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Properties

Label 32.0.87095214414...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{24}\cdot 17^{24}$
Root discriminant $48.49$
Ramified primes $3, 5, 17$
Class number $300$ (GRH)
Class group $[5, 60]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -17, -15, -75, 311, 1476, 111, -8701, 9977, 19105, -16109, 113360, -246837, -284709, 1584805, 284709, -246837, -113360, -16109, -19105, 9977, 8701, 111, -1476, 311, 75, -15, 17, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 6*x^30 + 17*x^29 - 15*x^28 + 75*x^27 + 311*x^26 - 1476*x^25 + 111*x^24 + 8701*x^23 + 9977*x^22 - 19105*x^21 - 16109*x^20 - 113360*x^19 - 246837*x^18 + 284709*x^17 + 1584805*x^16 - 284709*x^15 - 246837*x^14 + 113360*x^13 - 16109*x^12 + 19105*x^11 + 9977*x^10 - 8701*x^9 + 111*x^8 + 1476*x^7 + 311*x^6 - 75*x^5 - 15*x^4 - 17*x^3 - 6*x^2 + x + 1)
 
gp: K = bnfinit(x^32 - x^31 - 6*x^30 + 17*x^29 - 15*x^28 + 75*x^27 + 311*x^26 - 1476*x^25 + 111*x^24 + 8701*x^23 + 9977*x^22 - 19105*x^21 - 16109*x^20 - 113360*x^19 - 246837*x^18 + 284709*x^17 + 1584805*x^16 - 284709*x^15 - 246837*x^14 + 113360*x^13 - 16109*x^12 + 19105*x^11 + 9977*x^10 - 8701*x^9 + 111*x^8 + 1476*x^7 + 311*x^6 - 75*x^5 - 15*x^4 - 17*x^3 - 6*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 6 x^{30} + 17 x^{29} - 15 x^{28} + 75 x^{27} + 311 x^{26} - 1476 x^{25} + 111 x^{24} + 8701 x^{23} + 9977 x^{22} - 19105 x^{21} - 16109 x^{20} - 113360 x^{19} - 246837 x^{18} + 284709 x^{17} + 1584805 x^{16} - 284709 x^{15} - 246837 x^{14} + 113360 x^{13} - 16109 x^{12} + 19105 x^{11} + 9977 x^{10} - 8701 x^{9} + 111 x^{8} + 1476 x^{7} + 311 x^{6} - 75 x^{5} - 15 x^{4} - 17 x^{3} - 6 x^{2} + x + 1 \)

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:  \(870952144140052907613964449886024608671665191650390625=3^{16}\cdot 5^{24}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.49$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(4,·)$, $\chi_{255}(137,·)$, $\chi_{255}(13,·)$, $\chi_{255}(16,·)$, $\chi_{255}(149,·)$, $\chi_{255}(217,·)$, $\chi_{255}(152,·)$, $\chi_{255}(154,·)$, $\chi_{255}(157,·)$, $\chi_{255}(38,·)$, $\chi_{255}(169,·)$, $\chi_{255}(47,·)$, $\chi_{255}(52,·)$, $\chi_{255}(188,·)$, $\chi_{255}(191,·)$, $\chi_{255}(64,·)$, $\chi_{255}(67,·)$, $\chi_{255}(203,·)$, $\chi_{255}(208,·)$, $\chi_{255}(86,·)$, $\chi_{255}(89,·)$, $\chi_{255}(98,·)$, $\chi_{255}(101,·)$, $\chi_{255}(103,·)$, $\chi_{255}(106,·)$, $\chi_{255}(239,·)$, $\chi_{255}(242,·)$, $\chi_{255}(118,·)$, $\chi_{255}(251,·)$, $\chi_{255}(254,·)$, $\chi_{255}(166,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{354} a^{20} - \frac{1}{118} a^{15} + \frac{64}{177} a^{10} + \frac{1}{118} a^{5} + \frac{1}{354}$, $\frac{1}{354} a^{21} - \frac{1}{118} a^{16} + \frac{64}{177} a^{11} + \frac{1}{118} a^{6} + \frac{1}{354} a$, $\frac{1}{354} a^{22} - \frac{1}{118} a^{17} - \frac{49}{354} a^{12} - \frac{1}{2} a^{9} + \frac{1}{118} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{354} a^{2} - \frac{1}{2}$, $\frac{1}{354} a^{23} - \frac{1}{118} a^{18} - \frac{49}{354} a^{13} - \frac{1}{2} a^{10} + \frac{1}{118} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} + \frac{1}{354} a^{3} - \frac{1}{2} a$, $\frac{1}{1416} a^{24} - \frac{15}{118} a^{19} + \frac{1}{8} a^{18} + \frac{16}{177} a^{14} + \frac{1}{8} a^{12} - \frac{22}{59} a^{9} - \frac{3}{8} a^{6} + \frac{133}{354} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8}$, $\frac{1}{1416} a^{25} + \frac{1}{8} a^{19} + \frac{37}{177} a^{15} + \frac{1}{8} a^{13} - \frac{6}{59} a^{10} - \frac{3}{8} a^{7} - \frac{43}{177} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a - \frac{22}{59}$, $\frac{1}{1969759368} a^{26} + \frac{134503}{1969759368} a^{25} - \frac{205081}{984879684} a^{24} - \frac{57137}{54715538} a^{23} + \frac{161603}{246219921} a^{22} - \frac{184448}{246219921} a^{21} + \frac{2612893}{1969759368} a^{20} - \frac{120142355}{656586456} a^{19} + \frac{50383913}{328293228} a^{18} - \frac{11765725}{164146614} a^{17} - \frac{25435175}{246219921} a^{16} + \frac{117696515}{492439842} a^{15} - \frac{216416011}{1969759368} a^{14} - \frac{133462003}{656586456} a^{13} + \frac{181703029}{984879684} a^{12} - \frac{80763233}{492439842} a^{11} - \frac{54783368}{246219921} a^{10} - \frac{23821691}{164146614} a^{9} - \frac{292272209}{656586456} a^{8} - \frac{277797295}{656586456} a^{7} - \frac{359971435}{984879684} a^{6} - \frac{69458855}{492439842} a^{5} - \frac{94452479}{492439842} a^{4} + \frac{54544127}{164146614} a^{3} + \frac{362552833}{1969759368} a^{2} - \frac{474685081}{1969759368} a + \frac{353564387}{984879684}$, $\frac{1}{1969759368} a^{27} - \frac{31267}{109431076} a^{25} + \frac{180593}{1969759368} a^{24} + \frac{3286}{2437821} a^{23} - \frac{123191}{492439842} a^{22} + \frac{356297}{1969759368} a^{21} - \frac{1568}{246219921} a^{20} - \frac{20207735}{109431076} a^{19} + \frac{121145851}{656586456} a^{18} - \frac{4083443}{246219921} a^{17} + \frac{3082367}{82073307} a^{16} + \frac{149465}{218862152} a^{15} + \frac{56315522}{246219921} a^{14} + \frac{42493597}{984879684} a^{13} + \frac{388675553}{1969759368} a^{12} + \frac{86272922}{246219921} a^{11} + \frac{107997893}{246219921} a^{10} + \frac{19239485}{218862152} a^{9} + \frac{19483109}{82073307} a^{8} + \frac{403172735}{984879684} a^{7} + \frac{10130827}{656586456} a^{6} + \frac{11311132}{27357769} a^{5} - \frac{45821687}{246219921} a^{4} + \frac{692208325}{1969759368} a^{3} - \frac{24410}{2437821} a^{2} + \frac{106804453}{984879684} a - \frac{805760455}{1969759368}$, $\frac{1}{25606871784} a^{28} - \frac{1}{4267811964} a^{27} - \frac{1}{25606871784} a^{26} + \frac{81421}{6401717946} a^{25} - \frac{1296901}{4267811964} a^{24} + \frac{646277}{3200858973} a^{23} + \frac{18737989}{25606871784} a^{22} - \frac{4240651}{4267811964} a^{21} + \frac{16170119}{25606871784} a^{20} + \frac{62330467}{355650997} a^{19} - \frac{70713191}{12803435892} a^{18} + \frac{37921748}{1066952991} a^{17} - \frac{1519504295}{25606871784} a^{16} + \frac{1649816989}{12803435892} a^{15} - \frac{274631085}{2845207976} a^{14} + \frac{30719422}{3200858973} a^{13} + \frac{2924553211}{12803435892} a^{12} - \frac{5500903}{12055966} a^{11} - \frac{5901659471}{25606871784} a^{10} + \frac{628568691}{1422603988} a^{9} + \frac{3397582231}{25606871784} a^{8} + \frac{148301765}{355650997} a^{7} + \frac{5245023061}{12803435892} a^{6} + \frac{741044153}{3200858973} a^{5} - \frac{2278440037}{8535623928} a^{4} + \frac{1535535647}{12803435892} a^{3} + \frac{2694533347}{25606871784} a^{2} - \frac{432946439}{1066952991} a - \frac{696453257}{12803435892}$, $\frac{1}{6134100528985416} a^{29} - \frac{1907}{511175044082118} a^{28} - \frac{5849}{78642314474172} a^{27} + \frac{308207}{3067050264492708} a^{26} - \frac{163062511435}{681566725442824} a^{25} + \frac{349409562487}{2044700176328472} a^{24} + \frac{756083168693}{6134100528985416} a^{23} + \frac{46110199958}{58981735855629} a^{22} - \frac{2523388302349}{3067050264492708} a^{21} - \frac{4319458631795}{3067050264492708} a^{20} - \frac{624175389815395}{6134100528985416} a^{19} + \frac{32000591769395}{157284628948344} a^{18} - \frac{52289714689}{2044700176328472} a^{17} + \frac{21302226179009}{1533525132246354} a^{16} - \frac{9050114788609}{1022350088164236} a^{15} + \frac{197143145890501}{1022350088164236} a^{14} - \frac{1024998188490983}{6134100528985416} a^{13} - \frac{643679194760335}{6134100528985416} a^{12} - \frac{1741107009625195}{6134100528985416} a^{11} + \frac{379024375251413}{766762566123177} a^{10} + \frac{7570673309567}{235926943422516} a^{9} - \frac{91399697073091}{1022350088164236} a^{8} - \frac{334588244110607}{681566725442824} a^{7} - \frac{2171159704532759}{6134100528985416} a^{6} - \frac{444441410084657}{2044700176328472} a^{5} + \frac{14011854280675}{39321157237086} a^{4} + \frac{1453122383333749}{3067050264492708} a^{3} - \frac{141461722907659}{3067050264492708} a^{2} + \frac{7465894582141}{471853886845032} a - \frac{1423077410287063}{6134100528985416}$, $\frac{1}{6134100528985416} a^{30} - \frac{233}{3067050264492708} a^{28} - \frac{62689}{766762566123177} a^{27} + \frac{2157}{26214104824724} a^{26} - \frac{1352274328655}{6134100528985416} a^{25} + \frac{1487328217873}{6134100528985416} a^{24} + \frac{305435164300}{255587522041059} a^{23} - \frac{4322374284299}{3067050264492708} a^{22} - \frac{17195950114}{19660578618543} a^{21} - \frac{658547564555}{3067050264492708} a^{20} + \frac{155913960627599}{2044700176328472} a^{19} + \frac{854513081232769}{6134100528985416} a^{18} + \frac{102787172245679}{1533525132246354} a^{17} + \frac{233500128846695}{1022350088164236} a^{16} - \frac{129088000748065}{1533525132246354} a^{15} + \frac{758157204477877}{3067050264492708} a^{14} - \frac{291541824640753}{2044700176328472} a^{13} + \frac{88789974735289}{471853886845032} a^{12} - \frac{40205677750895}{255587522041059} a^{11} + \frac{3350517859529}{3067050264492708} a^{10} - \frac{6241751188795}{511175044082118} a^{9} - \frac{83751265021955}{235926943422516} a^{8} + \frac{1772789142801205}{6134100528985416} a^{7} - \frac{602933191327637}{2044700176328472} a^{6} + \frac{5312301942677}{766762566123177} a^{5} - \frac{1313089819543039}{3067050264492708} a^{4} - \frac{3043979331703}{6553526206181} a^{3} + \frac{185504834312719}{3067050264492708} a^{2} - \frac{317144638281677}{2044700176328472} a - \frac{1603532328557}{13107052412362}$, $\frac{1}{6134100528985416} a^{31} + \frac{93347}{6134100528985416} a^{28} + \frac{34711}{340783362721412} a^{27} + \frac{375293}{2044700176328472} a^{26} - \frac{766519517291}{6134100528985416} a^{25} - \frac{112694595788}{766762566123177} a^{24} - \frac{380070130799}{766762566123177} a^{23} + \frac{5495320160287}{6134100528985416} a^{22} + \frac{2474053998017}{3067050264492708} a^{21} - \frac{1996231597331}{2044700176328472} a^{20} + \frac{76997339687639}{2044700176328472} a^{19} + \frac{30011372993789}{766762566123177} a^{18} - \frac{69035891915539}{511175044082118} a^{17} - \frac{414075287279423}{2044700176328472} a^{16} + \frac{537611288896777}{3067050264492708} a^{15} + \frac{481970193599803}{6134100528985416} a^{14} + \frac{112454782291861}{6134100528985416} a^{13} + \frac{3172106701271}{1533525132246354} a^{12} + \frac{143889633680165}{1533525132246354} a^{11} + \frac{638843772263285}{2044700176328472} a^{10} + \frac{117178195741231}{1022350088164236} a^{9} + \frac{747924234355655}{6134100528985416} a^{8} + \frac{574656446369891}{2044700176328472} a^{7} - \frac{101476792501979}{255587522041059} a^{6} - \frac{326756683566880}{766762566123177} a^{5} - \frac{2287896755200705}{6134100528985416} a^{4} + \frac{781086356201881}{3067050264492708} a^{3} - \frac{1216191503296769}{6134100528985416} a^{2} - \frac{12096672494867}{766762566123177} a - \frac{67573754726689}{255587522041059}$

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

Class group and class number

$C_{5}\times C_{60}$, which has order $300$ (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:  \( -\frac{734694447181295}{6134100528985416} a^{31} + \frac{1467544286039665}{6134100528985416} a^{30} + \frac{913295386088575}{1533525132246354} a^{29} - \frac{1405601230661515}{511175044082118} a^{28} + \frac{908070429201515}{235926943422516} a^{27} - \frac{7383495044137575}{681566725442824} a^{26} - \frac{173211422005225795}{6134100528985416} a^{25} + \frac{655312902394844795}{3067050264492708} a^{24} - \frac{97518739981560695}{511175044082118} a^{23} - \frac{120762802501129175}{117963471711258} a^{22} - \frac{476815056625518685}{3067050264492708} a^{21} + \frac{7053868649898594755}{2044700176328472} a^{20} - \frac{2409723649869904195}{6134100528985416} a^{19} + \frac{11973647161856610395}{1022350088164236} a^{18} + \frac{24659955313970377085}{1533525132246354} a^{17} - \frac{32286390464798943085}{511175044082118} a^{16} - \frac{474952209144674283235}{3067050264492708} a^{15} + \frac{1365392934602793967505}{6134100528985416} a^{14} - \frac{6899433435810351167}{681566725442824} a^{13} - \frac{131541287373149824745}{3067050264492708} a^{12} + \frac{23645197823405592845}{1533525132246354} a^{11} - \frac{1071632566757249270}{255587522041059} a^{10} + \frac{256220057214321035}{235926943422516} a^{9} + \frac{4547907605312721095}{2044700176328472} a^{8} - \frac{6435101090273447315}{6134100528985416} a^{7} - \frac{166139622284313575}{1022350088164236} a^{6} + \frac{106356179053171225}{766762566123177} a^{5} + \frac{5421365453083735}{117963471711258} a^{4} - \frac{7306660913527325}{1022350088164236} a^{3} + \frac{1457706374984065}{6134100528985416} a^{2} - \frac{2008266058289305}{1533525132246354} a - \frac{5112117658220315}{6134100528985416} \) (order $30$)
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:  \( 2940410246815.5933 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{85})\), 4.0.614125.2, 4.4.5527125.2, \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{-15}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-15}, \sqrt{-51})\), 4.0.614125.1, 4.4.5527125.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\zeta_{15})^+\), 4.0.36125.1, 4.4.325125.1, \(\Q(\zeta_{5})\), 4.0.44217.1, 4.4.122825.1, 4.0.1105425.1, 4.4.4913.1, 8.0.30549110765625.4, 8.0.4228250625.1, 8.0.30549110765625.3, 8.0.377149515625.1, 8.0.30549110765625.1, 8.8.30549110765625.1, 8.0.30549110765625.2, 8.0.105706265625.1, 8.0.105706265625.3, 8.0.1221964430625.1, 8.0.1221964430625.2, 8.8.105706265625.1, 8.0.1305015625.1, 8.0.1221964430625.3, 8.8.15085980625.1, \(\Q(\zeta_{15})\), 8.0.105706265625.2, 8.0.1955143089.1, 8.0.1221964430625.4, 16.0.933248168570425273681640625.2, 16.0.11173814592383056640625.1, 16.0.1493197069712680437890625.1, 16.0.933248168570425273681640625.1, 16.0.142241757136172119140625.1, 16.16.933248168570425273681640625.1, 16.0.933248168570425273681640625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$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}$
5Data not computed
17Data not computed