Normalized defining polynomial
\( x^{21} - 3 x^{20} - 72 x^{19} + 262 x^{18} + 1785 x^{17} - 7923 x^{16} - 17502 x^{15} + 107772 x^{14} + 38604 x^{13} - 717609 x^{12} + 440313 x^{11} + 2251605 x^{10} - 3001988 x^{9} - 2337081 x^{8} + 6238701 x^{7} - 1893950 x^{6} - 3157635 x^{5} + 2597691 x^{4} - 495851 x^{3} - 66246 x^{2} + 18912 x + 251 \)
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: | \(5778662528422377251527626979988196021835689=3^{28}\cdot 43^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $108.71$ | 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$$\chi_{387}(256,·)$, $\chi_{387}(1,·)$, $\chi_{387}(130,·)$, $\chi_{387}(259,·)$, $\chi_{387}(4,·)$, $\chi_{387}(133,·)$, $\chi_{387}(262,·)$, $\chi_{387}(385,·)$, $\chi_{387}(64,·)$, $\chi_{387}(322,·)$, $\chi_{387}(16,·)$, $\chi_{387}(145,·)$, $\chi_{387}(274,·)$, $\chi_{387}(97,·)$, $\chi_{387}(226,·)$, $\chi_{387}(355,·)$, $\chi_{387}(193,·)$, $\chi_{387}(121,·)$, $\chi_{387}(250,·)$, $\chi_{387}(379,·)$, $\chi_{387}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{259} a^{18} + \frac{8}{259} a^{17} - \frac{4}{259} a^{16} - \frac{2}{259} a^{15} + \frac{68}{259} a^{14} - \frac{13}{259} a^{13} - \frac{52}{259} a^{12} + \frac{39}{259} a^{11} - \frac{69}{259} a^{10} - \frac{41}{259} a^{9} - \frac{29}{259} a^{8} - \frac{88}{259} a^{7} + \frac{118}{259} a^{6} - \frac{25}{259} a^{5} + \frac{1}{259} a^{4} - \frac{2}{37} a^{3} - \frac{61}{259} a^{2} + \frac{25}{259} a + \frac{27}{259}$, $\frac{1}{259} a^{19} + \frac{6}{259} a^{17} - \frac{1}{37} a^{16} + \frac{10}{259} a^{15} - \frac{113}{259} a^{14} + \frac{18}{37} a^{13} - \frac{9}{37} a^{12} + \frac{9}{37} a^{11} + \frac{67}{259} a^{10} + \frac{3}{259} a^{9} - \frac{78}{259} a^{8} + \frac{8}{259} a^{7} - \frac{1}{37} a^{6} - \frac{58}{259} a^{5} + \frac{52}{259} a^{4} + \frac{88}{259} a^{3} + \frac{69}{259} a^{2} + \frac{12}{259} a - \frac{15}{37}$, $\frac{1}{32150934052884446784525141950738152033910521} a^{20} + \frac{20011761867310078764086123860907492259033}{32150934052884446784525141950738152033910521} a^{19} + \frac{44755235088556389352232322407153927193397}{32150934052884446784525141950738152033910521} a^{18} - \frac{71468685398735175689947736450717503745970}{4592990578983492397789305992962593147701503} a^{17} + \frac{5665887946806777826522379576940689561998}{868944163591471534716895728398328433348933} a^{16} + \frac{1032601348363763847497696951630977853094190}{32150934052884446784525141950738152033910521} a^{15} - \frac{11192621643465453023031850996594250917448783}{32150934052884446784525141950738152033910521} a^{14} + \frac{5882564682874840704856948923078799116428018}{32150934052884446784525141950738152033910521} a^{13} + \frac{14118220301851052519482625780496948241543473}{32150934052884446784525141950738152033910521} a^{12} - \frac{12282557587302134183910095678515257433939065}{32150934052884446784525141950738152033910521} a^{11} + \frac{13375333880529023098350527313659460279517036}{32150934052884446784525141950738152033910521} a^{10} - \frac{8936648697948973741272001907570095973162753}{32150934052884446784525141950738152033910521} a^{9} - \frac{13272073365237331946300797872344198030044930}{32150934052884446784525141950738152033910521} a^{8} + \frac{11523434967271330469027722306474536817353043}{32150934052884446784525141950738152033910521} a^{7} + \frac{273948571575666313749087433344639539782247}{4592990578983492397789305992962593147701503} a^{6} - \frac{9378287354337518915727500615325553857319729}{32150934052884446784525141950738152033910521} a^{5} - \frac{1347945709658695405077741680412244719106015}{4592990578983492397789305992962593147701503} a^{4} + \frac{6006148198863892284495168192632910077783757}{32150934052884446784525141950738152033910521} a^{3} + \frac{2335544878640766323764541903580233093007447}{32150934052884446784525141950738152033910521} a^{2} + \frac{1900429616477932613550323777334379996263487}{4592990578983492397789305992962593147701503} a - \frac{13496987488554564300929578064666754069325644}{32150934052884446784525141950738152033910521}$
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: | \( 520470899652557.25 \) (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
| \(\Q(\zeta_{9})^+\), 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 | $21$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{21}$ | $21$ | R | $21$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 43 | Data not computed | ||||||