Normalized defining polynomial
\( x^{22} - 9 x^{21} - 36 x^{20} + 429 x^{19} + 561 x^{18} - 8496 x^{17} - 6139 x^{16} + 89196 x^{15} + 57905 x^{14} - 526311 x^{13} - 390451 x^{12} + 1730819 x^{11} + 1575659 x^{10} - 2917939 x^{9} - 3465949 x^{8} + 1790924 x^{7} + 3593930 x^{6} + 695789 x^{5} - 1086909 x^{4} - 737116 x^{3} - 188316 x^{2} - 20940 x - 841 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(162243049887845980095628744560672832080078125=5^{11}\cdot 67^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $102.22$ | 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: | $5, 67$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(335=5\cdot 67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{335}(64,·)$, $\chi_{335}(1,·)$, $\chi_{335}(216,·)$, $\chi_{335}(131,·)$, $\chi_{335}(196,·)$, $\chi_{335}(129,·)$, $\chi_{335}(9,·)$, $\chi_{335}(76,·)$, $\chi_{335}(269,·)$, $\chi_{335}(14,·)$, $\chi_{335}(81,·)$, $\chi_{335}(149,·)$, $\chi_{335}(24,·)$, $\chi_{335}(89,·)$, $\chi_{335}(91,·)$, $\chi_{335}(156,·)$, $\chi_{335}(159,·)$, $\chi_{335}(226,·)$, $\chi_{335}(174,·)$, $\chi_{335}(241,·)$, $\chi_{335}(59,·)$, $\chi_{335}(126,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{29} a^{17} - \frac{8}{29} a^{16} + \frac{9}{29} a^{15} + \frac{8}{29} a^{14} - \frac{4}{29} a^{13} + \frac{7}{29} a^{12} + \frac{9}{29} a^{11} - \frac{1}{29} a^{10} - \frac{10}{29} a^{9} - \frac{14}{29} a^{8} - \frac{9}{29} a^{7} - \frac{10}{29} a^{6} + \frac{3}{29} a^{5} + \frac{8}{29} a^{4} + \frac{13}{29} a^{3} - \frac{4}{29} a^{2} + \frac{3}{29} a$, $\frac{1}{29} a^{18} + \frac{3}{29} a^{16} - \frac{7}{29} a^{15} + \frac{2}{29} a^{14} + \frac{4}{29} a^{13} + \frac{7}{29} a^{12} + \frac{13}{29} a^{11} + \frac{11}{29} a^{10} - \frac{7}{29} a^{9} - \frac{5}{29} a^{8} + \frac{5}{29} a^{7} + \frac{10}{29} a^{6} + \frac{3}{29} a^{5} - \frac{10}{29} a^{4} + \frac{13}{29} a^{3} - \frac{5}{29} a$, $\frac{1}{6931} a^{19} - \frac{25}{6931} a^{18} + \frac{50}{6931} a^{17} - \frac{719}{6931} a^{16} - \frac{879}{6931} a^{15} - \frac{511}{6931} a^{14} - \frac{3355}{6931} a^{13} - \frac{964}{6931} a^{12} + \frac{1095}{6931} a^{11} + \frac{425}{6931} a^{10} - \frac{561}{6931} a^{9} - \frac{3051}{6931} a^{8} - \frac{3351}{6931} a^{7} - \frac{1239}{6931} a^{6} - \frac{3163}{6931} a^{5} + \frac{2263}{6931} a^{4} + \frac{3273}{6931} a^{3} + \frac{445}{6931} a^{2} + \frac{3021}{6931} a - \frac{17}{239}$, $\frac{1}{7436963} a^{20} - \frac{79}{7436963} a^{19} + \frac{1507}{256447} a^{18} - \frac{96390}{7436963} a^{17} + \frac{2495823}{7436963} a^{16} + \frac{29236}{256447} a^{15} - \frac{2582534}{7436963} a^{14} - \frac{491}{31117} a^{13} - \frac{1486726}{7436963} a^{12} + \frac{2974444}{7436963} a^{11} - \frac{2535640}{7436963} a^{10} + \frac{1083623}{7436963} a^{9} - \frac{1597159}{7436963} a^{8} - \frac{2847459}{7436963} a^{7} + \frac{1811550}{7436963} a^{6} + \frac{1871638}{7436963} a^{5} + \frac{3482801}{7436963} a^{4} + \frac{2914690}{7436963} a^{3} - \frac{300639}{7436963} a^{2} - \frac{1215466}{7436963} a + \frac{1853}{8843}$, $\frac{1}{70509301182793819572007629808938403} a^{21} + \frac{126040748693736203765863153}{1905656788724157285729935940782119} a^{20} + \frac{1816993965036258770173659481120}{70509301182793819572007629808938403} a^{19} - \frac{174938452762878946390526720595809}{70509301182793819572007629808938403} a^{18} + \frac{20490569737393212837657996300555}{1905656788724157285729935940782119} a^{17} + \frac{10237921646600561104421290721600738}{70509301182793819572007629808938403} a^{16} - \frac{27124294154146339557756382599578510}{70509301182793819572007629808938403} a^{15} - \frac{12909308995801783910622802689735266}{70509301182793819572007629808938403} a^{14} + \frac{23326862527696193378527978282997325}{70509301182793819572007629808938403} a^{13} + \frac{20224904667074444596598228786655001}{70509301182793819572007629808938403} a^{12} - \frac{26151420761182531956165800175662200}{70509301182793819572007629808938403} a^{11} - \frac{13710457599632165707879474503828950}{70509301182793819572007629808938403} a^{10} + \frac{34847632986797414788462786428840981}{70509301182793819572007629808938403} a^{9} - \frac{27877372592464159498887358108093954}{70509301182793819572007629808938403} a^{8} - \frac{19784768923332939422377826697985067}{70509301182793819572007629808938403} a^{7} - \frac{19751623407487622925830833987081988}{70509301182793819572007629808938403} a^{6} - \frac{26214873486753486396592811680182983}{70509301182793819572007629808938403} a^{5} + \frac{5819298955527797473709453818298330}{70509301182793819572007629808938403} a^{4} + \frac{24739719344441769034772040773534422}{70509301182793819572007629808938403} a^{3} - \frac{8772427458196899928247323189995564}{70509301182793819572007629808938403} a^{2} - \frac{21177873420333601926331179237518082}{70509301182793819572007629808938403} a - \frac{30120314283409044085764208200007}{83839834937923685579081605004683}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $21$ | 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: | \( 1590137231203360.8 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 11.11.1822837804551761449.1 |
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 | $22$ | $22$ | R | $22$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 67 | Data not computed | ||||||