Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*x + 256)
gp: K = bnfinit(y^32 - 2*y^31 - 20*y^30 + 24*y^29 + 286*y^28 - 250*y^27 - 2682*y^26 + 1817*y^25 + 19614*y^24 - 14252*y^23 - 106581*y^22 + 99261*y^21 + 427695*y^20 - 593179*y^19 - 962589*y^18 + 2289866*y^17 - 85839*y^16 - 3519833*y^15 + 7206071*y^14 - 1530693*y^13 + 140965*y^12 + 5408365*y^11 + 271976*y^10 + 335065*y^9 + 2939829*y^8 - 637412*y^7 + 550944*y^6 + 18112*y^5 + 35760*y^4 + 7872*y^3 + 4288*y^2 + 384*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*x + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*x + 256)
\( x^{32} - 2 x^{31} - 20 x^{30} + 24 x^{29} + 286 x^{28} - 250 x^{27} - 2682 x^{26} + 1817 x^{25} + \cdots + 256 \)
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: |
\(38187227411894497653750080270198291885967163238525390625\)
\(\medspace = 3^{16}\cdot 5^{16}\cdot 7^{16}\cdot 29^{8}\cdot 769^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.57\) | 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: | $3^{1/2}5^{1/2}7^{1/2}29^{1/2}769^{1/2}\approx 1530.2303748128907$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(29\), \(769\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{22}-\frac{1}{4}a^{20}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{21}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{24}-\frac{1}{4}a^{20}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{25}-\frac{1}{8}a^{24}-\frac{1}{8}a^{22}+\frac{1}{8}a^{21}-\frac{1}{4}a^{20}+\frac{1}{8}a^{19}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}+\frac{1}{8}a^{15}+\frac{1}{8}a^{14}-\frac{3}{8}a^{13}-\frac{1}{8}a^{12}+\frac{3}{8}a^{11}-\frac{1}{8}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{3}$, $\frac{1}{8}a^{26}-\frac{1}{8}a^{24}-\frac{1}{8}a^{23}-\frac{1}{8}a^{21}-\frac{1}{8}a^{20}+\frac{1}{8}a^{19}-\frac{1}{8}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{8}a^{11}-\frac{1}{2}a^{10}+\frac{3}{8}a^{9}-\frac{3}{8}a^{8}+\frac{1}{4}a^{7}-\frac{1}{8}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{88}a^{27}-\frac{1}{44}a^{26}-\frac{5}{88}a^{25}-\frac{1}{8}a^{24}-\frac{5}{44}a^{23}-\frac{7}{88}a^{22}+\frac{21}{88}a^{21}+\frac{1}{88}a^{20}+\frac{21}{88}a^{19}-\frac{1}{11}a^{17}+\frac{1}{22}a^{16}-\frac{1}{11}a^{15}+\frac{5}{44}a^{14}+\frac{3}{11}a^{13}-\frac{7}{88}a^{12}-\frac{3}{22}a^{11}-\frac{31}{88}a^{10}+\frac{37}{88}a^{9}+\frac{5}{44}a^{8}+\frac{31}{88}a^{7}+\frac{39}{88}a^{6}-\frac{39}{88}a^{5}-\frac{13}{88}a^{4}-\frac{7}{22}a^{3}-\frac{1}{44}a^{2}-\frac{7}{22}a+\frac{5}{11}$, $\frac{1}{176}a^{28}+\frac{1}{88}a^{26}-\frac{5}{88}a^{25}-\frac{5}{88}a^{24}-\frac{1}{11}a^{23}-\frac{1}{44}a^{22}-\frac{23}{176}a^{21}+\frac{17}{88}a^{20}-\frac{3}{22}a^{19}+\frac{25}{176}a^{18}-\frac{23}{176}a^{17}+\frac{1}{16}a^{16}+\frac{5}{176}a^{15}+\frac{5}{16}a^{14}+\frac{15}{88}a^{13}-\frac{37}{176}a^{12}+\frac{3}{16}a^{11}-\frac{3}{176}a^{10}-\frac{81}{176}a^{9}-\frac{81}{176}a^{8}-\frac{9}{176}a^{7}+\frac{9}{22}a^{6}+\frac{41}{176}a^{5}-\frac{43}{176}a^{4}+\frac{15}{88}a^{3}-\frac{2}{11}a^{2}-\frac{1}{11}a+\frac{5}{11}$, $\frac{1}{352}a^{29}+\frac{1}{22}a^{26}-\frac{1}{16}a^{25}+\frac{3}{176}a^{24}-\frac{3}{176}a^{23}-\frac{31}{352}a^{22}+\frac{9}{88}a^{21}+\frac{5}{44}a^{20}-\frac{17}{352}a^{19}-\frac{45}{352}a^{18}-\frac{83}{352}a^{17}+\frac{63}{352}a^{16}+\frac{49}{352}a^{15}+\frac{41}{88}a^{14}+\frac{25}{352}a^{13}-\frac{19}{352}a^{12}+\frac{65}{352}a^{11}-\frac{63}{352}a^{10}+\frac{131}{352}a^{9}+\frac{59}{352}a^{8}-\frac{83}{176}a^{7}-\frac{103}{352}a^{6}-\frac{141}{352}a^{5}+\frac{39}{176}a^{4}-\frac{27}{88}a^{3}-\frac{7}{44}a^{2}-\frac{4}{11}a+\frac{3}{11}$, $\frac{1}{12\!\cdots\!64}a^{30}+\frac{45\!\cdots\!37}{60\!\cdots\!32}a^{29}-\frac{77\!\cdots\!81}{15\!\cdots\!08}a^{28}-\frac{17\!\cdots\!93}{15\!\cdots\!08}a^{27}-\frac{33\!\cdots\!01}{60\!\cdots\!32}a^{26}+\frac{64\!\cdots\!19}{60\!\cdots\!32}a^{25}+\frac{55\!\cdots\!11}{60\!\cdots\!32}a^{24}-\frac{70\!\cdots\!35}{12\!\cdots\!64}a^{23}+\frac{59\!\cdots\!89}{60\!\cdots\!32}a^{22}-\frac{59\!\cdots\!53}{15\!\cdots\!08}a^{21}+\frac{17\!\cdots\!03}{12\!\cdots\!64}a^{20}-\frac{17\!\cdots\!55}{12\!\cdots\!64}a^{19}+\frac{95\!\cdots\!53}{10\!\cdots\!24}a^{18}-\frac{18\!\cdots\!31}{12\!\cdots\!64}a^{17}+\frac{24\!\cdots\!27}{12\!\cdots\!64}a^{16}+\frac{39\!\cdots\!05}{60\!\cdots\!32}a^{15}+\frac{18\!\cdots\!81}{12\!\cdots\!64}a^{14}+\frac{52\!\cdots\!65}{10\!\cdots\!24}a^{13}+\frac{18\!\cdots\!31}{12\!\cdots\!64}a^{12}-\frac{13\!\cdots\!13}{12\!\cdots\!64}a^{11}+\frac{40\!\cdots\!85}{12\!\cdots\!64}a^{10}+\frac{66\!\cdots\!33}{12\!\cdots\!64}a^{9}+\frac{35\!\cdots\!90}{18\!\cdots\!01}a^{8}-\frac{22\!\cdots\!27}{12\!\cdots\!64}a^{7}+\frac{56\!\cdots\!29}{12\!\cdots\!64}a^{6}-\frac{12\!\cdots\!25}{30\!\cdots\!16}a^{5}+\frac{12\!\cdots\!09}{30\!\cdots\!16}a^{4}-\frac{24\!\cdots\!75}{75\!\cdots\!04}a^{3}+\frac{29\!\cdots\!77}{68\!\cdots\!64}a^{2}+\frac{39\!\cdots\!61}{18\!\cdots\!01}a-\frac{83\!\cdots\!38}{18\!\cdots\!01}$, $\frac{1}{19\!\cdots\!68}a^{31}-\frac{35\!\cdots\!21}{89\!\cdots\!44}a^{30}-\frac{68\!\cdots\!37}{49\!\cdots\!92}a^{29}+\frac{52\!\cdots\!25}{24\!\cdots\!96}a^{28}+\frac{24\!\cdots\!67}{98\!\cdots\!84}a^{27}-\frac{27\!\cdots\!45}{98\!\cdots\!84}a^{26}+\frac{10\!\cdots\!55}{98\!\cdots\!84}a^{25}-\frac{71\!\cdots\!15}{19\!\cdots\!68}a^{24}+\frac{57\!\cdots\!01}{98\!\cdots\!84}a^{23}-\frac{12\!\cdots\!41}{44\!\cdots\!72}a^{22}-\frac{24\!\cdots\!77}{19\!\cdots\!68}a^{21}+\frac{13\!\cdots\!81}{19\!\cdots\!68}a^{20}+\frac{12\!\cdots\!03}{19\!\cdots\!68}a^{19}+\frac{47\!\cdots\!09}{19\!\cdots\!68}a^{18}-\frac{17\!\cdots\!85}{19\!\cdots\!68}a^{17}-\frac{18\!\cdots\!53}{98\!\cdots\!84}a^{16}-\frac{47\!\cdots\!15}{19\!\cdots\!68}a^{15}-\frac{12\!\cdots\!03}{17\!\cdots\!88}a^{14}-\frac{51\!\cdots\!73}{18\!\cdots\!52}a^{13}-\frac{27\!\cdots\!73}{19\!\cdots\!68}a^{12}-\frac{29\!\cdots\!31}{19\!\cdots\!68}a^{11}+\frac{17\!\cdots\!01}{19\!\cdots\!68}a^{10}+\frac{21\!\cdots\!37}{49\!\cdots\!92}a^{9}+\frac{85\!\cdots\!61}{19\!\cdots\!68}a^{8}-\frac{41\!\cdots\!77}{17\!\cdots\!88}a^{7}-\frac{39\!\cdots\!13}{24\!\cdots\!96}a^{6}+\frac{50\!\cdots\!45}{34\!\cdots\!76}a^{5}+\frac{85\!\cdots\!67}{11\!\cdots\!68}a^{4}-\frac{11\!\cdots\!95}{61\!\cdots\!24}a^{3}-\frac{16\!\cdots\!07}{61\!\cdots\!24}a^{2}-\frac{56\!\cdots\!65}{30\!\cdots\!62}a-\frac{34\!\cdots\!82}{15\!\cdots\!81}$
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
$C_{2}\times C_{2}\times C_{64}$, which has order $256$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $256$ (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{6316431889618417040785100776953905963421799934108988606355357}{6298506191864642495415701393810178443998231679109013525284884416} a^{31} + \frac{628130092812516670268472456129631546992712130101252647369921}{286295735993847386155259154264099020181737803595864251149312928} a^{30} + \frac{7775610426990796235204277564994277734613429971051200334912831}{393656636991540155963481337113136152749889479944313345330305276} a^{29} - \frac{44060225047631684903598711253111693345203033465689617799657021}{1574626547966160623853925348452544610999557919777253381321221104} a^{28} - \frac{893551129462945466467758133437618038357889535370831908234780071}{3149253095932321247707850696905089221999115839554506762642442208} a^{27} + \frac{964843324438177314615652349188993642495934701296830777005498005}{3149253095932321247707850696905089221999115839554506762642442208} a^{26} + \frac{8387077626946712447665019626276199376392205544177893750544147145}{3149253095932321247707850696905089221999115839554506762642442208} a^{25} - \frac{14778231404474885462350421428738149033971574495614232506149657517}{6298506191864642495415701393810178443998231679109013525284884416} a^{24} - \frac{61479191481251671626582396930225061367588678189291601053669222949}{3149253095932321247707850696905089221999115839554506762642442208} a^{23} + \frac{648731220261679561599974555602484429099366303704506451532427159}{35786966999230923269407394283012377522717225449483031393664116} a^{22} + \frac{665278330750201789703659078422614278431984794351531685834210887077}{6298506191864642495415701393810178443998231679109013525284884416} a^{21} - \frac{760353519532506044252173703807255319024466145581611958030653261661}{6298506191864642495415701393810178443998231679109013525284884416} a^{20} - \frac{2632661032020302828304120034029686974388973878542182574447211894603}{6298506191864642495415701393810178443998231679109013525284884416} a^{19} + \frac{4302064030111858408891619698491413602149851668620413240906907697787}{6298506191864642495415701393810178443998231679109013525284884416} a^{18} + \frac{5573124120966449263578819889923014802567178010104640626133102425053}{6298506191864642495415701393810178443998231679109013525284884416} a^{17} - \frac{7944404540904370668444314123121205809532512892873569620783312141259}{3149253095932321247707850696905089221999115839554506762642442208} a^{16} + \frac{2815245207209419070046513245077201366309731774204846322651542511235}{6298506191864642495415701393810178443998231679109013525284884416} a^{15} + \frac{191909522489853621015549006184387555826673499700312607036382644885}{52053770180699524755501664411654367305770509744702591118056896} a^{14} - \frac{456313494067990291518240745631634598755327845824149926801501096999}{57784460475822408214822948567065857284387446597330399314540224} a^{13} + \frac{16501449140753103785944926447540426109569570782939867282355987167021}{6298506191864642495415701393810178443998231679109013525284884416} a^{12} + \frac{658077898862305960395842026870819635112446043530491935287193541091}{6298506191864642495415701393810178443998231679109013525284884416} a^{11} - \frac{34656780824129533939095105128930380548037731876056586689749995438589}{6298506191864642495415701393810178443998231679109013525284884416} a^{10} + \frac{1104456735596687415622496338849876287968445329550069153614881471675}{1574626547966160623853925348452544610999557919777253381321221104} a^{9} + \frac{778499057631527780623279869440312312105382103277665549826396700491}{6298506191864642495415701393810178443998231679109013525284884416} a^{8} - \frac{149764706780411040838188637448468186141374415765234105486065837217}{52053770180699524755501664411654367305770509744702591118056896} a^{7} + \frac{1848999091803279006522738265046326423878205398567598431962660586537}{1574626547966160623853925348452544610999557919777253381321221104} a^{6} - \frac{1266742729663451572241766192988049346205684561169423093225720857}{2772229837968592647630150261360113751759785070030375671340178} a^{5} + \frac{5037325923612884045677115469273669155748887122017712577746268277}{143147867996923693077629577132049510090868901797932125574656464} a^{4} - \frac{2442543593906218133730822660958682710310541588471421629199699987}{196828318495770077981740668556568076374944739972156672665152638} a^{3} + \frac{695589830020701910319430584878487658247282746129427243779440692}{98414159247885038990870334278284038187472369986078336332576319} a^{2} - \frac{437409499744626237106535422083142423042099221100352261445915379}{196828318495770077981740668556568076374944739972156672665152638} a + \frac{110728746966283131900792903881454121058361979088756628515636748}{98414159247885038990870334278284038187472369986078336332576319} \)
(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: |
$\frac{23\!\cdots\!61}{21\!\cdots\!16}a^{31}-\frac{40\!\cdots\!67}{21\!\cdots\!16}a^{30}-\frac{15\!\cdots\!39}{67\!\cdots\!38}a^{29}+\frac{27\!\cdots\!71}{13\!\cdots\!76}a^{28}+\frac{35\!\cdots\!91}{10\!\cdots\!08}a^{27}-\frac{49\!\cdots\!65}{27\!\cdots\!52}a^{26}-\frac{16\!\cdots\!37}{54\!\cdots\!04}a^{25}+\frac{24\!\cdots\!43}{21\!\cdots\!16}a^{24}+\frac{48\!\cdots\!97}{21\!\cdots\!16}a^{23}-\frac{25\!\cdots\!21}{27\!\cdots\!52}a^{22}-\frac{26\!\cdots\!49}{21\!\cdots\!16}a^{21}+\frac{80\!\cdots\!49}{10\!\cdots\!08}a^{20}+\frac{13\!\cdots\!29}{27\!\cdots\!52}a^{19}-\frac{13\!\cdots\!09}{27\!\cdots\!52}a^{18}-\frac{69\!\cdots\!61}{54\!\cdots\!04}a^{17}+\frac{48\!\cdots\!01}{21\!\cdots\!16}a^{16}+\frac{15\!\cdots\!69}{21\!\cdots\!16}a^{15}-\frac{43\!\cdots\!95}{10\!\cdots\!08}a^{14}+\frac{73\!\cdots\!93}{10\!\cdots\!08}a^{13}+\frac{88\!\cdots\!17}{10\!\cdots\!08}a^{12}-\frac{12\!\cdots\!07}{27\!\cdots\!52}a^{11}+\frac{63\!\cdots\!15}{10\!\cdots\!08}a^{10}+\frac{46\!\cdots\!97}{21\!\cdots\!16}a^{9}+\frac{66\!\cdots\!23}{21\!\cdots\!16}a^{8}+\frac{17\!\cdots\!47}{54\!\cdots\!04}a^{7}+\frac{57\!\cdots\!21}{21\!\cdots\!16}a^{6}+\frac{34\!\cdots\!51}{10\!\cdots\!08}a^{5}+\frac{37\!\cdots\!47}{27\!\cdots\!52}a^{4}+\frac{14\!\cdots\!33}{27\!\cdots\!52}a^{3}+\frac{16\!\cdots\!83}{13\!\cdots\!76}a^{2}+\frac{20\!\cdots\!95}{67\!\cdots\!38}a+\frac{25\!\cdots\!74}{33\!\cdots\!69}$, $\frac{21\!\cdots\!71}{36\!\cdots\!64}a^{31}-\frac{33\!\cdots\!87}{33\!\cdots\!24}a^{30}-\frac{56\!\cdots\!99}{45\!\cdots\!08}a^{29}+\frac{24\!\cdots\!91}{22\!\cdots\!04}a^{28}+\frac{32\!\cdots\!81}{18\!\cdots\!32}a^{27}-\frac{45\!\cdots\!65}{45\!\cdots\!08}a^{26}-\frac{15\!\cdots\!85}{91\!\cdots\!16}a^{25}+\frac{22\!\cdots\!37}{36\!\cdots\!64}a^{24}+\frac{44\!\cdots\!55}{36\!\cdots\!64}a^{23}-\frac{20\!\cdots\!71}{41\!\cdots\!28}a^{22}-\frac{24\!\cdots\!47}{36\!\cdots\!64}a^{21}+\frac{73\!\cdots\!55}{18\!\cdots\!32}a^{20}+\frac{12\!\cdots\!75}{45\!\cdots\!08}a^{19}-\frac{63\!\cdots\!73}{22\!\cdots\!04}a^{18}-\frac{63\!\cdots\!87}{91\!\cdots\!16}a^{17}+\frac{44\!\cdots\!83}{36\!\cdots\!64}a^{16}+\frac{14\!\cdots\!75}{36\!\cdots\!64}a^{15}-\frac{35\!\cdots\!99}{16\!\cdots\!12}a^{14}+\frac{61\!\cdots\!11}{16\!\cdots\!48}a^{13}+\frac{81\!\cdots\!35}{18\!\cdots\!32}a^{12}-\frac{11\!\cdots\!17}{45\!\cdots\!08}a^{11}+\frac{58\!\cdots\!85}{18\!\cdots\!32}a^{10}+\frac{42\!\cdots\!47}{36\!\cdots\!64}a^{9}+\frac{60\!\cdots\!21}{36\!\cdots\!64}a^{8}+\frac{14\!\cdots\!79}{82\!\cdots\!56}a^{7}+\frac{52\!\cdots\!43}{36\!\cdots\!64}a^{6}+\frac{43\!\cdots\!27}{25\!\cdots\!92}a^{5}+\frac{30\!\cdots\!59}{41\!\cdots\!28}a^{4}+\frac{64\!\cdots\!59}{22\!\cdots\!04}a^{3}+\frac{14\!\cdots\!09}{22\!\cdots\!04}a^{2}+\frac{18\!\cdots\!61}{11\!\cdots\!02}a+\frac{31\!\cdots\!45}{57\!\cdots\!51}$, $\frac{26\!\cdots\!17}{17\!\cdots\!88}a^{31}-\frac{26\!\cdots\!19}{89\!\cdots\!44}a^{30}-\frac{64\!\cdots\!15}{22\!\cdots\!36}a^{29}+\frac{82\!\cdots\!21}{22\!\cdots\!36}a^{28}+\frac{37\!\cdots\!75}{89\!\cdots\!44}a^{27}-\frac{34\!\cdots\!73}{89\!\cdots\!44}a^{26}-\frac{34\!\cdots\!77}{89\!\cdots\!44}a^{25}+\frac{46\!\cdots\!79}{16\!\cdots\!08}a^{24}+\frac{25\!\cdots\!29}{89\!\cdots\!44}a^{23}-\frac{31\!\cdots\!48}{14\!\cdots\!71}a^{22}-\frac{27\!\cdots\!81}{17\!\cdots\!88}a^{21}+\frac{25\!\cdots\!95}{16\!\cdots\!08}a^{20}+\frac{11\!\cdots\!07}{17\!\cdots\!88}a^{19}-\frac{16\!\cdots\!79}{17\!\cdots\!88}a^{18}-\frac{24\!\cdots\!85}{17\!\cdots\!88}a^{17}+\frac{30\!\cdots\!05}{89\!\cdots\!44}a^{16}-\frac{43\!\cdots\!15}{17\!\cdots\!88}a^{15}-\frac{94\!\cdots\!09}{17\!\cdots\!88}a^{14}+\frac{17\!\cdots\!15}{16\!\cdots\!32}a^{13}-\frac{45\!\cdots\!21}{17\!\cdots\!88}a^{12}-\frac{21\!\cdots\!19}{17\!\cdots\!88}a^{11}+\frac{14\!\cdots\!25}{17\!\cdots\!88}a^{10}+\frac{45\!\cdots\!85}{28\!\cdots\!42}a^{9}+\frac{18\!\cdots\!65}{17\!\cdots\!88}a^{8}+\frac{75\!\cdots\!01}{17\!\cdots\!88}a^{7}-\frac{51\!\cdots\!01}{44\!\cdots\!72}a^{6}+\frac{41\!\cdots\!29}{63\!\cdots\!32}a^{5}+\frac{70\!\cdots\!39}{22\!\cdots\!36}a^{4}+\frac{46\!\cdots\!85}{11\!\cdots\!68}a^{3}-\frac{35\!\cdots\!77}{28\!\cdots\!42}a^{2}+\frac{73\!\cdots\!05}{14\!\cdots\!71}a+\frac{46\!\cdots\!70}{14\!\cdots\!71}$, $\frac{19\!\cdots\!13}{98\!\cdots\!84}a^{31}-\frac{14\!\cdots\!25}{44\!\cdots\!72}a^{30}-\frac{25\!\cdots\!67}{61\!\cdots\!24}a^{29}+\frac{81\!\cdots\!09}{24\!\cdots\!96}a^{28}+\frac{28\!\cdots\!19}{49\!\cdots\!92}a^{27}-\frac{14\!\cdots\!89}{49\!\cdots\!92}a^{26}-\frac{27\!\cdots\!53}{49\!\cdots\!92}a^{25}+\frac{17\!\cdots\!09}{98\!\cdots\!84}a^{24}+\frac{20\!\cdots\!53}{49\!\cdots\!92}a^{23}-\frac{83\!\cdots\!93}{56\!\cdots\!84}a^{22}-\frac{22\!\cdots\!89}{98\!\cdots\!84}a^{21}+\frac{12\!\cdots\!49}{98\!\cdots\!84}a^{20}+\frac{93\!\cdots\!83}{98\!\cdots\!84}a^{19}-\frac{90\!\cdots\!47}{98\!\cdots\!84}a^{18}-\frac{24\!\cdots\!65}{98\!\cdots\!84}a^{17}+\frac{20\!\cdots\!39}{49\!\cdots\!92}a^{16}+\frac{16\!\cdots\!85}{98\!\cdots\!84}a^{15}-\frac{74\!\cdots\!43}{89\!\cdots\!44}a^{14}+\frac{11\!\cdots\!11}{90\!\cdots\!76}a^{13}+\frac{40\!\cdots\!75}{98\!\cdots\!84}a^{12}-\frac{51\!\cdots\!19}{98\!\cdots\!84}a^{11}+\frac{13\!\cdots\!25}{98\!\cdots\!84}a^{10}+\frac{12\!\cdots\!49}{24\!\cdots\!96}a^{9}-\frac{16\!\cdots\!27}{98\!\cdots\!84}a^{8}+\frac{72\!\cdots\!35}{89\!\cdots\!44}a^{7}+\frac{42\!\cdots\!47}{24\!\cdots\!96}a^{6}-\frac{28\!\cdots\!87}{43\!\cdots\!22}a^{5}+\frac{40\!\cdots\!87}{22\!\cdots\!36}a^{4}-\frac{41\!\cdots\!01}{30\!\cdots\!62}a^{3}+\frac{16\!\cdots\!28}{15\!\cdots\!81}a^{2}+\frac{11\!\cdots\!71}{30\!\cdots\!62}a+\frac{18\!\cdots\!25}{15\!\cdots\!81}$, $\frac{88\!\cdots\!51}{89\!\cdots\!44}a^{31}-\frac{14\!\cdots\!69}{89\!\cdots\!44}a^{30}-\frac{91\!\cdots\!39}{44\!\cdots\!72}a^{29}+\frac{18\!\cdots\!05}{11\!\cdots\!68}a^{28}+\frac{12\!\cdots\!17}{44\!\cdots\!72}a^{27}-\frac{16\!\cdots\!11}{11\!\cdots\!68}a^{26}-\frac{69\!\cdots\!65}{25\!\cdots\!22}a^{25}+\frac{75\!\cdots\!29}{89\!\cdots\!44}a^{24}+\frac{17\!\cdots\!39}{89\!\cdots\!44}a^{23}-\frac{31\!\cdots\!67}{44\!\cdots\!72}a^{22}-\frac{97\!\cdots\!87}{89\!\cdots\!44}a^{21}+\frac{26\!\cdots\!39}{44\!\cdots\!72}a^{20}+\frac{20\!\cdots\!03}{44\!\cdots\!72}a^{19}-\frac{19\!\cdots\!77}{44\!\cdots\!72}a^{18}-\frac{50\!\cdots\!97}{44\!\cdots\!72}a^{17}+\frac{16\!\cdots\!69}{89\!\cdots\!44}a^{16}+\frac{59\!\cdots\!97}{89\!\cdots\!44}a^{15}-\frac{14\!\cdots\!69}{44\!\cdots\!72}a^{14}+\frac{29\!\cdots\!67}{51\!\cdots\!76}a^{13}+\frac{35\!\cdots\!91}{56\!\cdots\!84}a^{12}+\frac{15\!\cdots\!39}{40\!\cdots\!52}a^{11}+\frac{10\!\cdots\!03}{22\!\cdots\!36}a^{10}+\frac{18\!\cdots\!73}{89\!\cdots\!44}a^{9}+\frac{66\!\cdots\!35}{89\!\cdots\!44}a^{8}+\frac{14\!\cdots\!31}{56\!\cdots\!84}a^{7}+\frac{16\!\cdots\!33}{89\!\cdots\!44}a^{6}+\frac{85\!\cdots\!49}{19\!\cdots\!01}a^{5}-\frac{24\!\cdots\!25}{22\!\cdots\!36}a^{4}+\frac{43\!\cdots\!81}{11\!\cdots\!68}a^{3}-\frac{63\!\cdots\!41}{28\!\cdots\!42}a^{2}-\frac{34\!\cdots\!84}{14\!\cdots\!71}a-\frac{15\!\cdots\!87}{14\!\cdots\!71}$, $\frac{94\!\cdots\!79}{19\!\cdots\!68}a^{31}+\frac{28\!\cdots\!31}{89\!\cdots\!44}a^{30}-\frac{89\!\cdots\!43}{49\!\cdots\!92}a^{29}-\frac{42\!\cdots\!89}{61\!\cdots\!24}a^{28}+\frac{24\!\cdots\!57}{98\!\cdots\!84}a^{27}+\frac{10\!\cdots\!09}{98\!\cdots\!84}a^{26}-\frac{23\!\cdots\!39}{98\!\cdots\!84}a^{25}-\frac{19\!\cdots\!17}{19\!\cdots\!68}a^{24}+\frac{17\!\cdots\!97}{98\!\cdots\!84}a^{23}+\frac{32\!\cdots\!65}{44\!\cdots\!72}a^{22}-\frac{23\!\cdots\!11}{19\!\cdots\!68}a^{21}-\frac{75\!\cdots\!89}{19\!\cdots\!68}a^{20}+\frac{12\!\cdots\!41}{19\!\cdots\!68}a^{19}+\frac{27\!\cdots\!35}{19\!\cdots\!68}a^{18}-\frac{60\!\cdots\!59}{19\!\cdots\!68}a^{17}-\frac{25\!\cdots\!45}{98\!\cdots\!84}a^{16}+\frac{18\!\cdots\!15}{19\!\cdots\!68}a^{15}-\frac{48\!\cdots\!39}{16\!\cdots\!08}a^{14}-\frac{19\!\cdots\!91}{18\!\cdots\!52}a^{13}+\frac{59\!\cdots\!09}{19\!\cdots\!68}a^{12}-\frac{18\!\cdots\!85}{19\!\cdots\!68}a^{11}+\frac{92\!\cdots\!55}{19\!\cdots\!68}a^{10}+\frac{51\!\cdots\!03}{24\!\cdots\!96}a^{9}-\frac{23\!\cdots\!65}{19\!\cdots\!68}a^{8}+\frac{45\!\cdots\!31}{16\!\cdots\!08}a^{7}+\frac{51\!\cdots\!43}{49\!\cdots\!92}a^{6}-\frac{13\!\cdots\!61}{34\!\cdots\!76}a^{5}+\frac{52\!\cdots\!11}{20\!\cdots\!76}a^{4}-\frac{81\!\cdots\!67}{12\!\cdots\!48}a^{3}+\frac{14\!\cdots\!41}{61\!\cdots\!24}a^{2}-\frac{47\!\cdots\!38}{15\!\cdots\!81}a+\frac{11\!\cdots\!74}{15\!\cdots\!81}$, $\frac{25\!\cdots\!39}{19\!\cdots\!68}a^{31}-\frac{95\!\cdots\!37}{44\!\cdots\!72}a^{30}-\frac{41\!\cdots\!00}{15\!\cdots\!81}a^{29}+\frac{54\!\cdots\!73}{24\!\cdots\!96}a^{28}+\frac{37\!\cdots\!69}{98\!\cdots\!84}a^{27}-\frac{18\!\cdots\!45}{98\!\cdots\!84}a^{26}-\frac{35\!\cdots\!29}{98\!\cdots\!84}a^{25}+\frac{22\!\cdots\!47}{19\!\cdots\!68}a^{24}+\frac{13\!\cdots\!65}{49\!\cdots\!92}a^{23}-\frac{21\!\cdots\!35}{22\!\cdots\!36}a^{22}-\frac{28\!\cdots\!91}{19\!\cdots\!68}a^{21}+\frac{15\!\cdots\!09}{19\!\cdots\!68}a^{20}+\frac{11\!\cdots\!31}{19\!\cdots\!68}a^{19}-\frac{11\!\cdots\!59}{19\!\cdots\!68}a^{18}-\frac{30\!\cdots\!65}{19\!\cdots\!68}a^{17}+\frac{12\!\cdots\!09}{49\!\cdots\!92}a^{16}+\frac{19\!\cdots\!07}{19\!\cdots\!68}a^{15}-\frac{83\!\cdots\!63}{17\!\cdots\!88}a^{14}+\frac{13\!\cdots\!91}{18\!\cdots\!52}a^{13}+\frac{27\!\cdots\!39}{19\!\cdots\!68}a^{12}-\frac{10\!\cdots\!15}{19\!\cdots\!68}a^{11}+\frac{13\!\cdots\!69}{19\!\cdots\!68}a^{10}+\frac{29\!\cdots\!25}{98\!\cdots\!84}a^{9}+\frac{10\!\cdots\!51}{19\!\cdots\!68}a^{8}+\frac{70\!\cdots\!47}{17\!\cdots\!88}a^{7}+\frac{58\!\cdots\!95}{98\!\cdots\!84}a^{6}+\frac{72\!\cdots\!29}{17\!\cdots\!88}a^{5}+\frac{59\!\cdots\!69}{22\!\cdots\!36}a^{4}+\frac{98\!\cdots\!17}{12\!\cdots\!48}a^{3}+\frac{31\!\cdots\!81}{15\!\cdots\!81}a^{2}+\frac{25\!\cdots\!85}{30\!\cdots\!62}a+\frac{29\!\cdots\!02}{15\!\cdots\!81}$, $\frac{26\!\cdots\!77}{98\!\cdots\!84}a^{31}-\frac{95\!\cdots\!59}{22\!\cdots\!36}a^{30}-\frac{28\!\cdots\!61}{49\!\cdots\!92}a^{29}+\frac{10\!\cdots\!29}{24\!\cdots\!96}a^{28}+\frac{40\!\cdots\!11}{49\!\cdots\!92}a^{27}-\frac{17\!\cdots\!55}{49\!\cdots\!92}a^{26}-\frac{38\!\cdots\!85}{49\!\cdots\!92}a^{25}+\frac{18\!\cdots\!49}{98\!\cdots\!84}a^{24}+\frac{35\!\cdots\!57}{61\!\cdots\!24}a^{23}-\frac{73\!\cdots\!47}{44\!\cdots\!72}a^{22}-\frac{31\!\cdots\!41}{98\!\cdots\!84}a^{21}+\frac{14\!\cdots\!79}{98\!\cdots\!84}a^{20}+\frac{13\!\cdots\!71}{98\!\cdots\!84}a^{19}-\frac{11\!\cdots\!95}{98\!\cdots\!84}a^{18}-\frac{34\!\cdots\!77}{98\!\cdots\!84}a^{17}+\frac{26\!\cdots\!13}{49\!\cdots\!92}a^{16}+\frac{29\!\cdots\!35}{98\!\cdots\!84}a^{15}-\frac{95\!\cdots\!89}{89\!\cdots\!44}a^{14}+\frac{13\!\cdots\!79}{90\!\cdots\!76}a^{13}+\frac{59\!\cdots\!07}{98\!\cdots\!84}a^{12}-\frac{43\!\cdots\!35}{98\!\cdots\!84}a^{11}+\frac{14\!\cdots\!53}{98\!\cdots\!84}a^{10}+\frac{17\!\cdots\!53}{24\!\cdots\!96}a^{9}-\frac{13\!\cdots\!21}{98\!\cdots\!84}a^{8}+\frac{67\!\cdots\!45}{89\!\cdots\!44}a^{7}+\frac{96\!\cdots\!25}{61\!\cdots\!24}a^{6}-\frac{45\!\cdots\!15}{69\!\cdots\!52}a^{5}+\frac{72\!\cdots\!33}{11\!\cdots\!68}a^{4}-\frac{35\!\cdots\!85}{12\!\cdots\!48}a^{3}+\frac{15\!\cdots\!25}{30\!\cdots\!62}a^{2}+\frac{16\!\cdots\!87}{15\!\cdots\!81}a+\frac{30\!\cdots\!50}{15\!\cdots\!81}$, $\frac{54\!\cdots\!77}{27\!\cdots\!08}a^{31}-\frac{56\!\cdots\!36}{19\!\cdots\!01}a^{30}-\frac{28\!\cdots\!61}{69\!\cdots\!52}a^{29}+\frac{88\!\cdots\!15}{34\!\cdots\!76}a^{28}+\frac{81\!\cdots\!31}{13\!\cdots\!04}a^{27}-\frac{25\!\cdots\!15}{13\!\cdots\!04}a^{26}-\frac{76\!\cdots\!83}{13\!\cdots\!04}a^{25}+\frac{18\!\cdots\!17}{27\!\cdots\!08}a^{24}+\frac{13\!\cdots\!35}{34\!\cdots\!76}a^{23}-\frac{42\!\cdots\!21}{63\!\cdots\!32}a^{22}-\frac{61\!\cdots\!45}{27\!\cdots\!08}a^{21}+\frac{22\!\cdots\!31}{27\!\cdots\!08}a^{20}+\frac{25\!\cdots\!05}{27\!\cdots\!08}a^{19}-\frac{19\!\cdots\!57}{27\!\cdots\!08}a^{18}-\frac{68\!\cdots\!31}{27\!\cdots\!08}a^{17}+\frac{23\!\cdots\!73}{69\!\cdots\!52}a^{16}+\frac{58\!\cdots\!69}{27\!\cdots\!08}a^{15}-\frac{17\!\cdots\!25}{25\!\cdots\!28}a^{14}+\frac{27\!\cdots\!65}{25\!\cdots\!12}a^{13}+\frac{11\!\cdots\!45}{27\!\cdots\!08}a^{12}-\frac{25\!\cdots\!37}{27\!\cdots\!08}a^{11}+\frac{32\!\cdots\!79}{27\!\cdots\!08}a^{10}+\frac{77\!\cdots\!19}{13\!\cdots\!04}a^{9}+\frac{44\!\cdots\!29}{27\!\cdots\!08}a^{8}+\frac{16\!\cdots\!21}{25\!\cdots\!28}a^{7}+\frac{20\!\cdots\!25}{13\!\cdots\!04}a^{6}+\frac{48\!\cdots\!27}{69\!\cdots\!52}a^{5}+\frac{11\!\cdots\!93}{15\!\cdots\!08}a^{4}-\frac{25\!\cdots\!33}{17\!\cdots\!88}a^{3}+\frac{52\!\cdots\!01}{43\!\cdots\!22}a^{2}-\frac{39\!\cdots\!19}{43\!\cdots\!22}a+\frac{12\!\cdots\!92}{21\!\cdots\!11}$, $\frac{11\!\cdots\!57}{12\!\cdots\!48}a^{31}-\frac{17\!\cdots\!79}{89\!\cdots\!44}a^{30}-\frac{87\!\cdots\!43}{49\!\cdots\!92}a^{29}+\frac{62\!\cdots\!43}{24\!\cdots\!96}a^{28}+\frac{31\!\cdots\!83}{12\!\cdots\!48}a^{27}-\frac{13\!\cdots\!71}{49\!\cdots\!92}a^{26}-\frac{11\!\cdots\!11}{49\!\cdots\!92}a^{25}+\frac{10\!\cdots\!29}{49\!\cdots\!92}a^{24}+\frac{17\!\cdots\!43}{98\!\cdots\!84}a^{23}-\frac{73\!\cdots\!85}{44\!\cdots\!72}a^{22}-\frac{23\!\cdots\!27}{24\!\cdots\!96}a^{21}+\frac{10\!\cdots\!69}{98\!\cdots\!84}a^{20}+\frac{36\!\cdots\!75}{98\!\cdots\!84}a^{19}-\frac{60\!\cdots\!03}{98\!\cdots\!84}a^{18}-\frac{77\!\cdots\!13}{98\!\cdots\!84}a^{17}+\frac{22\!\cdots\!01}{98\!\cdots\!84}a^{16}-\frac{20\!\cdots\!33}{49\!\cdots\!92}a^{15}-\frac{29\!\cdots\!79}{89\!\cdots\!44}a^{14}+\frac{64\!\cdots\!49}{90\!\cdots\!76}a^{13}-\frac{23\!\cdots\!31}{98\!\cdots\!84}a^{12}+\frac{33\!\cdots\!33}{98\!\cdots\!84}a^{11}+\frac{48\!\cdots\!23}{98\!\cdots\!84}a^{10}-\frac{60\!\cdots\!49}{98\!\cdots\!84}a^{9}-\frac{11\!\cdots\!17}{12\!\cdots\!48}a^{8}+\frac{21\!\cdots\!91}{81\!\cdots\!04}a^{7}-\frac{10\!\cdots\!73}{98\!\cdots\!84}a^{6}+\frac{16\!\cdots\!49}{34\!\cdots\!76}a^{5}-\frac{84\!\cdots\!11}{28\!\cdots\!42}a^{4}+\frac{30\!\cdots\!13}{12\!\cdots\!48}a^{3}-\frac{25\!\cdots\!13}{61\!\cdots\!24}a^{2}+\frac{46\!\cdots\!65}{15\!\cdots\!81}a+\frac{12\!\cdots\!77}{15\!\cdots\!81}$, $\frac{10\!\cdots\!97}{19\!\cdots\!68}a^{31}-\frac{12\!\cdots\!07}{11\!\cdots\!68}a^{30}-\frac{67\!\cdots\!45}{61\!\cdots\!24}a^{29}+\frac{33\!\cdots\!71}{24\!\cdots\!96}a^{28}+\frac{15\!\cdots\!99}{98\!\cdots\!84}a^{27}-\frac{14\!\cdots\!51}{98\!\cdots\!84}a^{26}-\frac{14\!\cdots\!47}{98\!\cdots\!84}a^{25}+\frac{20\!\cdots\!17}{19\!\cdots\!68}a^{24}+\frac{13\!\cdots\!75}{12\!\cdots\!48}a^{23}-\frac{18\!\cdots\!57}{22\!\cdots\!36}a^{22}-\frac{11\!\cdots\!41}{19\!\cdots\!68}a^{21}+\frac{11\!\cdots\!95}{19\!\cdots\!68}a^{20}+\frac{46\!\cdots\!01}{19\!\cdots\!68}a^{19}-\frac{66\!\cdots\!09}{19\!\cdots\!68}a^{18}-\frac{10\!\cdots\!51}{19\!\cdots\!68}a^{17}+\frac{80\!\cdots\!17}{61\!\cdots\!24}a^{16}-\frac{67\!\cdots\!63}{19\!\cdots\!68}a^{15}-\frac{36\!\cdots\!29}{17\!\cdots\!88}a^{14}+\frac{72\!\cdots\!33}{18\!\cdots\!52}a^{13}-\frac{15\!\cdots\!07}{19\!\cdots\!68}a^{12}-\frac{54\!\cdots\!73}{19\!\cdots\!68}a^{11}+\frac{58\!\cdots\!51}{19\!\cdots\!68}a^{10}+\frac{20\!\cdots\!57}{98\!\cdots\!84}a^{9}-\frac{31\!\cdots\!63}{19\!\cdots\!68}a^{8}+\frac{26\!\cdots\!29}{17\!\cdots\!88}a^{7}-\frac{44\!\cdots\!05}{98\!\cdots\!84}a^{6}+\frac{43\!\cdots\!51}{34\!\cdots\!76}a^{5}-\frac{17\!\cdots\!59}{22\!\cdots\!36}a^{4}-\frac{16\!\cdots\!00}{15\!\cdots\!81}a^{3}-\frac{33\!\cdots\!75}{61\!\cdots\!24}a^{2}-\frac{64\!\cdots\!61}{30\!\cdots\!62}a-\frac{44\!\cdots\!54}{15\!\cdots\!81}$, $\frac{21\!\cdots\!75}{19\!\cdots\!68}a^{31}-\frac{16\!\cdots\!31}{89\!\cdots\!44}a^{30}-\frac{27\!\cdots\!61}{12\!\cdots\!48}a^{29}+\frac{50\!\cdots\!97}{24\!\cdots\!96}a^{28}+\frac{31\!\cdots\!61}{98\!\cdots\!84}a^{27}-\frac{18\!\cdots\!87}{98\!\cdots\!84}a^{26}-\frac{30\!\cdots\!87}{98\!\cdots\!84}a^{25}+\frac{23\!\cdots\!63}{19\!\cdots\!68}a^{24}+\frac{22\!\cdots\!31}{98\!\cdots\!84}a^{23}-\frac{22\!\cdots\!83}{22\!\cdots\!36}a^{22}-\frac{24\!\cdots\!71}{19\!\cdots\!68}a^{21}+\frac{15\!\cdots\!79}{19\!\cdots\!68}a^{20}+\frac{10\!\cdots\!61}{19\!\cdots\!68}a^{19}-\frac{10\!\cdots\!53}{19\!\cdots\!68}a^{18}-\frac{25\!\cdots\!63}{19\!\cdots\!68}a^{17}+\frac{22\!\cdots\!79}{98\!\cdots\!84}a^{16}+\frac{15\!\cdots\!31}{19\!\cdots\!68}a^{15}-\frac{76\!\cdots\!89}{17\!\cdots\!88}a^{14}+\frac{12\!\cdots\!65}{18\!\cdots\!52}a^{13}+\frac{23\!\cdots\!53}{19\!\cdots\!68}a^{12}-\frac{32\!\cdots\!61}{19\!\cdots\!68}a^{11}+\frac{12\!\cdots\!59}{19\!\cdots\!68}a^{10}+\frac{49\!\cdots\!75}{24\!\cdots\!96}a^{9}-\frac{10\!\cdots\!73}{19\!\cdots\!68}a^{8}+\frac{54\!\cdots\!99}{16\!\cdots\!08}a^{7}+\frac{10\!\cdots\!71}{49\!\cdots\!92}a^{6}-\frac{72\!\cdots\!67}{69\!\cdots\!52}a^{5}+\frac{78\!\cdots\!37}{22\!\cdots\!36}a^{4}-\frac{99\!\cdots\!09}{30\!\cdots\!62}a^{3}+\frac{10\!\cdots\!25}{61\!\cdots\!24}a^{2}+\frac{16\!\cdots\!85}{30\!\cdots\!62}a+\frac{10\!\cdots\!96}{15\!\cdots\!81}$, $\frac{22\!\cdots\!71}{24\!\cdots\!96}a^{31}-\frac{11\!\cdots\!99}{44\!\cdots\!72}a^{30}-\frac{41\!\cdots\!69}{24\!\cdots\!96}a^{29}+\frac{42\!\cdots\!67}{12\!\cdots\!48}a^{28}+\frac{15\!\cdots\!31}{61\!\cdots\!24}a^{27}-\frac{10\!\cdots\!11}{24\!\cdots\!96}a^{26}-\frac{55\!\cdots\!09}{24\!\cdots\!96}a^{25}+\frac{41\!\cdots\!81}{12\!\cdots\!48}a^{24}+\frac{81\!\cdots\!47}{49\!\cdots\!92}a^{23}-\frac{57\!\cdots\!23}{22\!\cdots\!36}a^{22}-\frac{21\!\cdots\!23}{24\!\cdots\!96}a^{21}+\frac{78\!\cdots\!87}{49\!\cdots\!92}a^{20}+\frac{15\!\cdots\!13}{49\!\cdots\!92}a^{19}-\frac{40\!\cdots\!21}{49\!\cdots\!92}a^{18}-\frac{21\!\cdots\!47}{49\!\cdots\!92}a^{17}+\frac{13\!\cdots\!89}{49\!\cdots\!92}a^{16}-\frac{21\!\cdots\!11}{12\!\cdots\!48}a^{15}-\frac{12\!\cdots\!01}{44\!\cdots\!72}a^{14}+\frac{40\!\cdots\!71}{45\!\cdots\!88}a^{13}-\frac{32\!\cdots\!21}{49\!\cdots\!92}a^{12}+\frac{10\!\cdots\!07}{49\!\cdots\!92}a^{11}+\frac{25\!\cdots\!49}{49\!\cdots\!92}a^{10}-\frac{16\!\cdots\!73}{49\!\cdots\!92}a^{9}+\frac{26\!\cdots\!25}{24\!\cdots\!96}a^{8}+\frac{13\!\cdots\!23}{44\!\cdots\!72}a^{7}-\frac{11\!\cdots\!41}{49\!\cdots\!92}a^{6}+\frac{24\!\cdots\!47}{17\!\cdots\!88}a^{5}-\frac{14\!\cdots\!97}{56\!\cdots\!84}a^{4}+\frac{58\!\cdots\!30}{15\!\cdots\!81}a^{3}+\frac{10\!\cdots\!59}{61\!\cdots\!24}a^{2}-\frac{73\!\cdots\!23}{30\!\cdots\!62}a+\frac{30\!\cdots\!70}{15\!\cdots\!81}$, $\frac{76\!\cdots\!81}{19\!\cdots\!68}a^{31}-\frac{30\!\cdots\!65}{89\!\cdots\!44}a^{30}-\frac{10\!\cdots\!03}{12\!\cdots\!48}a^{29}+\frac{10\!\cdots\!13}{24\!\cdots\!96}a^{28}+\frac{11\!\cdots\!67}{98\!\cdots\!84}a^{27}+\frac{29\!\cdots\!91}{98\!\cdots\!84}a^{26}-\frac{11\!\cdots\!41}{98\!\cdots\!84}a^{25}-\frac{95\!\cdots\!99}{19\!\cdots\!68}a^{24}+\frac{80\!\cdots\!09}{98\!\cdots\!84}a^{23}+\frac{70\!\cdots\!13}{22\!\cdots\!36}a^{22}-\frac{90\!\cdots\!45}{19\!\cdots\!68}a^{21}-\frac{17\!\cdots\!95}{19\!\cdots\!68}a^{20}+\frac{39\!\cdots\!63}{19\!\cdots\!68}a^{19}-\frac{71\!\cdots\!43}{19\!\cdots\!68}a^{18}-\frac{11\!\cdots\!29}{19\!\cdots\!68}a^{17}+\frac{41\!\cdots\!21}{98\!\cdots\!84}a^{16}+\frac{17\!\cdots\!93}{19\!\cdots\!68}a^{15}-\frac{22\!\cdots\!87}{17\!\cdots\!88}a^{14}+\frac{22\!\cdots\!15}{18\!\cdots\!52}a^{13}+\frac{44\!\cdots\!55}{19\!\cdots\!68}a^{12}-\frac{27\!\cdots\!59}{19\!\cdots\!68}a^{11}+\frac{40\!\cdots\!73}{19\!\cdots\!68}a^{10}+\frac{73\!\cdots\!47}{30\!\cdots\!62}a^{9}+\frac{11\!\cdots\!65}{19\!\cdots\!68}a^{8}+\frac{22\!\cdots\!67}{17\!\cdots\!88}a^{7}+\frac{48\!\cdots\!07}{49\!\cdots\!92}a^{6}+\frac{73\!\cdots\!27}{69\!\cdots\!52}a^{5}+\frac{37\!\cdots\!51}{22\!\cdots\!36}a^{4}+\frac{40\!\cdots\!23}{30\!\cdots\!62}a^{3}+\frac{36\!\cdots\!54}{15\!\cdots\!81}a^{2}+\frac{80\!\cdots\!89}{15\!\cdots\!81}a+\frac{18\!\cdots\!11}{15\!\cdots\!81}$, $\frac{42\!\cdots\!05}{19\!\cdots\!68}a^{31}-\frac{43\!\cdots\!55}{89\!\cdots\!44}a^{30}-\frac{52\!\cdots\!05}{12\!\cdots\!48}a^{29}+\frac{15\!\cdots\!31}{24\!\cdots\!96}a^{28}+\frac{59\!\cdots\!03}{98\!\cdots\!84}a^{27}-\frac{67\!\cdots\!89}{98\!\cdots\!84}a^{26}-\frac{56\!\cdots\!33}{98\!\cdots\!84}a^{25}+\frac{10\!\cdots\!13}{19\!\cdots\!68}a^{24}+\frac{41\!\cdots\!07}{98\!\cdots\!84}a^{23}-\frac{91\!\cdots\!51}{22\!\cdots\!36}a^{22}-\frac{44\!\cdots\!81}{19\!\cdots\!68}a^{21}+\frac{53\!\cdots\!29}{19\!\cdots\!68}a^{20}+\frac{17\!\cdots\!47}{19\!\cdots\!68}a^{19}-\frac{29\!\cdots\!27}{19\!\cdots\!68}a^{18}-\frac{36\!\cdots\!89}{19\!\cdots\!68}a^{17}+\frac{54\!\cdots\!99}{98\!\cdots\!84}a^{16}-\frac{23\!\cdots\!75}{19\!\cdots\!68}a^{15}-\frac{14\!\cdots\!35}{17\!\cdots\!88}a^{14}+\frac{31\!\cdots\!83}{18\!\cdots\!52}a^{13}-\frac{12\!\cdots\!61}{19\!\cdots\!68}a^{12}-\frac{65\!\cdots\!83}{19\!\cdots\!68}a^{11}+\frac{24\!\cdots\!13}{19\!\cdots\!68}a^{10}-\frac{10\!\cdots\!95}{49\!\cdots\!92}a^{9}-\frac{66\!\cdots\!55}{19\!\cdots\!68}a^{8}+\frac{11\!\cdots\!35}{17\!\cdots\!88}a^{7}-\frac{34\!\cdots\!13}{12\!\cdots\!48}a^{6}+\frac{70\!\cdots\!37}{69\!\cdots\!52}a^{5}+\frac{48\!\cdots\!71}{22\!\cdots\!36}a^{4}-\frac{42\!\cdots\!91}{12\!\cdots\!48}a^{3}+\frac{90\!\cdots\!61}{61\!\cdots\!24}a^{2}+\frac{95\!\cdots\!25}{30\!\cdots\!62}a-\frac{15\!\cdots\!08}{15\!\cdots\!81}$
| 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: | \( 8904823132081.674 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 8904823132081.674 \cdot 256}{6\cdot\sqrt{38187227411894497653750080270198291885967163238525390625}}\cr\approx \mathstrut & 0.362771234422294 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*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 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*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 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*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 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*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 | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 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 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 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 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(769\)
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |