919$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{63} a^{17} + \frac{1}{63} a^{16} - \frac{1}{63} a^{15} - \frac{1}{9} a^{14} + \frac{5}{63} a^{13} + \frac{2}{63} a^{12} - \frac{1}{63} a^{11} - \frac{4}{63} a^{10} - \frac{1}{7} a^{9} - \frac{1}{21} a^{8} - \frac{20}{63} a^{7} + \frac{22}{63} a^{6} + \frac{10}{63} a^{5} + \frac{22}{63} a^{4} - \frac{2}{63} a^{3} + \frac{4}{9} a^{2} - \frac{22}{63} a + \frac{26}{63}$, $\frac{1}{1681146971469} a^{18} - \frac{12482114905}{1681146971469} a^{17} - \frac{26042434111}{560382323823} a^{16} + \frac{1134363586}{88481419551} a^{15} - \frac{142234021442}{1681146971469} a^{14} + \frac{119912754793}{1681146971469} a^{13} + \frac{198152955184}{1681146971469} a^{12} + \frac{230017054447}{1681146971469} a^{11} + \frac{53305188299}{1681146971469} a^{10} - \frac{15535359472}{186794107941} a^{9} + \frac{11712170908}{240163853067} a^{8} + \frac{16274440718}{88481419551} a^{7} + \frac{814706226419}{1681146971469} a^{6} - \frac{36869152762}{88481419551} a^{5} - \frac{418133673766}{1681146971469} a^{4} - \frac{396229955110}{1681146971469} a^{3} - \frac{174508750715}{560382323823} a^{2} + \frac{66551965369}{240163853067} a - \frac{310401455248}{1681146971469}$
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: | \( 1152808.1284 \) | 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 | $19$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | $19$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | $19$ | $19$ | $19$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $19$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $19$ |
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 |
---|---|---|---|---|---|---|---|
919 | Data not computed |