Normalized defining polynomial
\( x^{21} - 129 x^{19} - 86 x^{18} + 6579 x^{17} + 7998 x^{16} - 169635 x^{15} - 279414 x^{14} + 2373600 x^{13} + 4785298 x^{12} - 17868564 x^{11} - 43218999 x^{10} + 64742219 x^{9} + 202286448 x^{8} - 64709754 x^{7} - 440981770 x^{6} - 148287951 x^{5} + 334614390 x^{4} + 261995044 x^{3} + 27998289 x^{2} - 12075690 x + 521117 \)
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: | \(10684747015052975538074582285998174444374188961=3^{28}\cdot 43^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $155.54$ | 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: | $3, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(387=3^{2}\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{387}(64,·)$, $\chi_{387}(1,·)$, $\chi_{387}(67,·)$, $\chi_{387}(262,·)$, $\chi_{387}(139,·)$, $\chi_{387}(142,·)$, $\chi_{387}(79,·)$, $\chi_{387}(145,·)$, $\chi_{387}(25,·)$, $\chi_{387}(31,·)$, $\chi_{387}(226,·)$, $\chi_{387}(187,·)$, $\chi_{387}(358,·)$, $\chi_{387}(40,·)$, $\chi_{387}(238,·)$, $\chi_{387}(49,·)$, $\chi_{387}(52,·)$, $\chi_{387}(379,·)$, $\chi_{387}(232,·)$, $\chi_{387}(382,·)$, $\chi_{387}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{19} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{20} - \frac{725318113030508913271680559498032886294292418677324233973568053960013004}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{19} - \frac{550947092863307786557329075646818584631477421980763191100391648001133327}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{18} + \frac{471155410924866464068681801091702997570133849700696323469573518512803797}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{17} + \frac{804086825923226543520671745416269695649275108329378586728877754992874169}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{16} - \frac{790134731297429252699332755282720616290689059430210699174197875700029511}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{15} - \frac{5859928259965579715654775003554286863250700745416975146554830676930683216}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{14} - \frac{2408470915859637426576300836755412839474989710816556233355859576359656032}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{13} - \frac{1739744175450886914652434421481287417606078641786809908959131253014521743}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{12} + \frac{275039371380554169349139996088942144101120808833198207942103194180231523}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{11} + \frac{2730493003279429030300279922787726605504179777637520240687732504721603798}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{10} - \frac{6558175412487845427467099008550936174605317836944807775422977380351909414}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{9} - \frac{4024522073888009488804487397723709598885640909776983708372368918794673692}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{8} + \frac{1592575452622060302950790270567075744825371088207591002035658330677939633}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{7} + \frac{4970060045494866819292610560724456651827064538345582585242974328932733110}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{6} - \frac{4282684481898125964109848182298552739685248967212730338621031644267610910}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{5} + \frac{4870747391597768489183649055421593694372905342270590348216889337950751391}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{4} + \frac{1437508674214494204450003619712143597393099279393075238287490661613057911}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{3} + \frac{4834886979826386334176000848822485645141395169849055437835497807648968124}{13398628643109915520954804957967623005214200661192130028304530679292992403} a^{2} - \frac{810197072423912418658117873303010438511504632247412700530000727482464715}{13398628643109915520954804957967623005214200661192130028304530679292992403} a - \frac{3572664494625688782422495082180250313422776297241945720748673129953}{1105588632982087261403977634950707402031042219753455733006397448576037}$
Class group and class number
$C_{3}$, which has order $3$ (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: | \( 3551271732999900.0 \) (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
| 3.3.149769.2, 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 | $21$ | R | $21$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ | $21$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | $21$ | R | $21$ | $21$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 43 | Data not computed | ||||||