Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256)
gp: K = bnfinit(y^32 - 6*y^31 - 4*y^30 + 56*y^29 + 186*y^28 - 958*y^27 - 1098*y^26 + 6187*y^25 + 12914*y^24 - 53084*y^23 - 45771*y^22 + 250489*y^21 + 210183*y^20 - 1343675*y^19 + 432927*y^18 + 3735782*y^17 - 4733917*y^16 - 7945401*y^15 + 34353397*y^14 - 53496679*y^13 + 52302241*y^12 - 28481985*y^11 + 8401560*y^10 + 3267183*y^9 - 363465*y^8 - 2158930*y^7 + 4307708*y^6 - 2643232*y^5 + 1313520*y^4 - 282816*y^3 + 44416*y^2 - 4096*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256)
\( x^{32} - 6 x^{31} - 4 x^{30} + 56 x^{29} + 186 x^{28} - 958 x^{27} - 1098 x^{26} + 6187 x^{25} + \cdots + 256 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(301456350881363594925339414949408216309322070465087890625\)
\(\medspace = 3^{16}\cdot 5^{16}\cdot 7^{16}\cdot 29^{8}\cdot 1289^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(58.21\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}5^{1/2}7^{1/2}29^{1/2}1289^{1/2}\approx 1981.1625375016558$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(29\), \(1289\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{22}-\frac{1}{4}a^{20}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{21}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}+\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{24}-\frac{1}{4}a^{20}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}+\frac{1}{4}a^{9}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{25}-\frac{1}{8}a^{24}-\frac{1}{8}a^{22}+\frac{1}{8}a^{21}-\frac{1}{4}a^{20}+\frac{1}{8}a^{19}-\frac{1}{4}a^{18}+\frac{1}{8}a^{17}-\frac{1}{8}a^{16}+\frac{1}{8}a^{15}+\frac{1}{8}a^{14}-\frac{3}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}+\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{26}-\frac{1}{8}a^{24}-\frac{1}{8}a^{23}-\frac{1}{8}a^{21}-\frac{1}{8}a^{20}-\frac{1}{8}a^{19}-\frac{1}{8}a^{18}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{12}+\frac{3}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{4}a^{7}-\frac{3}{8}a^{6}+\frac{3}{8}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{27}+\frac{1}{8}a^{20}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}+\frac{1}{8}a^{15}+\frac{3}{8}a^{14}-\frac{3}{8}a^{13}+\frac{3}{8}a^{11}-\frac{3}{8}a^{10}+\frac{3}{8}a^{9}+\frac{1}{8}a^{8}-\frac{3}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{28}-\frac{1}{8}a^{24}-\frac{1}{8}a^{23}-\frac{1}{8}a^{22}-\frac{1}{16}a^{21}-\frac{1}{4}a^{20}+\frac{1}{16}a^{18}+\frac{3}{16}a^{17}+\frac{1}{16}a^{16}+\frac{3}{16}a^{15}-\frac{7}{16}a^{14}+\frac{1}{4}a^{13}-\frac{5}{16}a^{12}+\frac{5}{16}a^{11}+\frac{3}{16}a^{10}-\frac{1}{16}a^{9}-\frac{5}{16}a^{8}-\frac{3}{16}a^{7}+\frac{3}{8}a^{6}-\frac{5}{16}a^{5}-\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{29}-\frac{1}{16}a^{25}+\frac{1}{16}a^{24}+\frac{1}{16}a^{23}+\frac{3}{32}a^{22}-\frac{1}{4}a^{21}+\frac{1}{32}a^{19}+\frac{7}{32}a^{18}+\frac{1}{32}a^{17}-\frac{5}{32}a^{16}+\frac{1}{32}a^{15}+\frac{1}{8}a^{14}-\frac{5}{32}a^{13}+\frac{13}{32}a^{12}+\frac{11}{32}a^{11}+\frac{7}{32}a^{10}-\frac{9}{32}a^{9}+\frac{9}{32}a^{8}+\frac{5}{16}a^{7}-\frac{9}{32}a^{6}-\frac{11}{32}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{68\!\cdots\!88}a^{30}+\frac{24\!\cdots\!79}{26\!\cdots\!88}a^{29}+\frac{57\!\cdots\!11}{17\!\cdots\!72}a^{28}+\frac{52\!\cdots\!27}{85\!\cdots\!36}a^{27}+\frac{30\!\cdots\!45}{34\!\cdots\!44}a^{26}+\frac{91\!\cdots\!81}{34\!\cdots\!44}a^{25}-\frac{33\!\cdots\!65}{34\!\cdots\!44}a^{24}+\frac{31\!\cdots\!39}{68\!\cdots\!88}a^{23}-\frac{17\!\cdots\!85}{34\!\cdots\!44}a^{22}-\frac{12\!\cdots\!01}{17\!\cdots\!72}a^{21}+\frac{10\!\cdots\!85}{62\!\cdots\!32}a^{20}-\frac{22\!\cdots\!23}{68\!\cdots\!88}a^{19}+\frac{24\!\cdots\!31}{68\!\cdots\!88}a^{18}+\frac{31\!\cdots\!13}{68\!\cdots\!88}a^{17}-\frac{11\!\cdots\!01}{68\!\cdots\!88}a^{16}+\frac{68\!\cdots\!45}{34\!\cdots\!44}a^{15}+\frac{55\!\cdots\!27}{68\!\cdots\!88}a^{14}-\frac{21\!\cdots\!97}{52\!\cdots\!76}a^{13}+\frac{10\!\cdots\!05}{68\!\cdots\!88}a^{12}-\frac{45\!\cdots\!27}{52\!\cdots\!76}a^{11}+\frac{67\!\cdots\!65}{68\!\cdots\!88}a^{10}+\frac{29\!\cdots\!95}{68\!\cdots\!88}a^{9}+\frac{14\!\cdots\!27}{17\!\cdots\!72}a^{8}+\frac{48\!\cdots\!67}{68\!\cdots\!88}a^{7}-\frac{96\!\cdots\!09}{68\!\cdots\!88}a^{6}+\frac{71\!\cdots\!05}{34\!\cdots\!44}a^{5}+\frac{23\!\cdots\!29}{17\!\cdots\!72}a^{4}-\frac{35\!\cdots\!67}{85\!\cdots\!36}a^{3}+\frac{15\!\cdots\!83}{32\!\cdots\!36}a^{2}+\frac{40\!\cdots\!12}{10\!\cdots\!17}a+\frac{32\!\cdots\!86}{10\!\cdots\!17}$, $\frac{1}{12\!\cdots\!64}a^{31}+\frac{26\!\cdots\!95}{63\!\cdots\!32}a^{30}-\frac{44\!\cdots\!47}{15\!\cdots\!08}a^{29}+\frac{46\!\cdots\!63}{19\!\cdots\!26}a^{28}-\frac{85\!\cdots\!15}{63\!\cdots\!32}a^{27}+\frac{19\!\cdots\!33}{63\!\cdots\!32}a^{26}+\frac{25\!\cdots\!79}{63\!\cdots\!32}a^{25}+\frac{91\!\cdots\!79}{12\!\cdots\!64}a^{24}+\frac{37\!\cdots\!03}{63\!\cdots\!32}a^{23}+\frac{80\!\cdots\!79}{15\!\cdots\!08}a^{22}-\frac{53\!\cdots\!99}{12\!\cdots\!64}a^{21}+\frac{12\!\cdots\!41}{12\!\cdots\!64}a^{20}-\frac{23\!\cdots\!91}{16\!\cdots\!16}a^{19}-\frac{35\!\cdots\!03}{70\!\cdots\!44}a^{18}+\frac{17\!\cdots\!35}{12\!\cdots\!64}a^{17}+\frac{13\!\cdots\!03}{63\!\cdots\!32}a^{16}-\frac{36\!\cdots\!37}{12\!\cdots\!64}a^{15}+\frac{53\!\cdots\!87}{12\!\cdots\!64}a^{14}+\frac{55\!\cdots\!77}{12\!\cdots\!64}a^{13}+\frac{40\!\cdots\!57}{12\!\cdots\!64}a^{12}-\frac{33\!\cdots\!13}{11\!\cdots\!24}a^{11}-\frac{35\!\cdots\!45}{12\!\cdots\!64}a^{10}-\frac{35\!\cdots\!47}{79\!\cdots\!04}a^{9}-\frac{47\!\cdots\!09}{12\!\cdots\!64}a^{8}+\frac{11\!\cdots\!55}{12\!\cdots\!64}a^{7}+\frac{94\!\cdots\!47}{63\!\cdots\!32}a^{6}+\frac{72\!\cdots\!25}{39\!\cdots\!52}a^{5}-\frac{71\!\cdots\!59}{15\!\cdots\!08}a^{4}-\frac{17\!\cdots\!67}{79\!\cdots\!04}a^{3}-\frac{14\!\cdots\!50}{99\!\cdots\!63}a^{2}-\frac{56\!\cdots\!51}{19\!\cdots\!26}a-\frac{60\!\cdots\!76}{99\!\cdots\!63}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{380}$, which has order $380$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $380$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{7563502853423618614229312591907410190336041160320332568289997550541197}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{31} - \frac{22348801514898987208093386970195961103764375456662185566945042945778329}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{30} - \frac{8572487344072617701244030891336221157809648644327178606574657031571423}{4229629906637211956736175759204927828002552963807683741944822360420786016} a^{29} + \frac{3284608912300655540582642950831935015729433486679024948723932196953371}{132175934582412873648005492475153994625079780118990116935775698763149563} a^{28} + \frac{722443453726792090043128584374932972363157334328198198993725498968445249}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{27} - \frac{3557625821023452466411073399512555252537741920006230427498803822032785815}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{26} - \frac{4473980662933822027122251030321731522912624573661254809457766744193600709}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{25} + \frac{45983871480710353528875678721719394073166170363277208064137112542210948167}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{24} + \frac{50922147681401245723286402248196935934147659919044313350374617697941603255}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{23} - \frac{98074195908412187851250668621474386583596336728732235362202185001390788945}{4229629906637211956736175759204927828002552963807683741944822360420786016} a^{22} - \frac{381718641273439388251406208833069932394218429451212938461141179846019672151}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{21} + \frac{1859975908184534337124959822927100067783069011131554444362612061950530145185}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{20} + \frac{22262681638030969397766840101952273494132734695110708395326015475798154553}{214158476285428453505629152364806472303926732344692847693408727109913216} a^{19} - \frac{10004283106768938049690225760333567772105053003110061910089560325635471018723}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{18} + \frac{21705827736758625652657789996038865112651303484008379396072837437872002131}{155215776390356402082061495750639553321194604176428761172287059098010496} a^{17} + \frac{14235422383754010053694091451513383324294824485772601731239021268274659649293}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{16} - \frac{33205339346249212587888839817586724344393295726766377400783024821923309820665}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{15} - \frac{63130712781929339319535869094829186830991672281580614662201950221591926988265}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{14} + \frac{254063217623789856730902271894218308773284639234429287439466434602688433677989}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{13} - \frac{381501081186332126328120293051191504780504913146605125387726281924038054131079}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{12} + \frac{32837597743009399882775302281584942871734975312697314496182030456736371923059}{1538047238777167984267700276074519210182746532293703178889026312880285824} a^{11} - \frac{183674051714186611324270088548529477398904151276289174835772503663326828237017}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{10} + \frac{12165620995087356064653547808893520088416813740979598813569920204615806873653}{4229629906637211956736175759204927828002552963807683741944822360420786016} a^{9} + \frac{27396639462497561727812878718498875892816110960241884818700161063354180754043}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{8} + \frac{646790805589830491020794933528416457161522150246776738659469566855203160087}{16918519626548847826944703036819711312010211855230734967779289441683144064} a^{7} - \frac{8119825725794623720094608890004412550753728956023126337561184409526879660611}{8459259813274423913472351518409855656005105927615367483889644720841572032} a^{6} + \frac{593305661106778408516948101107888148970280391325725093255947464955949256531}{325356146664400919748936596861917525230965612600591057072678643109291232} a^{5} - \frac{2135264649142780384545582398718173430315561885567775528065488318547168461821}{2114814953318605978368087879602463914001276481903841870972411180210393008} a^{4} + \frac{66673868014562409553466967939690641605070200191771308085085916506569649368}{132175934582412873648005492475153994625079780118990116935775698763149563} a^{3} - \frac{46639886293268279359425055631095015717360319859037333008915559358804066873}{528703738329651494592021969900615978500319120475960467743102795052598252} a^{2} + \frac{174043426350074992066227740792898852641892564132615549530564507251048763}{10167379583262528742154268651934922663467675393768470533521207597165351} a - \frac{76728685857983117059595003115835874024474777675281256942705270137616500}{132175934582412873648005492475153994625079780118990116935775698763149563} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{25\!\cdots\!11}{65\!\cdots\!84}a^{31}-\frac{22\!\cdots\!76}{10\!\cdots\!31}a^{30}-\frac{72\!\cdots\!45}{32\!\cdots\!92}a^{29}+\frac{35\!\cdots\!91}{16\!\cdots\!96}a^{28}+\frac{25\!\cdots\!95}{32\!\cdots\!92}a^{27}-\frac{11\!\cdots\!47}{32\!\cdots\!92}a^{26}-\frac{17\!\cdots\!75}{32\!\cdots\!92}a^{25}+\frac{15\!\cdots\!09}{65\!\cdots\!84}a^{24}+\frac{11\!\cdots\!57}{20\!\cdots\!62}a^{23}-\frac{64\!\cdots\!63}{32\!\cdots\!92}a^{22}-\frac{15\!\cdots\!81}{65\!\cdots\!84}a^{21}+\frac{62\!\cdots\!73}{65\!\cdots\!84}a^{20}+\frac{72\!\cdots\!17}{65\!\cdots\!84}a^{19}-\frac{33\!\cdots\!13}{65\!\cdots\!84}a^{18}+\frac{10\!\cdots\!21}{65\!\cdots\!84}a^{17}+\frac{51\!\cdots\!99}{32\!\cdots\!92}a^{16}-\frac{95\!\cdots\!05}{65\!\cdots\!84}a^{15}-\frac{24\!\cdots\!65}{65\!\cdots\!84}a^{14}+\frac{83\!\cdots\!27}{65\!\cdots\!84}a^{13}-\frac{11\!\cdots\!93}{65\!\cdots\!84}a^{12}+\frac{89\!\cdots\!67}{65\!\cdots\!84}a^{11}-\frac{28\!\cdots\!15}{65\!\cdots\!84}a^{10}-\frac{56\!\cdots\!33}{16\!\cdots\!96}a^{9}+\frac{11\!\cdots\!59}{65\!\cdots\!84}a^{8}+\frac{40\!\cdots\!03}{65\!\cdots\!84}a^{7}-\frac{41\!\cdots\!71}{32\!\cdots\!92}a^{6}+\frac{41\!\cdots\!57}{32\!\cdots\!92}a^{5}-\frac{81\!\cdots\!97}{16\!\cdots\!96}a^{4}+\frac{42\!\cdots\!07}{40\!\cdots\!24}a^{3}-\frac{15\!\cdots\!80}{10\!\cdots\!31}a^{2}+\frac{26\!\cdots\!99}{20\!\cdots\!62}a-\frac{16\!\cdots\!33}{10\!\cdots\!31}$, $\frac{15\!\cdots\!79}{97\!\cdots\!52}a^{31}-\frac{80\!\cdots\!91}{97\!\cdots\!52}a^{30}-\frac{14\!\cdots\!59}{97\!\cdots\!52}a^{29}+\frac{10\!\cdots\!14}{12\!\cdots\!69}a^{28}+\frac{46\!\cdots\!65}{12\!\cdots\!69}a^{27}-\frac{15\!\cdots\!82}{12\!\cdots\!69}a^{26}-\frac{30\!\cdots\!33}{97\!\cdots\!52}a^{25}+\frac{10\!\cdots\!49}{12\!\cdots\!69}a^{24}+\frac{29\!\cdots\!57}{97\!\cdots\!52}a^{23}-\frac{16\!\cdots\!59}{24\!\cdots\!38}a^{22}-\frac{18\!\cdots\!48}{12\!\cdots\!69}a^{21}+\frac{80\!\cdots\!93}{24\!\cdots\!38}a^{20}+\frac{21\!\cdots\!03}{30\!\cdots\!22}a^{19}-\frac{17\!\cdots\!77}{97\!\cdots\!52}a^{18}-\frac{62\!\cdots\!09}{48\!\cdots\!76}a^{17}+\frac{62\!\cdots\!27}{97\!\cdots\!52}a^{16}-\frac{16\!\cdots\!09}{97\!\cdots\!52}a^{15}-\frac{18\!\cdots\!77}{97\!\cdots\!52}a^{14}+\frac{20\!\cdots\!71}{48\!\cdots\!76}a^{13}-\frac{18\!\cdots\!11}{48\!\cdots\!76}a^{12}+\frac{82\!\cdots\!07}{48\!\cdots\!76}a^{11}+\frac{76\!\cdots\!21}{97\!\cdots\!52}a^{10}-\frac{57\!\cdots\!47}{12\!\cdots\!69}a^{9}+\frac{29\!\cdots\!45}{97\!\cdots\!52}a^{8}+\frac{22\!\cdots\!95}{48\!\cdots\!76}a^{7}-\frac{64\!\cdots\!57}{48\!\cdots\!76}a^{6}+\frac{20\!\cdots\!39}{12\!\cdots\!69}a^{5}-\frac{48\!\cdots\!08}{12\!\cdots\!69}a^{4}-\frac{21\!\cdots\!85}{97\!\cdots\!52}a^{3}+\frac{13\!\cdots\!48}{12\!\cdots\!69}a^{2}-\frac{15\!\cdots\!84}{12\!\cdots\!69}a-\frac{89\!\cdots\!78}{12\!\cdots\!69}$, $\frac{99\!\cdots\!57}{79\!\cdots\!04}a^{31}-\frac{23\!\cdots\!97}{31\!\cdots\!16}a^{30}-\frac{43\!\cdots\!29}{79\!\cdots\!04}a^{29}+\frac{11\!\cdots\!99}{15\!\cdots\!08}a^{28}+\frac{47\!\cdots\!13}{19\!\cdots\!26}a^{27}-\frac{18\!\cdots\!27}{15\!\cdots\!08}a^{26}-\frac{23\!\cdots\!85}{15\!\cdots\!08}a^{25}+\frac{12\!\cdots\!87}{15\!\cdots\!08}a^{24}+\frac{40\!\cdots\!25}{24\!\cdots\!32}a^{23}-\frac{65\!\cdots\!30}{99\!\cdots\!63}a^{22}-\frac{99\!\cdots\!61}{15\!\cdots\!08}a^{21}+\frac{98\!\cdots\!03}{31\!\cdots\!16}a^{20}+\frac{11\!\cdots\!11}{40\!\cdots\!04}a^{19}-\frac{53\!\cdots\!55}{31\!\cdots\!16}a^{18}+\frac{13\!\cdots\!63}{31\!\cdots\!16}a^{17}+\frac{15\!\cdots\!89}{31\!\cdots\!16}a^{16}-\frac{88\!\cdots\!49}{15\!\cdots\!08}a^{15}-\frac{33\!\cdots\!89}{31\!\cdots\!16}a^{14}+\frac{13\!\cdots\!31}{31\!\cdots\!16}a^{13}-\frac{20\!\cdots\!29}{31\!\cdots\!16}a^{12}+\frac{17\!\cdots\!93}{28\!\cdots\!56}a^{11}-\frac{74\!\cdots\!37}{24\!\cdots\!32}a^{10}+\frac{19\!\cdots\!21}{24\!\cdots\!32}a^{9}+\frac{18\!\cdots\!97}{39\!\cdots\!52}a^{8}-\frac{36\!\cdots\!33}{31\!\cdots\!16}a^{7}-\frac{88\!\cdots\!57}{31\!\cdots\!16}a^{6}+\frac{81\!\cdots\!55}{15\!\cdots\!08}a^{5}-\frac{46\!\cdots\!91}{15\!\cdots\!08}a^{4}+\frac{13\!\cdots\!91}{99\!\cdots\!63}a^{3}-\frac{48\!\cdots\!77}{19\!\cdots\!26}a^{2}+\frac{45\!\cdots\!84}{99\!\cdots\!63}a-\frac{15\!\cdots\!04}{99\!\cdots\!63}$, $\frac{45\!\cdots\!83}{63\!\cdots\!32}a^{31}-\frac{14\!\cdots\!35}{31\!\cdots\!16}a^{30}-\frac{28\!\cdots\!89}{15\!\cdots\!08}a^{29}+\frac{63\!\cdots\!83}{15\!\cdots\!02}a^{28}+\frac{39\!\cdots\!75}{31\!\cdots\!16}a^{27}-\frac{23\!\cdots\!49}{31\!\cdots\!16}a^{26}-\frac{19\!\cdots\!39}{31\!\cdots\!16}a^{25}+\frac{30\!\cdots\!81}{63\!\cdots\!32}a^{24}+\frac{26\!\cdots\!05}{31\!\cdots\!16}a^{23}-\frac{65\!\cdots\!75}{15\!\cdots\!08}a^{22}-\frac{11\!\cdots\!69}{48\!\cdots\!64}a^{21}+\frac{12\!\cdots\!43}{63\!\cdots\!32}a^{20}+\frac{85\!\cdots\!51}{80\!\cdots\!08}a^{19}-\frac{65\!\cdots\!05}{63\!\cdots\!32}a^{18}+\frac{35\!\cdots\!53}{63\!\cdots\!32}a^{17}+\frac{85\!\cdots\!11}{31\!\cdots\!16}a^{16}-\frac{26\!\cdots\!31}{63\!\cdots\!32}a^{15}-\frac{32\!\cdots\!87}{63\!\cdots\!32}a^{14}+\frac{16\!\cdots\!23}{63\!\cdots\!32}a^{13}-\frac{28\!\cdots\!29}{63\!\cdots\!32}a^{12}+\frac{26\!\cdots\!53}{57\!\cdots\!12}a^{11}-\frac{17\!\cdots\!87}{63\!\cdots\!32}a^{10}+\frac{14\!\cdots\!55}{15\!\cdots\!08}a^{9}+\frac{11\!\cdots\!61}{63\!\cdots\!32}a^{8}-\frac{67\!\cdots\!71}{63\!\cdots\!32}a^{7}-\frac{53\!\cdots\!85}{31\!\cdots\!16}a^{6}+\frac{55\!\cdots\!49}{15\!\cdots\!08}a^{5}-\frac{20\!\cdots\!83}{79\!\cdots\!04}a^{4}+\frac{12\!\cdots\!00}{99\!\cdots\!63}a^{3}-\frac{67\!\cdots\!47}{19\!\cdots\!26}a^{2}+\frac{41\!\cdots\!82}{99\!\cdots\!63}a-\frac{28\!\cdots\!95}{99\!\cdots\!63}$, $\frac{94\!\cdots\!99}{31\!\cdots\!16}a^{31}-\frac{11\!\cdots\!91}{63\!\cdots\!32}a^{30}-\frac{28\!\cdots\!09}{31\!\cdots\!16}a^{29}+\frac{26\!\cdots\!81}{15\!\cdots\!08}a^{28}+\frac{83\!\cdots\!29}{15\!\cdots\!08}a^{27}-\frac{93\!\cdots\!53}{31\!\cdots\!16}a^{26}-\frac{88\!\cdots\!21}{31\!\cdots\!16}a^{25}+\frac{17\!\cdots\!75}{91\!\cdots\!07}a^{24}+\frac{22\!\cdots\!47}{63\!\cdots\!32}a^{23}-\frac{52\!\cdots\!69}{31\!\cdots\!16}a^{22}-\frac{34\!\cdots\!23}{31\!\cdots\!16}a^{21}+\frac{48\!\cdots\!95}{63\!\cdots\!32}a^{20}+\frac{39\!\cdots\!81}{80\!\cdots\!08}a^{19}-\frac{26\!\cdots\!75}{63\!\cdots\!32}a^{18}+\frac{12\!\cdots\!59}{63\!\cdots\!32}a^{17}+\frac{69\!\cdots\!63}{63\!\cdots\!32}a^{16}-\frac{25\!\cdots\!45}{15\!\cdots\!08}a^{15}-\frac{13\!\cdots\!07}{63\!\cdots\!32}a^{14}+\frac{67\!\cdots\!41}{63\!\cdots\!32}a^{13}-\frac{11\!\cdots\!57}{63\!\cdots\!32}a^{12}+\frac{10\!\cdots\!61}{57\!\cdots\!12}a^{11}-\frac{70\!\cdots\!81}{63\!\cdots\!32}a^{10}+\frac{23\!\cdots\!55}{63\!\cdots\!32}a^{9}+\frac{22\!\cdots\!35}{31\!\cdots\!16}a^{8}-\frac{23\!\cdots\!51}{63\!\cdots\!32}a^{7}-\frac{40\!\cdots\!77}{63\!\cdots\!32}a^{6}+\frac{44\!\cdots\!29}{31\!\cdots\!16}a^{5}-\frac{16\!\cdots\!31}{15\!\cdots\!08}a^{4}+\frac{20\!\cdots\!63}{39\!\cdots\!52}a^{3}-\frac{28\!\cdots\!95}{19\!\cdots\!26}a^{2}+\frac{33\!\cdots\!15}{19\!\cdots\!26}a-\frac{12\!\cdots\!56}{99\!\cdots\!63}$, $\frac{19\!\cdots\!67}{31\!\cdots\!16}a^{31}-\frac{58\!\cdots\!19}{15\!\cdots\!08}a^{30}-\frac{83\!\cdots\!47}{31\!\cdots\!16}a^{29}+\frac{20\!\cdots\!79}{61\!\cdots\!08}a^{28}+\frac{18\!\cdots\!41}{15\!\cdots\!08}a^{27}-\frac{92\!\cdots\!39}{15\!\cdots\!08}a^{26}-\frac{55\!\cdots\!87}{79\!\cdots\!04}a^{25}+\frac{11\!\cdots\!63}{31\!\cdots\!16}a^{24}+\frac{32\!\cdots\!69}{39\!\cdots\!52}a^{23}-\frac{10\!\cdots\!13}{31\!\cdots\!16}a^{22}-\frac{72\!\cdots\!01}{24\!\cdots\!32}a^{21}+\frac{48\!\cdots\!27}{31\!\cdots\!16}a^{20}+\frac{27\!\cdots\!57}{20\!\cdots\!52}a^{19}-\frac{11\!\cdots\!37}{14\!\cdots\!12}a^{18}+\frac{36\!\cdots\!41}{15\!\cdots\!08}a^{17}+\frac{72\!\cdots\!61}{31\!\cdots\!16}a^{16}-\frac{44\!\cdots\!35}{15\!\cdots\!08}a^{15}-\frac{15\!\cdots\!91}{31\!\cdots\!16}a^{14}+\frac{33\!\cdots\!25}{15\!\cdots\!08}a^{13}-\frac{31\!\cdots\!23}{99\!\cdots\!63}a^{12}+\frac{44\!\cdots\!73}{14\!\cdots\!28}a^{11}-\frac{12\!\cdots\!55}{79\!\cdots\!04}a^{10}+\frac{13\!\cdots\!43}{31\!\cdots\!16}a^{9}+\frac{38\!\cdots\!33}{15\!\cdots\!08}a^{8}-\frac{86\!\cdots\!37}{31\!\cdots\!16}a^{7}-\frac{40\!\cdots\!95}{31\!\cdots\!16}a^{6}+\frac{83\!\cdots\!77}{31\!\cdots\!16}a^{5}-\frac{68\!\cdots\!11}{43\!\cdots\!84}a^{4}+\frac{73\!\cdots\!82}{99\!\cdots\!63}a^{3}-\frac{51\!\cdots\!11}{39\!\cdots\!52}a^{2}+\frac{43\!\cdots\!25}{99\!\cdots\!63}a+\frac{15\!\cdots\!43}{99\!\cdots\!63}$, $\frac{91\!\cdots\!31}{12\!\cdots\!64}a^{31}-\frac{14\!\cdots\!31}{24\!\cdots\!32}a^{30}+\frac{21\!\cdots\!69}{31\!\cdots\!16}a^{29}+\frac{96\!\cdots\!53}{19\!\cdots\!26}a^{28}+\frac{26\!\cdots\!31}{63\!\cdots\!32}a^{27}-\frac{64\!\cdots\!07}{63\!\cdots\!32}a^{26}+\frac{47\!\cdots\!03}{63\!\cdots\!32}a^{25}+\frac{83\!\cdots\!93}{12\!\cdots\!64}a^{24}-\frac{28\!\cdots\!85}{39\!\cdots\!52}a^{23}-\frac{19\!\cdots\!43}{31\!\cdots\!16}a^{22}+\frac{65\!\cdots\!87}{12\!\cdots\!64}a^{21}+\frac{34\!\cdots\!45}{12\!\cdots\!64}a^{20}-\frac{40\!\cdots\!39}{16\!\cdots\!16}a^{19}-\frac{17\!\cdots\!03}{12\!\cdots\!64}a^{18}+\frac{31\!\cdots\!31}{12\!\cdots\!64}a^{17}+\frac{37\!\cdots\!57}{15\!\cdots\!08}a^{16}-\frac{12\!\cdots\!23}{12\!\cdots\!64}a^{15}+\frac{10\!\cdots\!35}{97\!\cdots\!28}a^{14}+\frac{50\!\cdots\!69}{12\!\cdots\!64}a^{13}-\frac{91\!\cdots\!75}{97\!\cdots\!28}a^{12}+\frac{13\!\cdots\!35}{11\!\cdots\!24}a^{11}-\frac{11\!\cdots\!21}{12\!\cdots\!64}a^{10}+\frac{23\!\cdots\!27}{63\!\cdots\!32}a^{9}-\frac{36\!\cdots\!27}{12\!\cdots\!64}a^{8}-\frac{96\!\cdots\!13}{12\!\cdots\!64}a^{7}-\frac{20\!\cdots\!85}{79\!\cdots\!04}a^{6}+\frac{21\!\cdots\!83}{31\!\cdots\!16}a^{5}-\frac{13\!\cdots\!77}{15\!\cdots\!08}a^{4}+\frac{59\!\cdots\!01}{15\!\cdots\!02}a^{3}-\frac{64\!\cdots\!09}{39\!\cdots\!52}a^{2}+\frac{12\!\cdots\!15}{99\!\cdots\!63}a-\frac{19\!\cdots\!01}{76\!\cdots\!51}$, $\frac{27\!\cdots\!83}{63\!\cdots\!32}a^{31}-\frac{41\!\cdots\!23}{15\!\cdots\!08}a^{30}-\frac{57\!\cdots\!93}{31\!\cdots\!16}a^{29}+\frac{39\!\cdots\!69}{15\!\cdots\!08}a^{28}+\frac{25\!\cdots\!79}{31\!\cdots\!16}a^{27}-\frac{13\!\cdots\!63}{31\!\cdots\!16}a^{26}-\frac{12\!\cdots\!63}{24\!\cdots\!32}a^{25}+\frac{17\!\cdots\!05}{63\!\cdots\!32}a^{24}+\frac{91\!\cdots\!13}{15\!\cdots\!08}a^{23}-\frac{56\!\cdots\!63}{24\!\cdots\!32}a^{22}-\frac{13\!\cdots\!09}{63\!\cdots\!32}a^{21}+\frac{70\!\cdots\!85}{63\!\cdots\!32}a^{20}+\frac{77\!\cdots\!63}{80\!\cdots\!08}a^{19}-\frac{37\!\cdots\!57}{63\!\cdots\!32}a^{18}+\frac{10\!\cdots\!93}{63\!\cdots\!32}a^{17}+\frac{53\!\cdots\!29}{31\!\cdots\!16}a^{16}-\frac{12\!\cdots\!57}{63\!\cdots\!32}a^{15}-\frac{23\!\cdots\!81}{63\!\cdots\!32}a^{14}+\frac{95\!\cdots\!15}{63\!\cdots\!32}a^{13}-\frac{14\!\cdots\!25}{63\!\cdots\!32}a^{12}+\frac{12\!\cdots\!61}{57\!\cdots\!12}a^{11}-\frac{69\!\cdots\!75}{63\!\cdots\!32}a^{10}+\frac{11\!\cdots\!49}{39\!\cdots\!52}a^{9}+\frac{94\!\cdots\!23}{63\!\cdots\!32}a^{8}-\frac{52\!\cdots\!89}{63\!\cdots\!32}a^{7}-\frac{32\!\cdots\!01}{31\!\cdots\!16}a^{6}+\frac{56\!\cdots\!25}{31\!\cdots\!16}a^{5}-\frac{13\!\cdots\!59}{12\!\cdots\!16}a^{4}+\frac{19\!\cdots\!71}{39\!\cdots\!52}a^{3}-\frac{10\!\cdots\!35}{99\!\cdots\!63}a^{2}+\frac{22\!\cdots\!69}{19\!\cdots\!26}a-\frac{92\!\cdots\!22}{99\!\cdots\!63}$, $\frac{12\!\cdots\!31}{11\!\cdots\!24}a^{31}-\frac{40\!\cdots\!85}{57\!\cdots\!12}a^{30}-\frac{55\!\cdots\!47}{28\!\cdots\!56}a^{29}+\frac{22\!\cdots\!61}{36\!\cdots\!32}a^{28}+\frac{10\!\cdots\!43}{57\!\cdots\!12}a^{27}-\frac{65\!\cdots\!97}{57\!\cdots\!12}a^{26}-\frac{46\!\cdots\!07}{57\!\cdots\!12}a^{25}+\frac{84\!\cdots\!33}{11\!\cdots\!24}a^{24}+\frac{51\!\cdots\!51}{44\!\cdots\!24}a^{23}-\frac{18\!\cdots\!01}{28\!\cdots\!56}a^{22}-\frac{33\!\cdots\!25}{11\!\cdots\!24}a^{21}+\frac{34\!\cdots\!39}{11\!\cdots\!24}a^{20}+\frac{18\!\cdots\!19}{14\!\cdots\!56}a^{19}-\frac{18\!\cdots\!33}{11\!\cdots\!24}a^{18}+\frac{11\!\cdots\!85}{11\!\cdots\!24}a^{17}+\frac{23\!\cdots\!13}{57\!\cdots\!12}a^{16}-\frac{78\!\cdots\!87}{11\!\cdots\!24}a^{15}-\frac{80\!\cdots\!75}{11\!\cdots\!24}a^{14}+\frac{47\!\cdots\!27}{11\!\cdots\!24}a^{13}-\frac{83\!\cdots\!77}{11\!\cdots\!24}a^{12}+\frac{89\!\cdots\!39}{11\!\cdots\!24}a^{11}-\frac{43\!\cdots\!39}{89\!\cdots\!48}a^{10}+\frac{19\!\cdots\!21}{11\!\cdots\!56}a^{9}+\frac{23\!\cdots\!05}{11\!\cdots\!24}a^{8}-\frac{26\!\cdots\!23}{11\!\cdots\!24}a^{7}-\frac{14\!\cdots\!51}{57\!\cdots\!12}a^{6}+\frac{16\!\cdots\!23}{28\!\cdots\!56}a^{5}-\frac{65\!\cdots\!39}{14\!\cdots\!28}a^{4}+\frac{15\!\cdots\!13}{72\!\cdots\!64}a^{3}-\frac{24\!\cdots\!21}{36\!\cdots\!32}a^{2}+\frac{13\!\cdots\!67}{18\!\cdots\!66}a-\frac{66\!\cdots\!66}{90\!\cdots\!33}$, $\frac{39\!\cdots\!11}{97\!\cdots\!28}a^{31}-\frac{16\!\cdots\!37}{63\!\cdots\!32}a^{30}-\frac{16\!\cdots\!45}{24\!\cdots\!32}a^{29}+\frac{37\!\cdots\!19}{15\!\cdots\!08}a^{28}+\frac{42\!\cdots\!63}{63\!\cdots\!32}a^{27}-\frac{26\!\cdots\!61}{63\!\cdots\!32}a^{26}-\frac{18\!\cdots\!19}{63\!\cdots\!32}a^{25}+\frac{34\!\cdots\!73}{12\!\cdots\!64}a^{24}+\frac{27\!\cdots\!03}{63\!\cdots\!32}a^{23}-\frac{75\!\cdots\!51}{31\!\cdots\!16}a^{22}-\frac{13\!\cdots\!69}{12\!\cdots\!64}a^{21}+\frac{14\!\cdots\!91}{12\!\cdots\!64}a^{20}+\frac{74\!\cdots\!35}{16\!\cdots\!16}a^{19}-\frac{74\!\cdots\!29}{12\!\cdots\!64}a^{18}+\frac{49\!\cdots\!45}{12\!\cdots\!64}a^{17}+\frac{94\!\cdots\!97}{63\!\cdots\!32}a^{16}-\frac{32\!\cdots\!31}{12\!\cdots\!64}a^{15}-\frac{32\!\cdots\!23}{12\!\cdots\!64}a^{14}+\frac{15\!\cdots\!87}{97\!\cdots\!28}a^{13}-\frac{34\!\cdots\!73}{12\!\cdots\!64}a^{12}+\frac{25\!\cdots\!01}{89\!\cdots\!48}a^{11}-\frac{23\!\cdots\!71}{12\!\cdots\!64}a^{10}+\frac{13\!\cdots\!61}{19\!\cdots\!26}a^{9}+\frac{87\!\cdots\!49}{12\!\cdots\!64}a^{8}-\frac{11\!\cdots\!19}{12\!\cdots\!64}a^{7}-\frac{58\!\cdots\!75}{63\!\cdots\!32}a^{6}+\frac{67\!\cdots\!35}{31\!\cdots\!16}a^{5}-\frac{13\!\cdots\!87}{79\!\cdots\!04}a^{4}+\frac{67\!\cdots\!53}{79\!\cdots\!04}a^{3}-\frac{39\!\cdots\!55}{15\!\cdots\!02}a^{2}+\frac{56\!\cdots\!11}{19\!\cdots\!26}a-\frac{26\!\cdots\!52}{99\!\cdots\!63}$, $\frac{13\!\cdots\!57}{12\!\cdots\!64}a^{31}-\frac{20\!\cdots\!25}{31\!\cdots\!16}a^{30}-\frac{12\!\cdots\!79}{31\!\cdots\!16}a^{29}+\frac{59\!\cdots\!36}{99\!\cdots\!63}a^{28}+\frac{12\!\cdots\!37}{63\!\cdots\!32}a^{27}-\frac{65\!\cdots\!41}{63\!\cdots\!32}a^{26}-\frac{71\!\cdots\!19}{63\!\cdots\!32}a^{25}+\frac{84\!\cdots\!59}{12\!\cdots\!64}a^{24}+\frac{41\!\cdots\!19}{30\!\cdots\!04}a^{23}-\frac{18\!\cdots\!51}{31\!\cdots\!16}a^{22}-\frac{59\!\cdots\!95}{12\!\cdots\!64}a^{21}+\frac{34\!\cdots\!51}{12\!\cdots\!64}a^{20}+\frac{34\!\cdots\!19}{16\!\cdots\!16}a^{19}-\frac{18\!\cdots\!73}{12\!\cdots\!64}a^{18}+\frac{64\!\cdots\!69}{12\!\cdots\!64}a^{17}+\frac{79\!\cdots\!19}{19\!\cdots\!26}a^{16}-\frac{65\!\cdots\!69}{12\!\cdots\!64}a^{15}-\frac{10\!\cdots\!35}{12\!\cdots\!64}a^{14}+\frac{46\!\cdots\!71}{12\!\cdots\!64}a^{13}-\frac{73\!\cdots\!49}{12\!\cdots\!64}a^{12}+\frac{65\!\cdots\!57}{11\!\cdots\!24}a^{11}-\frac{30\!\cdots\!99}{97\!\cdots\!28}a^{10}+\frac{42\!\cdots\!33}{48\!\cdots\!64}a^{9}+\frac{46\!\cdots\!07}{12\!\cdots\!64}a^{8}-\frac{84\!\cdots\!31}{12\!\cdots\!64}a^{7}-\frac{37\!\cdots\!85}{15\!\cdots\!08}a^{6}+\frac{14\!\cdots\!91}{31\!\cdots\!16}a^{5}-\frac{46\!\cdots\!33}{15\!\cdots\!08}a^{4}+\frac{55\!\cdots\!83}{39\!\cdots\!52}a^{3}-\frac{11\!\cdots\!11}{39\!\cdots\!52}a^{2}+\frac{32\!\cdots\!31}{99\!\cdots\!63}a-\frac{14\!\cdots\!60}{99\!\cdots\!63}$, $\frac{19\!\cdots\!91}{12\!\cdots\!64}a^{31}-\frac{16\!\cdots\!23}{15\!\cdots\!08}a^{30}+\frac{22\!\cdots\!17}{79\!\cdots\!04}a^{29}+\frac{14\!\cdots\!75}{15\!\cdots\!08}a^{28}+\frac{12\!\cdots\!47}{63\!\cdots\!32}a^{27}-\frac{85\!\cdots\!91}{48\!\cdots\!64}a^{26}-\frac{15\!\cdots\!73}{63\!\cdots\!32}a^{25}+\frac{14\!\cdots\!73}{12\!\cdots\!64}a^{24}+\frac{32\!\cdots\!25}{31\!\cdots\!16}a^{23}-\frac{16\!\cdots\!11}{15\!\cdots\!08}a^{22}+\frac{12\!\cdots\!03}{12\!\cdots\!64}a^{21}+\frac{57\!\cdots\!37}{12\!\cdots\!64}a^{20}-\frac{90\!\cdots\!79}{16\!\cdots\!16}a^{19}-\frac{30\!\cdots\!67}{12\!\cdots\!64}a^{18}+\frac{33\!\cdots\!71}{12\!\cdots\!64}a^{17}+\frac{45\!\cdots\!09}{87\!\cdots\!68}a^{16}-\frac{16\!\cdots\!47}{12\!\cdots\!64}a^{15}-\frac{67\!\cdots\!69}{12\!\cdots\!64}a^{14}+\frac{82\!\cdots\!17}{12\!\cdots\!64}a^{13}-\frac{16\!\cdots\!31}{12\!\cdots\!64}a^{12}+\frac{18\!\cdots\!47}{11\!\cdots\!24}a^{11}-\frac{14\!\cdots\!57}{12\!\cdots\!64}a^{10}+\frac{31\!\cdots\!35}{63\!\cdots\!32}a^{9}-\frac{37\!\cdots\!31}{97\!\cdots\!28}a^{8}-\frac{65\!\cdots\!45}{97\!\cdots\!28}a^{7}-\frac{46\!\cdots\!23}{15\!\cdots\!08}a^{6}+\frac{39\!\cdots\!23}{39\!\cdots\!52}a^{5}-\frac{40\!\cdots\!01}{39\!\cdots\!52}a^{4}+\frac{43\!\cdots\!51}{79\!\cdots\!04}a^{3}-\frac{81\!\cdots\!17}{39\!\cdots\!52}a^{2}+\frac{32\!\cdots\!32}{99\!\cdots\!63}a-\frac{22\!\cdots\!61}{99\!\cdots\!63}$, $\frac{79\!\cdots\!00}{99\!\cdots\!63}a^{31}-\frac{29\!\cdots\!05}{63\!\cdots\!32}a^{30}-\frac{32\!\cdots\!11}{79\!\cdots\!04}a^{29}+\frac{70\!\cdots\!87}{15\!\cdots\!08}a^{28}+\frac{12\!\cdots\!33}{79\!\cdots\!04}a^{27}-\frac{23\!\cdots\!09}{31\!\cdots\!16}a^{26}-\frac{32\!\cdots\!71}{31\!\cdots\!16}a^{25}+\frac{15\!\cdots\!91}{31\!\cdots\!16}a^{24}+\frac{72\!\cdots\!05}{63\!\cdots\!32}a^{23}-\frac{64\!\cdots\!43}{15\!\cdots\!08}a^{22}-\frac{71\!\cdots\!89}{15\!\cdots\!08}a^{21}+\frac{12\!\cdots\!15}{63\!\cdots\!32}a^{20}+\frac{16\!\cdots\!35}{80\!\cdots\!08}a^{19}-\frac{66\!\cdots\!53}{63\!\cdots\!32}a^{18}+\frac{87\!\cdots\!17}{63\!\cdots\!32}a^{17}+\frac{14\!\cdots\!91}{48\!\cdots\!64}a^{16}-\frac{50\!\cdots\!63}{15\!\cdots\!08}a^{15}-\frac{45\!\cdots\!59}{63\!\cdots\!32}a^{14}+\frac{16\!\cdots\!71}{63\!\cdots\!32}a^{13}-\frac{23\!\cdots\!55}{63\!\cdots\!32}a^{12}+\frac{19\!\cdots\!35}{57\!\cdots\!12}a^{11}-\frac{99\!\cdots\!15}{63\!\cdots\!32}a^{10}+\frac{20\!\cdots\!59}{63\!\cdots\!32}a^{9}+\frac{10\!\cdots\!89}{31\!\cdots\!16}a^{8}+\frac{19\!\cdots\!13}{63\!\cdots\!32}a^{7}-\frac{11\!\cdots\!01}{63\!\cdots\!32}a^{6}+\frac{48\!\cdots\!59}{15\!\cdots\!08}a^{5}-\frac{24\!\cdots\!71}{15\!\cdots\!08}a^{4}+\frac{70\!\cdots\!88}{99\!\cdots\!63}a^{3}-\frac{31\!\cdots\!33}{39\!\cdots\!52}a^{2}+\frac{12\!\cdots\!68}{99\!\cdots\!63}a-\frac{41\!\cdots\!01}{99\!\cdots\!63}$, $\frac{18\!\cdots\!71}{12\!\cdots\!64}a^{31}-\frac{27\!\cdots\!83}{31\!\cdots\!16}a^{30}-\frac{19\!\cdots\!31}{31\!\cdots\!16}a^{29}+\frac{32\!\cdots\!73}{39\!\cdots\!52}a^{28}+\frac{13\!\cdots\!63}{48\!\cdots\!64}a^{27}-\frac{87\!\cdots\!51}{63\!\cdots\!32}a^{26}-\frac{10\!\cdots\!81}{63\!\cdots\!32}a^{25}+\frac{11\!\cdots\!25}{12\!\cdots\!64}a^{24}+\frac{15\!\cdots\!99}{79\!\cdots\!04}a^{23}-\frac{24\!\cdots\!59}{31\!\cdots\!16}a^{22}-\frac{89\!\cdots\!77}{12\!\cdots\!64}a^{21}+\frac{45\!\cdots\!41}{12\!\cdots\!64}a^{20}+\frac{40\!\cdots\!97}{12\!\cdots\!32}a^{19}-\frac{24\!\cdots\!43}{12\!\cdots\!64}a^{18}+\frac{36\!\cdots\!27}{70\!\cdots\!44}a^{17}+\frac{86\!\cdots\!95}{15\!\cdots\!08}a^{16}-\frac{83\!\cdots\!55}{12\!\cdots\!64}a^{15}-\frac{15\!\cdots\!29}{12\!\cdots\!64}a^{14}+\frac{62\!\cdots\!53}{12\!\cdots\!64}a^{13}-\frac{95\!\cdots\!91}{12\!\cdots\!64}a^{12}+\frac{82\!\cdots\!71}{11\!\cdots\!24}a^{11}-\frac{46\!\cdots\!01}{12\!\cdots\!64}a^{10}+\frac{60\!\cdots\!19}{63\!\cdots\!32}a^{9}+\frac{70\!\cdots\!65}{12\!\cdots\!64}a^{8}-\frac{39\!\cdots\!61}{12\!\cdots\!64}a^{7}-\frac{25\!\cdots\!11}{79\!\cdots\!04}a^{6}+\frac{19\!\cdots\!51}{31\!\cdots\!16}a^{5}-\frac{55\!\cdots\!93}{15\!\cdots\!08}a^{4}+\frac{32\!\cdots\!67}{19\!\cdots\!26}a^{3}-\frac{57\!\cdots\!85}{19\!\cdots\!26}a^{2}+\frac{67\!\cdots\!69}{19\!\cdots\!26}a-\frac{26\!\cdots\!08}{99\!\cdots\!63}$, $\frac{23\!\cdots\!51}{57\!\cdots\!12}a^{31}-\frac{13\!\cdots\!81}{57\!\cdots\!12}a^{30}-\frac{16\!\cdots\!25}{72\!\cdots\!64}a^{29}+\frac{32\!\cdots\!83}{14\!\cdots\!28}a^{28}+\frac{23\!\cdots\!99}{28\!\cdots\!56}a^{27}-\frac{26\!\cdots\!71}{72\!\cdots\!64}a^{26}-\frac{20\!\cdots\!99}{36\!\cdots\!32}a^{25}+\frac{13\!\cdots\!95}{57\!\cdots\!12}a^{24}+\frac{34\!\cdots\!65}{57\!\cdots\!12}a^{23}-\frac{29\!\cdots\!09}{14\!\cdots\!28}a^{22}-\frac{14\!\cdots\!73}{57\!\cdots\!12}a^{21}+\frac{17\!\cdots\!45}{18\!\cdots\!66}a^{20}+\frac{10\!\cdots\!43}{91\!\cdots\!16}a^{19}-\frac{75\!\cdots\!03}{14\!\cdots\!28}a^{18}+\frac{37\!\cdots\!71}{14\!\cdots\!28}a^{17}+\frac{90\!\cdots\!35}{57\!\cdots\!12}a^{16}-\frac{86\!\cdots\!47}{57\!\cdots\!12}a^{15}-\frac{13\!\cdots\!93}{36\!\cdots\!32}a^{14}+\frac{18\!\cdots\!03}{14\!\cdots\!28}a^{13}-\frac{52\!\cdots\!27}{28\!\cdots\!56}a^{12}+\frac{45\!\cdots\!97}{28\!\cdots\!56}a^{11}-\frac{92\!\cdots\!39}{14\!\cdots\!28}a^{10}+\frac{50\!\cdots\!73}{57\!\cdots\!12}a^{9}+\frac{11\!\cdots\!71}{57\!\cdots\!12}a^{8}+\frac{23\!\cdots\!05}{72\!\cdots\!64}a^{7}-\frac{50\!\cdots\!17}{57\!\cdots\!12}a^{6}+\frac{21\!\cdots\!53}{14\!\cdots\!28}a^{5}-\frac{88\!\cdots\!01}{14\!\cdots\!28}a^{4}+\frac{26\!\cdots\!29}{90\!\cdots\!33}a^{3}-\frac{10\!\cdots\!97}{36\!\cdots\!32}a^{2}+\frac{30\!\cdots\!85}{90\!\cdots\!33}a+\frac{40\!\cdots\!74}{90\!\cdots\!33}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 14470191696026.596 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 14470191696026.596 \cdot 380}{6\cdot\sqrt{301456350881363594925339414949408216309322070465087890625}}\cr\approx \mathstrut & 0.311438504763485 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 6*x^31 - 4*x^30 + 56*x^29 + 186*x^28 - 958*x^27 - 1098*x^26 + 6187*x^25 + 12914*x^24 - 53084*x^23 - 45771*x^22 + 250489*x^21 + 210183*x^20 - 1343675*x^19 + 432927*x^18 + 3735782*x^17 - 4733917*x^16 - 7945401*x^15 + 34353397*x^14 - 53496679*x^13 + 52302241*x^12 - 28481985*x^11 + 8401560*x^10 + 3267183*x^9 - 363465*x^8 - 2158930*x^7 + 4307708*x^6 - 2643232*x^5 + 1313520*x^4 - 282816*x^3 + 44416*x^2 - 4096*x + 256);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$D_4^2:C_2^3$ (as 32T12882):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{8}$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.2.0.1}{2} }^{16}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1289\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |