Normalized defining polynomial
\(x^{19} - 3 x^{18} + 17 x^{17} - 47 x^{16} + 135 x^{15} - 310 x^{14} + 709 x^{13} - 1319 x^{12} + 2423 x^{11} - 3757 x^{10} + 5497 x^{9} - 6561 x^{8} + 7142 x^{7} - 5699 x^{6} + 4212 x^{5} - 1490 x^{4} + 482 x^{3} + 408 x^{2} - 210 x + 49\)
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: | \(-5121210743359411191500170799\)\(\medspace = -\,11^{9}\cdot 109^{9}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $28.73$ | 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: | $11, 109$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11} a^{11} - \frac{3}{11} a^{10} + \frac{3}{11} a^{8} - \frac{4}{11} a^{7} - \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{3}{11} a^{2} - \frac{2}{11}$, $\frac{1}{11} a^{12} + \frac{2}{11} a^{10} + \frac{3}{11} a^{9} + \frac{5}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{1}{11} a^{5} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{10} + \frac{5}{11} a^{9} + \frac{3}{11} a^{7} - \frac{4}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{11} a^{4} + \frac{2}{11} a^{3} + \frac{4}{11} a^{2} + \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{10} - \frac{2}{11} a^{8} - \frac{1}{11} a^{7} + \frac{2}{11} a^{4} + \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{15} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} + \frac{2}{11} a^{8} - \frac{4}{11} a^{7} - \frac{4}{11} a^{6} - \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{4}{11} a - \frac{2}{11}$, $\frac{1}{77} a^{16} - \frac{2}{77} a^{15} - \frac{2}{77} a^{14} - \frac{1}{77} a^{13} + \frac{1}{77} a^{11} - \frac{4}{77} a^{10} - \frac{10}{77} a^{9} - \frac{25}{77} a^{8} - \frac{5}{11} a^{7} - \frac{4}{11} a^{6} - \frac{16}{77} a^{5} - \frac{3}{11} a^{4} + \frac{26}{77} a^{3} + \frac{24}{77} a^{2} - \frac{18}{77} a$, $\frac{1}{847} a^{17} + \frac{3}{847} a^{16} + \frac{37}{847} a^{15} + \frac{38}{847} a^{14} + \frac{30}{847} a^{13} + \frac{1}{847} a^{12} - \frac{6}{847} a^{11} + \frac{17}{77} a^{10} + \frac{387}{847} a^{9} - \frac{335}{847} a^{8} + \frac{43}{121} a^{7} + \frac{152}{847} a^{6} + \frac{417}{847} a^{5} + \frac{3}{77} a^{4} + \frac{20}{121} a^{3} + \frac{263}{847} a^{2} + \frac{393}{847} a + \frac{24}{121}$, $\frac{1}{51727543335603443} a^{18} - \frac{29701844258160}{51727543335603443} a^{17} - \frac{172604384349723}{51727543335603443} a^{16} + \frac{842653333060995}{51727543335603443} a^{15} - \frac{268650353179353}{7389649047943349} a^{14} - \frac{66104475505676}{7389649047943349} a^{13} + \frac{1898574407964536}{51727543335603443} a^{12} + \frac{94415154718519}{4702503939600313} a^{11} - \frac{5289571219058366}{51727543335603443} a^{10} + \frac{146979307371132}{1202966124083801} a^{9} - \frac{8703660410832559}{51727543335603443} a^{8} - \frac{11008289411013897}{51727543335603443} a^{7} + \frac{18133680031107288}{51727543335603443} a^{6} + \frac{2003985898343274}{4702503939600313} a^{5} + \frac{800344130348002}{7389649047943349} a^{4} - \frac{18476274851913583}{51727543335603443} a^{3} - \frac{3933526836525905}{51727543335603443} a^{2} + \frac{2729455584365821}{51727543335603443} a + \frac{209747834143810}{671786277085759}$
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: | \( 2481320.5796 \) | 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 | $19$ | $19$ | $19$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | R | $19$ | $19$ | $19$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $19$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $19$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
$11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
$109$ | $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |