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Properties

Label 32.0.15729269092...0000.9
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number $3200$ (GRH)
Class group $[2, 2, 2, 2, 10, 20]$ (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, 87480, 0, 1108809, 0, 14027904, 0, 177460632, 0, 119220552, 0, 56712663, 0, 23604240, 0, 9141847, 0, 2274304, 0, 481887, 0, 91304, 0, 14936, 0, 1688, 0, 177, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 177*x^28 + 1688*x^26 + 14936*x^24 + 91304*x^22 + 481887*x^20 + 2274304*x^18 + 9141847*x^16 + 23604240*x^14 + 56712663*x^12 + 119220552*x^10 + 177460632*x^8 + 14027904*x^6 + 1108809*x^4 + 87480*x^2 + 6561)
 
gp: K = bnfinit(x^32 + 16*x^30 + 177*x^28 + 1688*x^26 + 14936*x^24 + 91304*x^22 + 481887*x^20 + 2274304*x^18 + 9141847*x^16 + 23604240*x^14 + 56712663*x^12 + 119220552*x^10 + 177460632*x^8 + 14027904*x^6 + 1108809*x^4 + 87480*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} + 16 x^{30} + 177 x^{28} + 1688 x^{26} + 14936 x^{24} + 91304 x^{22} + 481887 x^{20} + 2274304 x^{18} + 9141847 x^{16} + 23604240 x^{14} + 56712663 x^{12} + 119220552 x^{10} + 177460632 x^{8} + 14027904 x^{6} + 1108809 x^{4} + 87480 x^{2} + 6561 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(659,·)$, $\chi_{840}(407,·)$, $\chi_{840}(797,·)$, $\chi_{840}(673,·)$, $\chi_{840}(293,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(559,·)$, $\chi_{840}(307,·)$, $\chi_{840}(827,·)$, $\chi_{840}(629,·)$, $\chi_{840}(323,·)$, $\chi_{840}(71,·)$, $\chi_{840}(713,·)$, $\chi_{840}(461,·)$, $\chi_{840}(589,·)$, $\chi_{840}(337,·)$, $\chi_{840}(727,·)$, $\chi_{840}(223,·)$, $\chi_{840}(209,·)$, $\chi_{840}(421,·)$, $\chi_{840}(491,·)$, $\chi_{840}(743,·)$, $\chi_{840}(239,·)$, $\chi_{840}(757,·)$, $\chi_{840}(41,·)$, $\chi_{840}(377,·)$, $\chi_{840}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{7}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{8}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{9}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{18} - \frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{3} a^{8} - \frac{2}{9} a^{6} + \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{19} - \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{3} a^{9} - \frac{2}{9} a^{7} + \frac{1}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{27} a^{22} + \frac{1}{27} a^{20} - \frac{4}{27} a^{16} + \frac{2}{27} a^{14} - \frac{4}{27} a^{12} - \frac{1}{9} a^{10} + \frac{4}{27} a^{8} + \frac{1}{27} a^{6} + \frac{2}{9} a^{4}$, $\frac{1}{27} a^{23} + \frac{1}{27} a^{21} - \frac{4}{27} a^{17} + \frac{2}{27} a^{15} - \frac{4}{27} a^{13} - \frac{1}{9} a^{11} + \frac{4}{27} a^{9} + \frac{1}{27} a^{7} + \frac{2}{9} a^{5}$, $\frac{1}{405} a^{24} + \frac{7}{405} a^{22} - \frac{4}{135} a^{20} + \frac{10}{81} a^{18} + \frac{14}{405} a^{16} + \frac{44}{405} a^{14} - \frac{4}{45} a^{12} + \frac{58}{405} a^{10} + \frac{151}{405} a^{8} - \frac{44}{135} a^{6} - \frac{14}{45} a^{4} - \frac{2}{15} a^{2} + \frac{1}{5}$, $\frac{1}{1215} a^{25} - \frac{8}{1215} a^{23} + \frac{7}{135} a^{21} + \frac{28}{243} a^{19} - \frac{61}{1215} a^{17} + \frac{194}{1215} a^{15} + \frac{23}{405} a^{13} + \frac{13}{1215} a^{11} - \frac{44}{1215} a^{9} - \frac{19}{405} a^{7} + \frac{61}{135} a^{5} - \frac{7}{45} a^{3} + \frac{1}{15} a$, $\frac{1}{251171453418131480175} a^{26} - \frac{22498836232747277}{251171453418131480175} a^{24} + \frac{248581771079575528}{16744763561208765345} a^{22} + \frac{5621154584122610573}{251171453418131480175} a^{20} - \frac{39883150119177251311}{251171453418131480175} a^{18} + \frac{21400633945113374258}{251171453418131480175} a^{16} + \frac{2499802503347720912}{27907939268681275575} a^{14} + \frac{31530127067374130392}{251171453418131480175} a^{12} + \frac{35486400100442707249}{251171453418131480175} a^{10} - \frac{661121082753851807}{83723817806043826725} a^{8} + \frac{6330533491714797088}{27907939268681275575} a^{6} + \frac{36923533294197380}{372105856915750341} a^{4} - \frac{517658141937195728}{3100882140964586175} a^{2} - \frac{455705807501826388}{1033627380321528725}$, $\frac{1}{753514360254394440525} a^{27} - \frac{22498836232747277}{753514360254394440525} a^{25} - \frac{371594657113341707}{50234290683626296035} a^{23} - \frac{3681491838771147952}{753514360254394440525} a^{21} - \frac{123606967925221078036}{753514360254394440525} a^{19} + \frac{58611219636688408358}{753514360254394440525} a^{17} - \frac{8870098680189095063}{83723817806043826725} a^{15} - \frac{14983105047094662233}{753514360254394440525} a^{13} + \frac{63394339369123982824}{753514360254394440525} a^{11} - \frac{124696406721337298807}{251171453418131480175} a^{9} - \frac{22611033157288007212}{83723817806043826725} a^{7} + \frac{202303914145641976}{1116317570747251023} a^{5} + \frac{4650478759670447897}{9302646422893758525} a^{3} - \frac{455705807501826388}{3100882140964586175} a$, $\frac{1}{753514360254394440525} a^{28} + \frac{1}{753514360254394440525} a^{26} + \frac{18273888757537567}{27907939268681275575} a^{24} + \frac{4886536949236169123}{753514360254394440525} a^{22} + \frac{39206386467951075668}{753514360254394440525} a^{20} - \frac{1930403162889575962}{30140574410175777621} a^{18} - \frac{21964965788481348871}{251171453418131480175} a^{16} + \frac{108160354973199363196}{753514360254394440525} a^{14} - \frac{19589965901578018474}{150702872050878888105} a^{12} - \frac{970182647639520088}{251171453418131480175} a^{10} + \frac{25542187362194680211}{83723817806043826725} a^{8} + \frac{1068656891223918752}{3100882140964586175} a^{6} - \frac{1277355875987612573}{9302646422893758525} a^{4} + \frac{1206961134022874924}{3100882140964586175} a^{2} + \frac{19136063716618767}{1033627380321528725}$, $\frac{1}{2260543080763183321575} a^{29} + \frac{1}{2260543080763183321575} a^{27} + \frac{18273888757537567}{83723817806043826725} a^{25} - \frac{23021402319445106452}{2260543080763183321575} a^{23} + \frac{95022265005313626818}{2260543080763183321575} a^{21} - \frac{8628308587373082100}{90421723230527332863} a^{19} - \frac{68478197902950141496}{753514360254394440525} a^{17} + \frac{219792112047924465496}{2260543080763183321575} a^{15} + \frac{36225912635784532676}{452108616152636664315} a^{13} - \frac{970182647639520088}{753514360254394440525} a^{11} + \frac{13138658798336335511}{251171453418131480175} a^{9} - \frac{6920126064129190832}{83723817806043826725} a^{7} + \frac{789898884655444877}{27907939268681275575} a^{5} + \frac{1206961134022874924}{9302646422893758525} a^{3} - \frac{1014491316604909958}{3100882140964586175} a$, $\frac{1}{2260543080763183321575} a^{30} + \frac{1}{2260543080763183321575} a^{28} - \frac{724459023699572884}{2260543080763183321575} a^{24} + \frac{20383279657022970488}{2260543080763183321575} a^{22} + \frac{34631494240412819543}{2260543080763183321575} a^{20} + \frac{106772192663061328937}{753514360254394440525} a^{18} + \frac{202326860655534927874}{2260543080763183321575} a^{16} + \frac{282258060180224638483}{2260543080763183321575} a^{14} - \frac{111999569054501887339}{753514360254394440525} a^{12} + \frac{7312408848249061787}{251171453418131480175} a^{10} + \frac{975621380895912200}{3348952712241753069} a^{8} - \frac{3745125213128905936}{27907939268681275575} a^{6} - \frac{1378187248237547192}{3100882140964586175} a^{4} - \frac{303107764389432746}{620176428192917235} a^{2} + \frac{366664858798615388}{1033627380321528725}$, $\frac{1}{6781629242289549964725} a^{31} + \frac{1}{6781629242289549964725} a^{29} - \frac{724459023699572884}{6781629242289549964725} a^{25} + \frac{104107097463066797213}{6781629242289549964725} a^{23} + \frac{369526765464588126443}{6781629242289549964725} a^{21} - \frac{60675442949026324513}{2260543080763183321575} a^{19} - \frac{886082770823034819551}{6781629242289549964725} a^{17} + \frac{198534242374180811758}{6781629242289549964725} a^{15} - \frac{56183690517139336189}{2260543080763183321575} a^{13} + \frac{118944165922974164087}{753514360254394440525} a^{11} - \frac{760872618044256058}{10046858136725259207} a^{9} - \frac{27518554960524066611}{83723817806043826725} a^{7} + \frac{3789949653370096433}{9302646422893758525} a^{5} - \frac{169944413484579497}{620176428192917235} a^{3} + \frac{1400292239120144113}{3100882140964586175} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{20}$, which has order $3200$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{298507316350352}{452108616152636664315} a^{30} + \frac{4322688226643072}{452108616152636664315} a^{28} + \frac{15204980265744512}{150702872050878888105} a^{26} + \frac{84833441183203025}{90421723230527332863} a^{24} + \frac{3699136671376184176}{452108616152636664315} a^{22} + \frac{20539903023558002944}{452108616152636664315} a^{20} + \frac{34318286595875185232}{150702872050878888105} a^{18} + \frac{463499912587324422656}{452108616152636664315} a^{16} + \frac{1714073660343398142704}{452108616152636664315} a^{14} + \frac{110303849194591118464}{16744763561208765345} a^{12} + \frac{48423630143936620160}{3348952712241753069} a^{10} + \frac{131799549973661200384}{5581587853736255115} a^{8} + \frac{694566067969274096}{372105856915750341} a^{6} - \frac{297476532213100905392}{1860529284578751705} a^{4} + \frac{2406951398926256}{206725476064305745} a^{2} + \frac{181371533985024}{206725476064305745} \) (order $10$)
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:  \( 3297199900542.683 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{2}, \sqrt{105})\), \(\Q(\sqrt{2}, \sqrt{35})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{70})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{3}, \sqrt{70})\), \(\Q(\sqrt{6}, \sqrt{35})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{10}, \sqrt{42})\), \(\Q(\sqrt{14}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{10}, \sqrt{21})\), \(\Q(\sqrt{14}, \sqrt{15})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{21}, \sqrt{30})\), \(\Q(\sqrt{15}, \sqrt{42})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{30}, \sqrt{35})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), 4.0.392000.2, 4.0.98000.1, 4.0.72000.2, 4.0.18000.1, \(\Q(\zeta_{5})\), 4.0.8000.2, 4.0.55125.1, 4.0.3528000.1, 8.8.7965941760000.5, 8.8.497871360000.1, 8.8.7965941760000.4, 8.8.98344960000.1, 8.8.7965941760000.8, 8.8.3317760000.1, 8.8.12745506816.1, 8.8.497871360000.2, 8.8.7965941760000.6, 8.8.31116960000.1, 8.8.7965941760000.7, 8.8.7965941760000.2, 8.8.7965941760000.1, 8.8.7965941760000.9, 8.8.7965941760000.3, 8.0.2458624000000.6, 8.0.82944000000.6, 8.0.64000000.2, 8.0.12446784000000.14, 8.0.12446784000000.1, 8.0.777924000000.1, 8.0.3038765625.3, 8.0.12446784000000.5, 8.0.199148544000000.220, 8.0.199148544000000.140, 8.0.12446784000000.20, 8.0.12446784000000.11, 8.0.153664000000.6, 8.0.2458624000000.1, 8.0.12446784000000.4, 8.0.199148544000000.163, 8.0.2458624000000.3, 8.0.9604000000.3, 8.0.199148544000000.157, 8.0.777924000000.3, 8.0.12446784000000.9, 8.0.199148544000000.168, 8.0.5184000000.6, 8.0.82944000000.4, 8.0.199148544000000.137, 8.0.777924000000.5, 8.0.82944000000.1, 8.0.324000000.3, 16.16.63456228123711897600000000.1, 16.0.39660142577319936000000000000.32, 16.0.154922431942656000000000000.20, 16.0.6044831973376000000000000.5, 16.0.39660142577319936000000000000.10, 16.0.39660142577319936000000000000.36, 16.0.6879707136000000000000.4, 16.0.154922431942656000000000000.1, 16.0.39660142577319936000000000000.6, 16.0.39660142577319936000000000000.11, 16.0.605165749776000000000000.5, 16.0.39660142577319936000000000000.46, 16.0.39660142577319936000000000000.3, 16.0.39660142577319936000000000000.30, 16.0.39660142577319936000000000000.55

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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$