Normalized defining polynomial
\( x^{21} - 4 x^{20} - 262 x^{19} + 498 x^{18} + 28760 x^{17} - 1460 x^{16} - 1645358 x^{15} - 2346530 x^{14} + 50774140 x^{13} + 131954593 x^{12} - 804361037 x^{11} - 2979538158 x^{10} + 5552672841 x^{9} + 30745724450 x^{8} - 7081032431 x^{7} - 143924148549 x^{6} - 65958243269 x^{5} + 278406118274 x^{4} + 185139245847 x^{3} - 190583514332 x^{2} - 112640151504 x + 4073697629 \)
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: | \(135445515708155977295451660588195107344558789043889961=7^{14}\cdot 197^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $338.91$ | 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: | $7, 197$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1379=7\cdot 197\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1379}(1,·)$, $\chi_{1379}(1346,·)$, $\chi_{1379}(198,·)$, $\chi_{1379}(1089,·)$, $\chi_{1379}(585,·)$, $\chi_{1379}(1163,·)$, $\chi_{1379}(695,·)$, $\chi_{1379}(1296,·)$, $\chi_{1379}(1360,·)$, $\chi_{1379}(1373,·)$, $\chi_{1379}(361,·)$, $\chi_{1379}(592,·)$, $\chi_{1379}(36,·)$, $\chi_{1379}(233,·)$, $\chi_{1379}(498,·)$, $\chi_{1379}(114,·)$, $\chi_{1379}(627,·)$, $\chi_{1379}(375,·)$, $\chi_{1379}(508,·)$, $\chi_{1379}(1149,·)$, $\chi_{1379}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1021} a^{17} + \frac{180}{1021} a^{16} + \frac{291}{1021} a^{15} - \frac{65}{1021} a^{14} + \frac{173}{1021} a^{13} + \frac{459}{1021} a^{12} + \frac{455}{1021} a^{11} + \frac{233}{1021} a^{10} - \frac{233}{1021} a^{9} - \frac{489}{1021} a^{8} + \frac{206}{1021} a^{7} + \frac{425}{1021} a^{6} + \frac{360}{1021} a^{5} - \frac{5}{1021} a^{4} + \frac{245}{1021} a^{3} + \frac{379}{1021} a^{2} + \frac{473}{1021} a + \frac{213}{1021}$, $\frac{1}{1477387} a^{18} + \frac{170}{1477387} a^{17} + \frac{216985}{1477387} a^{16} - \frac{309275}{1477387} a^{15} - \frac{700604}{1477387} a^{14} - \frac{351474}{1477387} a^{13} + \frac{278682}{1477387} a^{12} - \frac{618959}{1477387} a^{11} + \frac{721326}{1477387} a^{10} - \frac{678145}{1477387} a^{9} + \frac{163351}{1477387} a^{8} + \frac{554810}{1477387} a^{7} - \frac{255056}{1477387} a^{6} + \frac{121978}{1477387} a^{5} + \frac{450556}{1477387} a^{4} + \frac{325670}{1477387} a^{3} - \frac{292260}{1477387} a^{2} - \frac{424148}{1477387} a - \frac{575932}{1477387}$, $\frac{1}{13856412673} a^{19} - \frac{2766}{13856412673} a^{18} + \frac{5552169}{13856412673} a^{17} + \frac{950566353}{13856412673} a^{16} - \frac{454555217}{13856412673} a^{15} - \frac{3514145808}{13856412673} a^{14} - \frac{6270238324}{13856412673} a^{13} + \frac{3290704228}{13856412673} a^{12} - \frac{735195905}{13856412673} a^{11} - \frac{6834113793}{13856412673} a^{10} - \frac{889904371}{13856412673} a^{9} + \frac{3063589941}{13856412673} a^{8} - \frac{414014109}{13856412673} a^{7} + \frac{2956706386}{13856412673} a^{6} + \frac{3515536988}{13856412673} a^{5} + \frac{6276070815}{13856412673} a^{4} + \frac{1911920038}{13856412673} a^{3} + \frac{2829515734}{13856412673} a^{2} + \frac{4107765478}{13856412673} a + \frac{5638738701}{13856412673}$, $\frac{1}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{20} + \frac{3036722973839359326338183300866297659337321225283615788636543698862416489}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{19} - \frac{29312369018111262399106993576503689069157258585261536561107409097730933656366}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{18} + \frac{67218987288090473600195547545899347006893210353619628011666694574275401978789645}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{17} + \frac{48356768902575190389431469243799062761934327327377666080842296313412404062186403447}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{16} - \frac{27781353913204427412525704790868879080633480039522162465755986035576222215947186265}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{15} + \frac{291888134824277907546672989811182682197751764845667303960392367386573316431736639}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{14} + \frac{60589891131337712761704687310193225424106356744276763171888555007611266165674826137}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{13} - \frac{61508012622544608822980393289505053162412313254597615135128813192692226518303534871}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{12} + \frac{33743731382975850903248600459062818068468711335241290412868497724989065564375505763}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{11} + \frac{17785819700443874866439779441657560072663872376345485092328876312194551456969277602}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{10} - \frac{23183426607791060872525247402407253768671514746807697583374432886353529177197764890}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{9} - \frac{5412101502979367650977833825301390300282291995029350668949459662267509776184644593}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{8} - \frac{27550622810949643749275345175187810148081968721132896690078618992693046288930862273}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{7} + \frac{37101770606071890706886989936867648774212231956428491597177223575151652206608193215}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{6} + \frac{46220753219528106809654611750766421969884751085099255008069612323949674494392374174}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{5} - \frac{24992935352335686602208439604930089999818220800276447518825704501539652806770668034}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{4} + \frac{50482075076381968118147465609709409763001151906727972585915390963029567719585365692}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{3} + \frac{3131809299154489300405619098193951091259444532745245100692455905988552204452397157}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a^{2} + \frac{73361128450657621683481575177228232729936092198892957725142175543679716342968523323}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643} a - \frac{23244408581579810534879724087371033658333478096817417462219710886074837780550291590}{153049134962093675124438537545344054378198542776402202824101935183974334717306525643}$
Class group and class number
$C_{7}$, which has order $7$ (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: | \( 9941186336646355000 \) (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
| \(\Q(\zeta_{7})^+\), 7.7.58451728309129.1 |
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 | $21$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | $21$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| $197$ | 197.7.6.1 | $x^{7} - 197$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 197.7.6.1 | $x^{7} - 197$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 197.7.6.1 | $x^{7} - 197$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |