Normalized defining polynomial
\( x^{21} - x^{20} - 278 x^{19} + 211 x^{18} + 32317 x^{17} - 11281 x^{16} - 2039432 x^{15} - 467981 x^{14} + 75279887 x^{13} + 73572952 x^{12} - 1596987413 x^{11} - 3083325600 x^{10} + 16530810553 x^{9} + 56124016529 x^{8} - 18817408458 x^{7} - 347856148950 x^{6} - 730940435110 x^{5} - 788186980737 x^{4} - 503882066680 x^{3} - 193054422458 x^{2} - 41100543009 x - 3748629959 \)
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: | \(483509712157573278109148029130668665453659848723054473746235392=2^{14}\cdot 29^{18}\cdot 3323\cdot 17923^{2}\cdot 102139^{2}\cdot 112213^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $965.99$ | 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: | $2, 29, 3323, 17923, 102139, 112213$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{2}{17} a^{17} + \frac{1}{17} a^{16} + \frac{3}{17} a^{15} - \frac{8}{17} a^{14} - \frac{6}{17} a^{13} + \frac{5}{17} a^{12} + \frac{8}{17} a^{11} - \frac{8}{17} a^{10} + \frac{2}{17} a^{9} - \frac{7}{17} a^{8} + \frac{5}{17} a^{6} - \frac{7}{17} a^{5} + \frac{3}{17} a^{4} - \frac{6}{17} a^{3} - \frac{2}{17} a^{2} + \frac{6}{17} a - \frac{5}{17}$, $\frac{1}{17} a^{19} - \frac{3}{17} a^{17} + \frac{1}{17} a^{16} + \frac{3}{17} a^{15} - \frac{7}{17} a^{14} - \frac{2}{17} a^{12} - \frac{7}{17} a^{11} + \frac{1}{17} a^{10} + \frac{6}{17} a^{9} - \frac{3}{17} a^{8} + \frac{5}{17} a^{7} + \frac{5}{17} a^{4} - \frac{7}{17} a^{3} - \frac{7}{17} a^{2} - \frac{7}{17}$, $\frac{1}{321145237513507002495418627484066647357650218335140340642074295762277} a^{20} + \frac{7215656124032939094574662693668385361047043066648532533952828862953}{321145237513507002495418627484066647357650218335140340642074295762277} a^{19} + \frac{3373441880725361768011488968702449886012794681575762316639741207314}{321145237513507002495418627484066647357650218335140340642074295762277} a^{18} - \frac{91533757265855271410035889536391896030505726524718316212831180442150}{321145237513507002495418627484066647357650218335140340642074295762277} a^{17} + \frac{73680128607322365576107613340709089452958385336242314487092401484734}{321145237513507002495418627484066647357650218335140340642074295762277} a^{16} + \frac{59126087152641609982489046549414327109322174511166086301104342519411}{321145237513507002495418627484066647357650218335140340642074295762277} a^{15} - \frac{46479418606093029546081006732513018651905554535109845608246258931604}{321145237513507002495418627484066647357650218335140340642074295762277} a^{14} + \frac{57059914146586533848623756250681558084088676439203349790515355809066}{321145237513507002495418627484066647357650218335140340642074295762277} a^{13} - \frac{17533917044387223857343395514951200070646664945019836470093887516860}{321145237513507002495418627484066647357650218335140340642074295762277} a^{12} + \frac{22238650258590080577132240441321977684191982139839216714736099623835}{321145237513507002495418627484066647357650218335140340642074295762277} a^{11} + \frac{67421204558657050165944395276123525299458605097325867342150756615798}{321145237513507002495418627484066647357650218335140340642074295762277} a^{10} + \frac{41995849316841727892970843455227583157136685834576324158999078131935}{321145237513507002495418627484066647357650218335140340642074295762277} a^{9} + \frac{72511690357494458311015044230441490902859299644762207976720254425700}{321145237513507002495418627484066647357650218335140340642074295762277} a^{8} - \frac{85429193950232144745211377097832586281531815013693144221659709746817}{321145237513507002495418627484066647357650218335140340642074295762277} a^{7} - \frac{97788548903069700478432543578061270971180494652949268269912022618988}{321145237513507002495418627484066647357650218335140340642074295762277} a^{6} + \frac{8028374574996191050400357099590884607744830778397310776003355973206}{18890896324323941323259919263768626315155895196184725920122017397781} a^{5} + \frac{19522331588558461017079981527425531789540697243300317618047603370278}{321145237513507002495418627484066647357650218335140340642074295762277} a^{4} + \frac{146943943930897932542840654932442774886458310161892058285712168018200}{321145237513507002495418627484066647357650218335140340642074295762277} a^{3} - \frac{7046684068312410706650723580832671132065091102194826863061321442665}{18890896324323941323259919263768626315155895196184725920122017397781} a^{2} + \frac{158928317123490908914420833584608319244384829496081415779394659220483}{321145237513507002495418627484066647357650218335140340642074295762277} a + \frac{6709402095707361574659343787244059127376777875946673307066835216}{95950175534361219747660181501065625144203829798368790152995009191}$
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: | \( 1567173051490000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1959552 |
| The 333 conjugacy class representatives for t21n123 are not computed |
| Character table for t21n123 is not computed |
Intermediate fields
| 7.7.594823321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | $21$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ | $21$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }$ | $21$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.24 | $x^{14} - 3 x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{2} + 2 x + 3$ | $2$ | $7$ | $14$ | 14T9 | $[2, 2, 2, 2]^{7}$ | |
| $29$ | 29.7.6.2 | $x^{7} - 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.14.12.1 | $x^{14} + 2407 x^{7} + 1839267$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ | |
| 3323 | Data not computed | ||||||
| 17923 | Data not computed | ||||||
| 102139 | Data not computed | ||||||
| 112213 | Data not computed | ||||||