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Properties

Label 42.0.89605127606...7984.1
Degree $42$
Signature $[0, 21]$
Discriminant $-\,2^{42}\cdot 7^{28}\cdot 29^{36}$
Root discriminant $131.19$
Ramified primes $2, 7, 29$
Class number Not computed
Class group Not computed
Galois group $C_{42}$ (as 42T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8082649, 0, 221604496, 0, 2568252188, 0, 16842157767, 0, 70396440790, 0, 199814257891, 0, 400104193619, 0, 579268843957, 0, 616608189604, 0, 488271862609, 0, 290044496138, 0, 129986543401, 0, 44091529459, 0, 11324272470, 0, 2195598458, 0, 318981566, 0, 34265834, 0, 2662832, 0, 144488, 0, 5156, 0, 108, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649)
 
gp: K = bnfinit(x^42 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649, 1)
 

Normalized defining polynomial

\( x^{42} + 108 x^{40} + 5156 x^{38} + 144488 x^{36} + 2662832 x^{34} + 34265834 x^{32} + 318981566 x^{30} + 2195598458 x^{28} + 11324272470 x^{26} + 44091529459 x^{24} + 129986543401 x^{22} + 290044496138 x^{20} + 488271862609 x^{18} + 616608189604 x^{16} + 579268843957 x^{14} + 400104193619 x^{12} + 199814257891 x^{10} + 70396440790 x^{8} + 16842157767 x^{6} + 2568252188 x^{4} + 221604496 x^{2} + 8082649 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 21]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-89605127606769749671156289869299627336734024539522082907017794058334138710349541732777984=-\,2^{42}\cdot 7^{28}\cdot 29^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.19$
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:  $2, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(812=2^{2}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{812}(1,·)$, $\chi_{812}(645,·)$, $\chi_{812}(513,·)$, $\chi_{812}(393,·)$, $\chi_{812}(779,·)$, $\chi_{812}(141,·)$, $\chi_{812}(401,·)$, $\chi_{812}(529,·)$, $\chi_{812}(277,·)$, $\chi_{812}(23,·)$, $\chi_{812}(25,·)$, $\chi_{812}(687,·)$, $\chi_{812}(799,·)$, $\chi_{812}(291,·)$, $\chi_{812}(165,·)$, $\chi_{812}(807,·)$, $\chi_{812}(169,·)$, $\chi_{812}(683,·)$, $\chi_{812}(429,·)$, $\chi_{812}(431,·)$, $\chi_{812}(53,·)$, $\chi_{812}(697,·)$, $\chi_{812}(571,·)$, $\chi_{812}(575,·)$, $\chi_{812}(65,·)$, $\chi_{812}(603,·)$, $\chi_{812}(407,·)$, $\chi_{812}(197,·)$, $\chi_{812}(459,·)$, $\chi_{812}(81,·)$, $\chi_{812}(547,·)$, $\chi_{812}(471,·)$, $\chi_{812}(219,·)$, $\chi_{812}(487,·)$, $\chi_{812}(233,·)$, $\chi_{812}(107,·)$, $\chi_{812}(239,·)$, $\chi_{812}(625,·)$, $\chi_{812}(373,·)$, $\chi_{812}(123,·)$, $\chi_{812}(281,·)$, $\chi_{812}(639,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{41} a^{28} - \frac{2}{41} a^{26} + \frac{12}{41} a^{24} - \frac{12}{41} a^{22} + \frac{16}{41} a^{20} + \frac{11}{41} a^{18} + \frac{6}{41} a^{16} + \frac{2}{41} a^{14} - \frac{10}{41} a^{12} - \frac{16}{41} a^{10} - \frac{10}{41} a^{8} - \frac{2}{41} a^{6} - \frac{12}{41} a^{4} - \frac{17}{41} a^{2} + \frac{2}{41}$, $\frac{1}{41} a^{29} - \frac{2}{41} a^{27} + \frac{12}{41} a^{25} - \frac{12}{41} a^{23} + \frac{16}{41} a^{21} + \frac{11}{41} a^{19} + \frac{6}{41} a^{17} + \frac{2}{41} a^{15} - \frac{10}{41} a^{13} - \frac{16}{41} a^{11} - \frac{10}{41} a^{9} - \frac{2}{41} a^{7} - \frac{12}{41} a^{5} - \frac{17}{41} a^{3} + \frac{2}{41} a$, $\frac{1}{41} a^{30} + \frac{8}{41} a^{26} + \frac{12}{41} a^{24} - \frac{8}{41} a^{22} + \frac{2}{41} a^{20} - \frac{13}{41} a^{18} + \frac{14}{41} a^{16} - \frac{6}{41} a^{14} + \frac{5}{41} a^{12} - \frac{1}{41} a^{10} + \frac{19}{41} a^{8} - \frac{16}{41} a^{6} + \frac{9}{41} a^{2} + \frac{4}{41}$, $\frac{1}{41} a^{31} + \frac{8}{41} a^{27} + \frac{12}{41} a^{25} - \frac{8}{41} a^{23} + \frac{2}{41} a^{21} - \frac{13}{41} a^{19} + \frac{14}{41} a^{17} - \frac{6}{41} a^{15} + \frac{5}{41} a^{13} - \frac{1}{41} a^{11} + \frac{19}{41} a^{9} - \frac{16}{41} a^{7} + \frac{9}{41} a^{3} + \frac{4}{41} a$, $\frac{1}{41} a^{32} - \frac{13}{41} a^{26} + \frac{19}{41} a^{24} + \frac{16}{41} a^{22} - \frac{18}{41} a^{20} + \frac{8}{41} a^{18} - \frac{13}{41} a^{16} - \frac{11}{41} a^{14} - \frac{3}{41} a^{12} - \frac{17}{41} a^{10} - \frac{18}{41} a^{8} + \frac{16}{41} a^{6} - \frac{18}{41} a^{4} + \frac{17}{41} a^{2} - \frac{16}{41}$, $\frac{1}{41} a^{33} - \frac{13}{41} a^{27} + \frac{19}{41} a^{25} + \frac{16}{41} a^{23} - \frac{18}{41} a^{21} + \frac{8}{41} a^{19} - \frac{13}{41} a^{17} - \frac{11}{41} a^{15} - \frac{3}{41} a^{13} - \frac{17}{41} a^{11} - \frac{18}{41} a^{9} + \frac{16}{41} a^{7} - \frac{18}{41} a^{5} + \frac{17}{41} a^{3} - \frac{16}{41} a$, $\frac{1}{41} a^{34} - \frac{7}{41} a^{26} + \frac{8}{41} a^{24} - \frac{10}{41} a^{22} + \frac{11}{41} a^{20} + \frac{7}{41} a^{18} - \frac{15}{41} a^{16} - \frac{18}{41} a^{14} + \frac{17}{41} a^{12} + \frac{20}{41} a^{10} + \frac{9}{41} a^{8} - \frac{3}{41} a^{6} - \frac{16}{41} a^{4} + \frac{9}{41} a^{2} - \frac{15}{41}$, $\frac{1}{41} a^{35} - \frac{7}{41} a^{27} + \frac{8}{41} a^{25} - \frac{10}{41} a^{23} + \frac{11}{41} a^{21} + \frac{7}{41} a^{19} - \frac{15}{41} a^{17} - \frac{18}{41} a^{15} + \frac{17}{41} a^{13} + \frac{20}{41} a^{11} + \frac{9}{41} a^{9} - \frac{3}{41} a^{7} - \frac{16}{41} a^{5} + \frac{9}{41} a^{3} - \frac{15}{41} a$, $\frac{1}{697} a^{36} - \frac{3}{697} a^{34} - \frac{2}{697} a^{32} + \frac{7}{697} a^{30} + \frac{5}{697} a^{28} + \frac{128}{697} a^{26} + \frac{156}{697} a^{24} - \frac{150}{697} a^{22} + \frac{11}{697} a^{20} - \frac{216}{697} a^{18} - \frac{146}{697} a^{16} + \frac{157}{697} a^{14} - \frac{233}{697} a^{12} - \frac{298}{697} a^{10} - \frac{309}{697} a^{8} + \frac{9}{41} a^{6} - \frac{10}{697} a^{4} - \frac{299}{697} a^{2} - \frac{322}{697}$, $\frac{1}{697} a^{37} - \frac{3}{697} a^{35} - \frac{2}{697} a^{33} + \frac{7}{697} a^{31} + \frac{5}{697} a^{29} + \frac{128}{697} a^{27} + \frac{156}{697} a^{25} - \frac{150}{697} a^{23} + \frac{11}{697} a^{21} - \frac{216}{697} a^{19} - \frac{146}{697} a^{17} + \frac{157}{697} a^{15} - \frac{233}{697} a^{13} - \frac{298}{697} a^{11} - \frac{309}{697} a^{9} + \frac{9}{41} a^{7} - \frac{10}{697} a^{5} - \frac{299}{697} a^{3} - \frac{322}{697} a$, $\frac{1}{187826832067837} a^{38} - \frac{50723618660}{187826832067837} a^{36} + \frac{1221832958179}{187826832067837} a^{34} + \frac{510713146435}{187826832067837} a^{32} + \frac{1608511785163}{187826832067837} a^{30} + \frac{774026028124}{187826832067837} a^{28} - \frac{44462739652157}{187826832067837} a^{26} - \frac{9249828091152}{187826832067837} a^{24} + \frac{84424648732937}{187826832067837} a^{22} + \frac{32995320682301}{187826832067837} a^{20} - \frac{10403622782925}{187826832067837} a^{18} + \frac{92644711871534}{187826832067837} a^{16} - \frac{17620678824946}{187826832067837} a^{14} - \frac{59400134441736}{187826832067837} a^{12} - \frac{86652089205468}{187826832067837} a^{10} - \frac{61428513121647}{187826832067837} a^{8} - \frac{23970126732222}{187826832067837} a^{6} + \frac{72070737010553}{187826832067837} a^{4} - \frac{77245390807947}{187826832067837} a^{2} + \frac{26125422954385}{187826832067837}$, $\frac{1}{187826832067837} a^{39} - \frac{50723618660}{187826832067837} a^{37} + \frac{1221832958179}{187826832067837} a^{35} + \frac{510713146435}{187826832067837} a^{33} + \frac{1608511785163}{187826832067837} a^{31} + \frac{774026028124}{187826832067837} a^{29} - \frac{44462739652157}{187826832067837} a^{27} - \frac{9249828091152}{187826832067837} a^{25} + \frac{84424648732937}{187826832067837} a^{23} + \frac{32995320682301}{187826832067837} a^{21} - \frac{10403622782925}{187826832067837} a^{19} + \frac{92644711871534}{187826832067837} a^{17} - \frac{17620678824946}{187826832067837} a^{15} - \frac{59400134441736}{187826832067837} a^{13} - \frac{86652089205468}{187826832067837} a^{11} - \frac{61428513121647}{187826832067837} a^{9} - \frac{23970126732222}{187826832067837} a^{7} + \frac{72070737010553}{187826832067837} a^{5} - \frac{77245390807947}{187826832067837} a^{3} + \frac{26125422954385}{187826832067837} a$, $\frac{1}{501667326021261135213832084050952333159085793092549158765718263} a^{40} - \frac{1243628270415200744066279498193925775039299671974}{501667326021261135213832084050952333159085793092549158765718263} a^{38} + \frac{186407575173566324770089807794548341920991353473072665047842}{501667326021261135213832084050952333159085793092549158765718263} a^{36} + \frac{72288998656260996525835991407536611257300729780938704615486}{29509842707133007953754828473585431362299164299561715221512839} a^{34} + \frac{1038281384082175196376835340155465612862221449872686002814420}{501667326021261135213832084050952333159085793092549158765718263} a^{32} + \frac{559312251689909877369336643207316294766788752890551098688852}{501667326021261135213832084050952333159085793092549158765718263} a^{30} - \frac{529855916933795890571779364770736683261537984924396350516048}{501667326021261135213832084050952333159085793092549158765718263} a^{28} - \frac{210353765831822577162738992288653310589793219916323492679382784}{501667326021261135213832084050952333159085793092549158765718263} a^{26} + \frac{93811002048910474631664125459522094717409296688338442033181341}{501667326021261135213832084050952333159085793092549158765718263} a^{24} - \frac{14550899322721259144925765954498367101454296896742127992441513}{501667326021261135213832084050952333159085793092549158765718263} a^{22} - \frac{106182384479409501835109193337383523510147814852113835671270222}{501667326021261135213832084050952333159085793092549158765718263} a^{20} - \frac{107025276285391146442635752179287389290282114656900936113020958}{501667326021261135213832084050952333159085793092549158765718263} a^{18} - \frac{10677518948680778671485740784212577246743318187103891023413469}{501667326021261135213832084050952333159085793092549158765718263} a^{16} - \frac{100561144304072873130548992699512236377465390455447717616355707}{501667326021261135213832084050952333159085793092549158765718263} a^{14} + \frac{103296294773322818298970910915412067113018959367449965550790836}{501667326021261135213832084050952333159085793092549158765718263} a^{12} + \frac{69710897224828543332563733395340847023310137546662316564620792}{501667326021261135213832084050952333159085793092549158765718263} a^{10} - \frac{112066122683081530200559389820334555397552433018648747467566606}{501667326021261135213832084050952333159085793092549158765718263} a^{8} - \frac{82527351896140082748514390523588734916701924262156240654499367}{501667326021261135213832084050952333159085793092549158765718263} a^{6} + \frac{241657470298800816331465054117783368234643749021272850626152707}{501667326021261135213832084050952333159085793092549158765718263} a^{4} + \frac{170848813588597641307451554479482565960379755178976685742332094}{501667326021261135213832084050952333159085793092549158765718263} a^{2} - \frac{66547463883041791662806900629257629836752794684686583293349825}{501667326021261135213832084050952333159085793092549158765718263}$, $\frac{1}{1426240207878445407412924614956857483171280909762117258370937021709} a^{41} - \frac{3748521110720980677608454875848669528811192058856471}{1426240207878445407412924614956857483171280909762117258370937021709} a^{39} - \frac{189047560017630698165126308193932779653494769746609687014190971}{1426240207878445407412924614956857483171280909762117258370937021709} a^{37} + \frac{15819741110195131442735759413148431474250777235824145261282369480}{1426240207878445407412924614956857483171280909762117258370937021709} a^{35} - \frac{2255347668550474340325516344295423363926201300728387071428786979}{1426240207878445407412924614956857483171280909762117258370937021709} a^{33} - \frac{10652108710693218376793770344312108752849833979701351835944102413}{1426240207878445407412924614956857483171280909762117258370937021709} a^{31} + \frac{5150128631282616966330423923708406643578800252374979501809095618}{1426240207878445407412924614956857483171280909762117258370937021709} a^{29} + \frac{519553195282846745027085505351867634164614664525988285175018371311}{1426240207878445407412924614956857483171280909762117258370937021709} a^{27} + \frac{298277659146998108160169952904380242359615450459900562808546018476}{1426240207878445407412924614956857483171280909762117258370937021709} a^{25} - \frac{452402396142455230569296143674232287203236547664611436390370507733}{1426240207878445407412924614956857483171280909762117258370937021709} a^{23} + \frac{548436469111230374223033867509413208231581876371321443547739888429}{1426240207878445407412924614956857483171280909762117258370937021709} a^{21} + \frac{544387599705356771641012177621877586788822611516786165907442910269}{1426240207878445407412924614956857483171280909762117258370937021709} a^{19} - \frac{604996418392878219195778545341181062464677650906395425016868639420}{1426240207878445407412924614956857483171280909762117258370937021709} a^{17} - \frac{15432056247211080258666462414177620799351335479069534528483993276}{83896482816379141612524977350403381363016524103653956374761001277} a^{15} + \frac{464728213665426109816498409597004551975979041247142934049629634711}{1426240207878445407412924614956857483171280909762117258370937021709} a^{13} - \frac{456769617203193900736014952081528154628089202917065256387801377194}{1426240207878445407412924614956857483171280909762117258370937021709} a^{11} - \frac{527060734971994666952468046916972073387888189970827796426133484344}{1426240207878445407412924614956857483171280909762117258370937021709} a^{9} - \frac{508651044832602474750103472186146222381693037339441738547403972448}{1426240207878445407412924614956857483171280909762117258370937021709} a^{7} - \frac{13034900492275051983447313275695747329417781566065338273626376193}{1426240207878445407412924614956857483171280909762117258370937021709} a^{5} - \frac{703254526303850303534075416980005670427355830222812202080419508525}{1426240207878445407412924614956857483171280909762117258370937021709} a^{3} + \frac{113077122384889800414337841993698271015310415717845787491166934494}{1426240207878445407412924614956857483171280909762117258370937021709} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2079519440197074364894543310912036}{617606824923260580261289005004530509} a^{41} - \frac{223792148505866147537082121599344301}{617606824923260580261289005004530509} a^{39} - \frac{10636344041198208571666816465416386860}{617606824923260580261289005004530509} a^{37} - \frac{296394453047644195643303005771952963837}{617606824923260580261289005004530509} a^{35} - \frac{5423963113863009604822895672333752239734}{617606824923260580261289005004530509} a^{33} - \frac{69180388157686556054042142415302571287068}{617606824923260580261289005004530509} a^{31} - \frac{636848736631251001186852339750665602939765}{617606824923260580261289005004530509} a^{29} - \frac{4322026997456446865556448691305118563200849}{617606824923260580261289005004530509} a^{27} - \frac{21894719750636794813227130896379825276415012}{617606824923260580261289005004530509} a^{25} - \frac{83308580934558081364287326503066953732978688}{617606824923260580261289005004530509} a^{23} - \frac{238421367470295726634472432867311376715621166}{617606824923260580261289005004530509} a^{21} - \frac{511891268852295488217807387920234964099740529}{617606824923260580261289005004530509} a^{19} - \frac{819428936512667177149992781397757308749218786}{617606824923260580261289005004530509} a^{17} - \frac{968582989840749820109151945190578012552926028}{617606824923260580261289005004530509} a^{15} - \frac{833834967171972015630785246764628319299537276}{617606824923260580261289005004530509} a^{13} - \frac{512833263667997644307112189948970515856585038}{617606824923260580261289005004530509} a^{11} - \frac{219201824971949301496786614413528476188460919}{617606824923260580261289005004530509} a^{9} - \frac{62477354549333550624539346301378908190900309}{617606824923260580261289005004530509} a^{7} - \frac{11106023942762739321171474383039323264278006}{617606824923260580261289005004530509} a^{5} - \frac{1089072856117519150253481080449568849408350}{617606824923260580261289005004530509} a^{3} - \frac{43906924214563157399060034454102838747056}{617606824923260580261289005004530509} a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{42}$ (as 42T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 6.0.153664.1, 7.7.594823321.1, 14.0.5796901408038404767744.1, 21.21.142736986105602839685204351151303673689.1

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 R $42$ $21^{2}$ R $42$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{14}$ $42$ $42$ R $42$ $21^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{42}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $21^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
7Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$