Normalized defining polynomial
\( x^{21} - x^{20} - 314 x^{19} + 641 x^{18} + 36738 x^{17} - 95806 x^{16} - 2134622 x^{15} + 6573588 x^{14} + 67487244 x^{13} - 249054401 x^{12} - 1162017486 x^{11} + 5450875699 x^{10} + 9285707168 x^{9} - 67555921271 x^{8} + 824221871 x^{7} + 429237041242 x^{6} - 490449416375 x^{5} - 989353991611 x^{4} + 2305780509364 x^{3} - 753973198685 x^{2} - 1273935520834 x + 833882527031 \)
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: | \(808066270618405716993861719647864148675120272481133649=7^{14}\cdot 127^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $369.00$ | 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: | $7, 127$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(889=7\cdot 127\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{889}(512,·)$, $\chi_{889}(1,·)$, $\chi_{889}(555,·)$, $\chi_{889}(711,·)$, $\chi_{889}(8,·)$, $\chi_{889}(778,·)$, $\chi_{889}(781,·)$, $\chi_{889}(849,·)$, $\chi_{889}(856,·)$, $\chi_{889}(25,·)$, $\chi_{889}(540,·)$, $\chi_{889}(354,·)$, $\chi_{889}(165,·)$, $\chi_{889}(625,·)$, $\chi_{889}(64,·)$, $\chi_{889}(107,·)$, $\chi_{889}(431,·)$, $\chi_{889}(200,·)$, $\chi_{889}(884,·)$, $\chi_{889}(569,·)$, $\chi_{889}(764,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{103} a^{18} - \frac{14}{103} a^{17} - \frac{4}{103} a^{16} + \frac{33}{103} a^{15} - \frac{33}{103} a^{14} + \frac{6}{103} a^{13} + \frac{44}{103} a^{12} + \frac{48}{103} a^{11} - \frac{49}{103} a^{10} - \frac{50}{103} a^{9} - \frac{22}{103} a^{8} + \frac{42}{103} a^{7} - \frac{39}{103} a^{6} - \frac{47}{103} a^{5} + \frac{15}{103} a^{4} + \frac{5}{103} a^{3} + \frac{30}{103} a^{2} - \frac{5}{103} a + \frac{39}{103}$, $\frac{1}{6077} a^{19} + \frac{25}{6077} a^{18} - \frac{1889}{6077} a^{17} + \frac{804}{6077} a^{16} + \frac{2696}{6077} a^{15} + \frac{2118}{6077} a^{14} + \frac{2132}{6077} a^{13} - \frac{1738}{6077} a^{12} - \frac{11}{103} a^{11} - \frac{107}{6077} a^{10} + \frac{603}{6077} a^{9} + \frac{626}{6077} a^{8} + \frac{1290}{6077} a^{7} + \frac{595}{6077} a^{6} - \frac{170}{6077} a^{5} - \frac{2397}{6077} a^{4} + \frac{1667}{6077} a^{3} + \frac{1577}{6077} a^{2} - \frac{2731}{6077} a + \frac{1624}{6077}$, $\frac{1}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{20} - \frac{388198803005312649842917773145151604718672130588625910889764334742049256528669884472773099879}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{19} + \frac{41790754324291206277272823458581623501151008366780793826829175787857462263087244061465616584516}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{18} - \frac{594790771539208758422869901270036069849420188831099529319150571485853504481446655087354137523121}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{17} - \frac{2006496043195778423375722622798360148476372142172143965471329364877756545170396771956926072697620}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{16} + \frac{475995791020272133857535292995082972973354335299926304220823935896023276067079158877546042085238}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{15} + \frac{576068925161639726098399485354617352195986950388079332740960641889585762650017854356894118183}{153908848482914691716074027159714479410221294559890289857111072428532794962358707792905307121541} a^{14} + \frac{3199292461334122885556032458623505720878798422345936963742884541127216479650292741868611982008253}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{13} + \frac{3451478595587508938387732992578407579306617451727505547235844017098565991707640353444024313993129}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{12} + \frac{2063384045998831168594625340359593456648398679647423516716588374849218146097629669037825768154417}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{11} - \frac{2255874697432176712519365691208808463713287425816025014850540666948374895391370377882661994867809}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{10} - \frac{1229356623187360378638118360958192281205044336919566844039239313154840327048863045493488805255406}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{9} - \frac{2067175746452948452993971600230473988379049268490336482281250120188647905303162735351617173120576}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{8} + \frac{804961423689196231012294245404996757293623753117762440677721136243745757498911003892366874687716}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{7} - \frac{4400681996973029150438201614488391746031901207646017735631400739940595595920656552998160928804814}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{6} - \frac{2640905374730635231066663198490282204093292793602879244167639046520187470078448548992872548454595}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{5} + \frac{1592417110106784794119037145734345068354853840409282145431252182539489835348162561601171350200364}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{4} + \frac{993184782934977996365950829340947205436451003230984435133262785270744570067408658547541977705811}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{3} - \frac{1297895964576249394222435668733775862542197855819027209957774792621150134300277774134373763790385}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a^{2} - \frac{1190998920041473920910164690822724276422533506709790985484966822245325365609632002617323724734646}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919} a + \frac{2405900021923461448844928903784641140884626686847459910400707547643845508636586745218867924548703}{9080622060491966811248367602423154285203056379033527101569553273283434902779163759781413120170919}$
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: | \( 45298926008459270000 \) (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
| 3.3.790321.2, 7.7.4195872914689.1 |
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 | $21$ | $21$ | $21$ | R | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{21}$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 127 | Data not computed | ||||||