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Properties

Label 32.0.15729269092...000.22
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 Not computed
Class group Not computed
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![43046721, 0, 38263752, 0, 4782969, 0, -17478504, 0, -15536448, 0, -11395728, 0, -1365417, 0, 5205456, 0, 4620511, 0, 578384, 0, -16857, 0, -15632, 0, -2368, 0, -296, 0, 9, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 8*x^30 + 9*x^28 - 296*x^26 - 2368*x^24 - 15632*x^22 - 16857*x^20 + 578384*x^18 + 4620511*x^16 + 5205456*x^14 - 1365417*x^12 - 11395728*x^10 - 15536448*x^8 - 17478504*x^6 + 4782969*x^4 + 38263752*x^2 + 43046721)
 
gp: K = bnfinit(x^32 + 8*x^30 + 9*x^28 - 296*x^26 - 2368*x^24 - 15632*x^22 - 16857*x^20 + 578384*x^18 + 4620511*x^16 + 5205456*x^14 - 1365417*x^12 - 11395728*x^10 - 15536448*x^8 - 17478504*x^6 + 4782969*x^4 + 38263752*x^2 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} + 8 x^{30} + 9 x^{28} - 296 x^{26} - 2368 x^{24} - 15632 x^{22} - 16857 x^{20} + 578384 x^{18} + 4620511 x^{16} + 5205456 x^{14} - 1365417 x^{12} - 11395728 x^{10} - 15536448 x^{8} - 17478504 x^{6} + 4782969 x^{4} + 38263752 x^{2} + 43046721 \)

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$$\chi_{840}(1,·)$, $\chi_{840}(517,·)$, $\chi_{840}(391,·)$, $\chi_{840}(13,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(797,·)$, $\chi_{840}(671,·)$, $\chi_{840}(673,·)$, $\chi_{840}(547,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(43,·)$, $\chi_{840}(559,·)$, $\chi_{840}(181,·)$, $\chi_{840}(827,·)$, $\chi_{840}(449,·)$, $\chi_{840}(323,·)$, $\chi_{840}(839,·)$, $\chi_{840}(461,·)$, $\chi_{840}(337,·)$, $\chi_{840}(211,·)$, $\chi_{840}(727,·)$, $\chi_{840}(349,·)$, $\chi_{840}(223,·)$, $\chi_{840}(617,·)$, $\chi_{840}(491,·)$, $\chi_{840}(113,·)$, $\chi_{840}(629,·)$, $\chi_{840}(503,·)$, $\chi_{840}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{16} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{19} - \frac{1}{27} a^{17} - \frac{1}{3} a^{15} + \frac{1}{27} a^{13} - \frac{1}{27} a^{11} + \frac{10}{27} a^{9} + \frac{1}{3} a^{7} - \frac{10}{27} a^{5} + \frac{10}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{198369} a^{20} - \frac{1}{81} a^{18} + \frac{4}{9} a^{16} + \frac{10}{81} a^{14} - \frac{28}{81} a^{12} + \frac{49843}{198369} a^{10} - \frac{4}{9} a^{8} - \frac{19}{81} a^{6} + \frac{37}{81} a^{4} - \frac{4}{9} a^{2} + \frac{729}{2449}$, $\frac{1}{595107} a^{21} - \frac{1}{243} a^{19} + \frac{4}{27} a^{17} + \frac{91}{243} a^{15} + \frac{53}{243} a^{13} + \frac{248212}{595107} a^{11} - \frac{13}{27} a^{9} + \frac{62}{243} a^{7} - \frac{44}{243} a^{5} - \frac{13}{27} a^{3} - \frac{1720}{7347} a$, $\frac{1}{1785321} a^{22} - \frac{1}{1785321} a^{20} + \frac{2}{81} a^{18} + \frac{253}{729} a^{16} - \frac{10}{729} a^{14} + \frac{799237}{1785321} a^{12} - \frac{95879}{198369} a^{10} - \frac{100}{729} a^{8} - \frac{143}{729} a^{6} + \frac{7}{81} a^{4} - \frac{2206}{7347} a^{2} - \frac{81}{2449}$, $\frac{1}{5355963} a^{23} - \frac{1}{5355963} a^{21} + \frac{2}{243} a^{19} + \frac{253}{2187} a^{17} - \frac{10}{2187} a^{15} + \frac{2584558}{5355963} a^{13} - \frac{294248}{595107} a^{11} - \frac{100}{2187} a^{9} - \frac{143}{2187} a^{7} + \frac{88}{243} a^{5} + \frac{5141}{22041} a^{3} - \frac{2530}{7347} a$, $\frac{1}{883733895} a^{24} - \frac{1}{16067889} a^{22} - \frac{1}{1785321} a^{20} - \frac{557}{360855} a^{18} - \frac{2926}{6561} a^{16} - \frac{4805234}{16067889} a^{14} + \frac{11022301}{98192655} a^{12} + \frac{5808608}{16067889} a^{10} + \frac{2773}{6561} a^{8} - \frac{17342}{40095} a^{6} - \frac{92648}{198369} a^{4} + \frac{2422}{7347} a^{2} - \frac{6904}{134695}$, $\frac{1}{2651201685} a^{25} - \frac{1}{48203667} a^{23} - \frac{1}{5355963} a^{21} - \frac{557}{1082565} a^{19} - \frac{2926}{19683} a^{17} - \frac{20873123}{48203667} a^{15} + \frac{11022301}{294577965} a^{13} + \frac{21876497}{48203667} a^{11} + \frac{2773}{19683} a^{9} + \frac{22753}{120285} a^{7} - \frac{291017}{595107} a^{5} + \frac{2422}{22041} a^{3} - \frac{6904}{404085} a$, $\frac{1}{24310551762834975} a^{26} + \frac{11245202}{24310551762834975} a^{24} - \frac{5375614}{49112225783505} a^{22} - \frac{21905917478}{24310551762834975} a^{20} - \frac{177909782464}{9926725913775} a^{18} + \frac{138801095824231}{442010032051545} a^{16} + \frac{1026254499457576}{2701172418092775} a^{14} + \frac{9419489160016598}{24310551762834975} a^{12} + \frac{117104555875978}{442010032051545} a^{10} + \frac{534137011303}{1102969545975} a^{8} - \frac{8825379323894}{300130268676975} a^{6} + \frac{137902481302}{606323775105} a^{4} - \frac{1162480479079}{3705311958975} a^{2} - \frac{149147510877}{411701328775}$, $\frac{1}{72931655288504925} a^{27} + \frac{11245202}{72931655288504925} a^{25} - \frac{5375614}{147336677350515} a^{23} - \frac{21905917478}{72931655288504925} a^{21} - \frac{177909782464}{29780177741325} a^{19} + \frac{138801095824231}{1326030096154635} a^{17} + \frac{3727426917550351}{8103517254278325} a^{15} + \frac{9419489160016598}{72931655288504925} a^{13} - \frac{324905476175567}{1326030096154635} a^{11} + \frac{1637106557278}{3308908637925} a^{9} - \frac{8825379323894}{900390806030925} a^{7} + \frac{137902481302}{1818971325315} a^{5} - \frac{1162480479079}{11115935876925} a^{3} + \frac{262553817898}{1235103986325} a$, $\frac{1}{218794965865514775} a^{28} - \frac{1}{218794965865514775} a^{26} + \frac{1467461}{24310551762834975} a^{24} + \frac{12631037887}{218794965865514775} a^{22} + \frac{100321909748}{218794965865514775} a^{20} + \frac{8686717370884558}{218794965865514775} a^{18} + \frac{11025890755063441}{24310551762834975} a^{16} - \frac{88177302643222279}{218794965865514775} a^{14} - \frac{61460414230697639}{218794965865514775} a^{12} + \frac{8696456262320092}{24310551762834975} a^{10} - \frac{92463867179602}{300130268676975} a^{8} - \frac{2715931551761}{11115935876925} a^{6} + \frac{9150573089}{3705311958975} a^{4} - \frac{1639186410409}{3705311958975} a^{2} - \frac{128501189081}{411701328775}$, $\frac{1}{656384897596544325} a^{29} - \frac{1}{656384897596544325} a^{27} + \frac{1467461}{72931655288504925} a^{25} + \frac{12631037887}{656384897596544325} a^{23} + \frac{100321909748}{656384897596544325} a^{21} + \frac{8686717370884558}{656384897596544325} a^{19} + \frac{11025890755063441}{72931655288504925} a^{17} - \frac{88177302643222279}{656384897596544325} a^{15} - \frac{280255380096212414}{656384897596544325} a^{13} - \frac{15614095500514883}{72931655288504925} a^{11} + \frac{207666401497373}{900390806030925} a^{9} - \frac{13831867428686}{33347807630775} a^{7} - \frac{3696161385886}{11115935876925} a^{5} - \frac{5344498369384}{11115935876925} a^{3} - \frac{128501189081}{1235103986325} a$, $\frac{1}{1969154692789632975} a^{30} - \frac{1}{1969154692789632975} a^{28} + \frac{2}{218794965865514775} a^{26} + \frac{8083352}{393830938557926595} a^{24} + \frac{276503910578}{1969154692789632975} a^{22} + \frac{2933185586626}{1969154692789632975} a^{20} - \frac{1874126957307061}{43758993173102955} a^{18} + \frac{622187989537459166}{1969154692789632975} a^{16} + \frac{175399518913886332}{1969154692789632975} a^{14} - \frac{599396405999251}{43758993173102955} a^{12} + \frac{8970170763803492}{24310551762834975} a^{10} + \frac{961154789649481}{2701172418092775} a^{8} - \frac{167292215543}{444637435077} a^{6} + \frac{14132739728681}{33347807630775} a^{4} - \frac{1142894396527}{3705311958975} a^{2} - \frac{63571117733}{411701328775}$, $\frac{1}{5907464078368898925} a^{31} - \frac{1}{5907464078368898925} a^{29} + \frac{2}{656384897596544325} a^{27} + \frac{8083352}{1181492815673779785} a^{25} + \frac{276503910578}{5907464078368898925} a^{23} + \frac{2933185586626}{5907464078368898925} a^{21} - \frac{1874126957307061}{131276979519308865} a^{19} + \frac{622187989537459166}{5907464078368898925} a^{17} + \frac{175399518913886332}{5907464078368898925} a^{15} - \frac{44358389579102206}{131276979519308865} a^{13} + \frac{8970170763803492}{72931655288504925} a^{11} - \frac{1740017628443294}{8103517254278325} a^{9} - \frac{167292215543}{1333912305231} a^{7} + \frac{14132739728681}{100043422892325} a^{5} - \frac{4848206355502}{11115935876925} a^{3} - \frac{63571117733}{1235103986325} a$

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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{74573424008}{1969154692789632975} a^{30} - \frac{9321678001}{35802812596175145} a^{28} + \frac{8}{411701328775} a^{26} + \frac{9321678001}{804064799015775} a^{24} + \frac{2759216688296}{35802812596175145} a^{22} + \frac{967068162535744}{1969154692789632975} a^{20} - \frac{74573424008}{2701172418092775} a^{18} - \frac{809932636472887}{35802812596175145} a^{16} - \frac{120883520316968}{804064799015775} a^{14} - \frac{9321678001}{33347807630775} a^{12} + \frac{120883520316968}{442010032051545} a^{10} + \frac{1008596238030199}{2701172418092775} a^{8} + \frac{3430377504368}{33347807630775} a^{6} - \frac{3430377504368}{3705311958975} a^{2} - \frac{9321678001}{7485478705} \) (order $30$)
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

$C_2^3\times C_4$ (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

\(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(\sqrt{6}, \sqrt{-70})\), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{35})\), \(\Q(\sqrt{6}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-70})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-10}, \sqrt{42})\), \(\Q(\sqrt{-14}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{-14}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-21}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{-10})\), \(\Q(\sqrt{-15}, \sqrt{42})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-15}, \sqrt{-21})\), \(\Q(\sqrt{-10}, \sqrt{-14})\), \(\Q(\sqrt{30}, \sqrt{35})\), 4.4.392000.1, 4.0.72000.2, 4.0.98000.1, \(\Q(\zeta_{15})^+\), 4.4.882000.1, \(\Q(\zeta_{5})\), 4.0.3528000.1, 4.4.8000.1, 8.0.7965941760000.66, 8.0.7965941760000.55, 8.0.7965941760000.10, 8.0.7965941760000.27, 8.0.7965941760000.30, 8.0.31116960000.9, 8.0.7965941760000.1, 8.0.207360000.2, 8.0.12745506816.5, 8.0.98344960000.6, 8.0.7965941760000.34, 8.8.7965941760000.9, 8.0.7965941760000.15, 8.0.497871360000.19, 8.0.7965941760000.48, 8.0.199148544000000.118, 8.0.777924000000.7, 8.0.777924000000.10, 8.0.199148544000000.224, 8.0.2458624000000.5, 8.0.5184000000.2, 8.0.199148544000000.169, 8.0.64000000.1, 8.8.12446784000000.6, 8.0.199148544000000.140, 8.8.199148544000000.4, 8.0.12446784000000.20, 8.8.199148544000000.10, 8.0.5184000000.6, 8.0.199148544000000.168, 8.8.5184000000.2, 8.0.153664000000.5, 8.0.199148544000000.16, 8.0.2458624000000.2, 8.0.12446784000000.16, 8.0.12446784000000.18, 8.0.5184000000.4, 8.0.777924000000.9, \(\Q(\zeta_{15})\), 8.8.2458624000000.1, 8.0.199148544000000.157, 8.0.9604000000.3, 8.8.777924000000.3, 16.0.63456228123711897600000000.15, 16.0.39660142577319936000000000000.28, 16.0.39660142577319936000000000000.59, 16.0.39660142577319936000000000000.13, 16.0.39660142577319936000000000000.29, 16.0.39660142577319936000000000000.38, 16.0.605165749776000000000000.9, 16.0.39660142577319936000000000000.53, 16.0.26873856000000000000.1, 16.0.6044831973376000000000000.7, 16.0.39660142577319936000000000000.19, 16.16.39660142577319936000000000000.9, 16.0.39660142577319936000000000000.55, 16.0.154922431942656000000000000.2, 16.0.39660142577319936000000000000.23

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}$