Normalized defining polynomial
\(x^{21} - 5 x^{20} + 11 x^{19} - 20 x^{18} + 37 x^{17} - 68 x^{16} + 120 x^{15} - 168 x^{14} + 237 x^{13} - 331 x^{12} + 362 x^{11} - 431 x^{10} + 452 x^{9} - 381 x^{8} + 350 x^{7} - 238 x^{6} + 169 x^{5} - 90 x^{4} + 43 x^{3} + 6 x^{2} + 27\) 
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-3960879820606443846053527\)\(\medspace = -\,13^{2}\cdot 1801^{2}\cdot 193327^{3}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $14.84$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $13, 1801, 193327$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $3$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{860153495355557217} a^{20} + \frac{181578608182792}{860153495355557217} a^{19} + \frac{52575980687782442}{860153495355557217} a^{18} + \frac{314489200548185155}{860153495355557217} a^{17} - \frac{397364443881064868}{860153495355557217} a^{16} + \frac{260490419820728083}{860153495355557217} a^{15} + \frac{40292578962412640}{95572610595061913} a^{14} + \frac{105710859436732222}{286717831785185739} a^{13} - \frac{44911611440406269}{286717831785185739} a^{12} - \frac{267143808405119254}{860153495355557217} a^{11} - \frac{412540822785471286}{860153495355557217} a^{10} - \frac{216459893359979384}{860153495355557217} a^{9} - \frac{129466399558311343}{860153495355557217} a^{8} - \frac{23402351702819640}{95572610595061913} a^{7} + \frac{160713392039199359}{860153495355557217} a^{6} - \frac{321587809763497516}{860153495355557217} a^{5} - \frac{368615916599978978}{860153495355557217} a^{4} + \frac{142990715126005727}{286717831785185739} a^{3} + \frac{196925523467236531}{860153495355557217} a^{2} - \frac{89655634718498909}{286717831785185739} a + \frac{31206378164293990}{95572610595061913}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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)];
| |
| Regulator: | \( 6158.71427835 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
| A non-solvable group of order 11022480 |
| The 429 conjugacy class representatives for t21n139 are not computed |
| Character table for t21n139 is not computed |
Intermediate fields
| 7.1.193327.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ | $15{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ | $21$ | R | $18{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | $15{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ | $18{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.9.0.1 | $x^{9} - 2 x + 2$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 1801 | Data not computed | ||||||
| 193327 | Data not computed | ||||||