Normalized defining polynomial
\( x^{21} - 4 x^{20} - 78 x^{19} + 218 x^{18} + 2534 x^{17} - 4502 x^{16} - 43774 x^{15} + 43542 x^{14} + 429292 x^{13} - 196925 x^{12} - 2399473 x^{11} + 392282 x^{10} + 7477031 x^{9} - 739098 x^{8} - 12783623 x^{7} + 2345155 x^{6} + 10795899 x^{5} - 3647782 x^{4} - 2993437 x^{3} + 1434512 x^{2} - 61380 x - 23557 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[21, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(994578651055668861859035058398460943925772561=13^{14}\cdot 43^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $138.91$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $13, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(559=13\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{559}(256,·)$, $\chi_{559}(1,·)$, $\chi_{559}(451,·)$, $\chi_{559}(133,·)$, $\chi_{559}(391,·)$, $\chi_{559}(87,·)$, $\chi_{559}(269,·)$, $\chi_{559}(16,·)$, $\chi_{559}(274,·)$, $\chi_{559}(471,·)$, $\chi_{559}(477,·)$, $\chi_{559}(35,·)$, $\chi_{559}(484,·)$, $\chi_{559}(360,·)$, $\chi_{559}(170,·)$, $\chi_{559}(107,·)$, $\chi_{559}(365,·)$, $\chi_{559}(302,·)$, $\chi_{559}(183,·)$, $\chi_{559}(250,·)$, $\chi_{559}(508,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{3} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{2359} a^{19} + \frac{16}{337} a^{18} - \frac{5}{337} a^{17} + \frac{25}{2359} a^{16} - \frac{8}{2359} a^{15} - \frac{243}{2359} a^{14} + \frac{289}{2359} a^{13} - \frac{1080}{2359} a^{12} + \frac{836}{2359} a^{11} + \frac{234}{2359} a^{10} + \frac{34}{337} a^{9} + \frac{900}{2359} a^{8} - \frac{1066}{2359} a^{7} + \frac{3}{7} a^{6} + \frac{923}{2359} a^{5} + \frac{548}{2359} a^{4} + \frac{94}{2359} a^{3} - \frac{22}{337} a^{2} + \frac{843}{2359} a - \frac{933}{2359}$, $\frac{1}{41741673481512110235679620268566346291836806648449} a^{20} - \frac{7550672483780801602279273544606106534554230295}{41741673481512110235679620268566346291836806648449} a^{19} + \frac{1313718214338338584507640931276895033914204439261}{41741673481512110235679620268566346291836806648449} a^{18} - \frac{133051298552079538996011222550667277868237382010}{5963096211644587176525660038366620898833829521207} a^{17} + \frac{1956137197694311139794471493194499806386813495877}{41741673481512110235679620268566346291836806648449} a^{16} + \frac{1892077354486054544633809881413419899495069549920}{41741673481512110235679620268566346291836806648449} a^{15} + \frac{19944203360314900828860643418970818627537620322103}{41741673481512110235679620268566346291836806648449} a^{14} - \frac{18392279456543195047784370972653814335916086035072}{41741673481512110235679620268566346291836806648449} a^{13} + \frac{16107306871636512582729270250562114497661304738642}{41741673481512110235679620268566346291836806648449} a^{12} + \frac{6745760378278429848121665956366622290195228637022}{41741673481512110235679620268566346291836806648449} a^{11} + \frac{9755361351018878199756896325028546584222193785032}{41741673481512110235679620268566346291836806648449} a^{10} - \frac{1086866470352955370341122419898502442803203610203}{41741673481512110235679620268566346291836806648449} a^{9} - \frac{12056834858825879018746145261892885997398431866429}{41741673481512110235679620268566346291836806648449} a^{8} + \frac{17813483751846560484458057578490419104723112548510}{41741673481512110235679620268566346291836806648449} a^{7} - \frac{10958030567056764914537824140834468118180883202731}{41741673481512110235679620268566346291836806648449} a^{6} + \frac{126097867424484817758777819315970040066970105619}{5963096211644587176525660038366620898833829521207} a^{5} - \frac{9170786333793420006291218231149569216696841243555}{41741673481512110235679620268566346291836806648449} a^{4} + \frac{14162204966064165575094833758982841562141016115318}{41741673481512110235679620268566346291836806648449} a^{3} + \frac{6408340020404012314679352857397741300560335169913}{41741673481512110235679620268566346291836806648449} a^{2} - \frac{4819316964311830926943079480551337240971168461768}{41741673481512110235679620268566346291836806648449} a + \frac{15965434241713217641589771132796539108742861111928}{41741673481512110235679620268566346291836806648449}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $20$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 4188365147703962.5 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ is not computed |
Intermediate fields
| 3.3.169.1, 7.7.6321363049.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | $21$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ | $21$ | R | $21$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | $21$ | R | ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | $21$ |
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 | Data not computed | ||||||
| 43 | Data not computed | ||||||