Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*x + 256)
gp: K = bnfinit(y^32 - 4*y^31 - 12*y^30 + 32*y^29 + 238*y^28 - 436*y^27 - 1752*y^26 + 2180*y^25 + 12832*y^24 - 19600*y^23 - 49714*y^22 + 123516*y^21 + 139502*y^20 - 681160*y^19 - 49208*y^18 + 2842960*y^17 - 4085902*y^16 - 1791724*y^15 + 15066016*y^14 - 16647948*y^13 + 54648*y^12 + 31538976*y^11 - 37592014*y^10 + 17627156*y^9 + 19990921*y^8 - 35629052*y^7 + 31635244*y^6 - 15719312*y^5 + 5344480*y^4 - 815552*y^3 + 89408*y^2 - 5888*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*x + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*x + 256)
\( x^{32} - 4 x^{31} - 12 x^{30} + 32 x^{29} + 238 x^{28} - 436 x^{27} - 1752 x^{26} + 2180 x^{25} + \cdots + 256 \)
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: |
\(524243597623110466962067813334456972481513316626261344256\)
\(\medspace = 2^{48}\cdot 3^{16}\cdot 7^{16}\cdot 41^{8}\cdot 113^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.22\) | 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: | $2^{3/2}3^{1/2}7^{1/2}41^{1/2}113^{1/2}\approx 882.2380631099522$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(41\), \(113\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{18}-\frac{1}{8}a^{16}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{20}-\frac{1}{8}a^{18}-\frac{1}{8}a^{17}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{15}-\frac{1}{8}a^{9}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{22}-\frac{1}{8}a^{17}-\frac{1}{16}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{16}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}+\frac{1}{16}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{23}-\frac{1}{8}a^{18}-\frac{1}{16}a^{17}-\frac{1}{4}a^{12}-\frac{1}{16}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{8}a^{6}+\frac{1}{16}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{24}+\frac{1}{16}a^{18}-\frac{1}{8}a^{16}-\frac{1}{8}a^{15}-\frac{1}{16}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{25}-\frac{1}{32}a^{24}-\frac{1}{32}a^{23}-\frac{1}{16}a^{20}+\frac{1}{32}a^{19}-\frac{1}{32}a^{18}+\frac{1}{32}a^{17}-\frac{1}{8}a^{16}-\frac{5}{32}a^{13}-\frac{7}{32}a^{12}+\frac{5}{32}a^{11}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}-\frac{3}{16}a^{8}+\frac{3}{32}a^{7}-\frac{15}{32}a^{6}-\frac{5}{32}a^{5}+\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{26}-\frac{1}{32}a^{23}-\frac{1}{16}a^{21}-\frac{1}{32}a^{20}+\frac{1}{16}a^{18}-\frac{3}{32}a^{17}-\frac{1}{8}a^{15}+\frac{3}{32}a^{14}+\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{3}{32}a^{11}+\frac{1}{8}a^{10}+\frac{3}{16}a^{9}+\frac{5}{32}a^{8}+\frac{1}{8}a^{7}+\frac{5}{16}a^{6}+\frac{15}{32}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{27}-\frac{1}{32}a^{24}-\frac{1}{32}a^{21}-\frac{1}{16}a^{19}+\frac{1}{32}a^{18}-\frac{1}{8}a^{17}-\frac{1}{16}a^{16}-\frac{1}{32}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{3}{32}a^{12}+\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{3}{32}a^{9}-\frac{1}{8}a^{8}-\frac{1}{16}a^{7}-\frac{5}{32}a^{6}+\frac{1}{4}a^{5}+\frac{7}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{28}-\frac{1}{64}a^{26}+\frac{1}{64}a^{24}-\frac{1}{64}a^{22}-\frac{1}{32}a^{21}-\frac{3}{64}a^{20}-\frac{1}{32}a^{19}-\frac{1}{64}a^{18}-\frac{3}{32}a^{17}-\frac{5}{64}a^{16}+\frac{1}{16}a^{15}-\frac{7}{64}a^{14}-\frac{1}{16}a^{13}-\frac{1}{64}a^{12}+\frac{3}{16}a^{11}-\frac{15}{64}a^{10}-\frac{5}{32}a^{9}+\frac{3}{64}a^{8}+\frac{3}{32}a^{7}+\frac{9}{64}a^{6}-\frac{3}{32}a^{5}-\frac{1}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{640}a^{29}-\frac{3}{640}a^{28}+\frac{3}{640}a^{27}+\frac{7}{640}a^{26}-\frac{9}{640}a^{25}-\frac{1}{640}a^{24}-\frac{3}{640}a^{23}+\frac{1}{128}a^{22}-\frac{33}{640}a^{21}-\frac{1}{640}a^{20}+\frac{11}{640}a^{19}+\frac{31}{640}a^{18}-\frac{13}{128}a^{17}+\frac{79}{640}a^{16}-\frac{23}{640}a^{15}+\frac{29}{640}a^{14}+\frac{77}{640}a^{13}-\frac{83}{640}a^{12}+\frac{103}{640}a^{11}-\frac{97}{640}a^{10}+\frac{77}{640}a^{9}+\frac{17}{128}a^{8}-\frac{27}{128}a^{7}-\frac{307}{640}a^{6}-\frac{39}{160}a^{5}-\frac{3}{80}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{10}a+\frac{1}{10}$, $\frac{1}{13\!\cdots\!60}a^{30}-\frac{13\!\cdots\!77}{43\!\cdots\!80}a^{29}-\frac{27\!\cdots\!57}{43\!\cdots\!80}a^{28}-\frac{11\!\cdots\!67}{20\!\cdots\!20}a^{27}+\frac{55\!\cdots\!57}{69\!\cdots\!80}a^{26}+\frac{21\!\cdots\!17}{34\!\cdots\!40}a^{25}+\frac{11\!\cdots\!67}{34\!\cdots\!40}a^{24}-\frac{45\!\cdots\!43}{25\!\cdots\!40}a^{23}+\frac{69\!\cdots\!59}{17\!\cdots\!20}a^{22}+\frac{75\!\cdots\!73}{34\!\cdots\!40}a^{21}+\frac{11\!\cdots\!13}{40\!\cdots\!40}a^{20}+\frac{20\!\cdots\!71}{69\!\cdots\!28}a^{19}-\frac{57\!\cdots\!63}{69\!\cdots\!80}a^{18}-\frac{35\!\cdots\!99}{34\!\cdots\!40}a^{17}+\frac{28\!\cdots\!71}{37\!\cdots\!20}a^{16}+\frac{27\!\cdots\!83}{34\!\cdots\!40}a^{15}-\frac{72\!\cdots\!41}{69\!\cdots\!80}a^{14}-\frac{61\!\cdots\!17}{69\!\cdots\!28}a^{13}+\frac{33\!\cdots\!39}{34\!\cdots\!40}a^{12}-\frac{13\!\cdots\!49}{34\!\cdots\!64}a^{11}+\frac{11\!\cdots\!41}{21\!\cdots\!90}a^{10}+\frac{79\!\cdots\!87}{34\!\cdots\!40}a^{9}-\frac{31\!\cdots\!57}{13\!\cdots\!56}a^{8}-\frac{37\!\cdots\!01}{15\!\cdots\!80}a^{7}+\frac{58\!\cdots\!41}{13\!\cdots\!60}a^{6}-\frac{32\!\cdots\!87}{34\!\cdots\!40}a^{5}+\frac{16\!\cdots\!23}{17\!\cdots\!20}a^{4}-\frac{18\!\cdots\!67}{17\!\cdots\!32}a^{3}-\frac{39\!\cdots\!37}{94\!\cdots\!30}a^{2}-\frac{37\!\cdots\!61}{94\!\cdots\!30}a+\frac{55\!\cdots\!09}{21\!\cdots\!90}$, $\frac{1}{12\!\cdots\!20}a^{31}+\frac{19\!\cdots\!67}{64\!\cdots\!60}a^{30}+\frac{44\!\cdots\!29}{80\!\cdots\!20}a^{29}+\frac{22\!\cdots\!73}{32\!\cdots\!80}a^{28}+\frac{60\!\cdots\!83}{64\!\cdots\!60}a^{27}-\frac{33\!\cdots\!65}{64\!\cdots\!76}a^{26}+\frac{18\!\cdots\!99}{16\!\cdots\!40}a^{25}-\frac{17\!\cdots\!01}{16\!\cdots\!40}a^{24}+\frac{12\!\cdots\!23}{40\!\cdots\!60}a^{23}-\frac{81\!\cdots\!93}{32\!\cdots\!80}a^{22}-\frac{15\!\cdots\!01}{64\!\cdots\!60}a^{21}-\frac{66\!\cdots\!37}{14\!\cdots\!60}a^{20}-\frac{31\!\cdots\!53}{64\!\cdots\!60}a^{19}+\frac{96\!\cdots\!39}{20\!\cdots\!80}a^{18}-\frac{47\!\cdots\!51}{40\!\cdots\!36}a^{17}+\frac{19\!\cdots\!99}{32\!\cdots\!80}a^{16}+\frac{62\!\cdots\!97}{64\!\cdots\!60}a^{15}-\frac{60\!\cdots\!59}{32\!\cdots\!80}a^{14}-\frac{49\!\cdots\!77}{40\!\cdots\!60}a^{13}+\frac{18\!\cdots\!53}{80\!\cdots\!20}a^{12}+\frac{29\!\cdots\!23}{16\!\cdots\!40}a^{11}+\frac{25\!\cdots\!79}{28\!\cdots\!12}a^{10}+\frac{36\!\cdots\!57}{64\!\cdots\!60}a^{9}-\frac{48\!\cdots\!73}{32\!\cdots\!80}a^{8}-\frac{45\!\cdots\!35}{25\!\cdots\!04}a^{7}+\frac{54\!\cdots\!49}{12\!\cdots\!52}a^{6}-\frac{57\!\cdots\!77}{32\!\cdots\!88}a^{5}+\frac{23\!\cdots\!19}{10\!\cdots\!90}a^{4}-\frac{76\!\cdots\!71}{20\!\cdots\!80}a^{3}-\frac{35\!\cdots\!07}{87\!\cdots\!60}a^{2}+\frac{62\!\cdots\!83}{40\!\cdots\!36}a-\frac{33\!\cdots\!79}{10\!\cdots\!90}$
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
$C_{2}\times C_{6}\times C_{108}$, which has order $1296$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $1296$ (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{39459987521698632841466323333184658871777908855371583934791}{115903461036401810928802386637217363344143272548379385589479040} a^{31} + \frac{15513495603973984873167421665857584039794747711862531071413}{11590346103640181092880238663721736334414327254837938558947904} a^{30} + \frac{483932686171702021156932540348594511875487378788989182164617}{115903461036401810928802386637217363344143272548379385589479040} a^{29} - \frac{1228705172072574619410309678267249130758014630030975033355379}{115903461036401810928802386637217363344143272548379385589479040} a^{28} - \frac{9472871765699610647914745425497089264923594385704568069991999}{115903461036401810928802386637217363344143272548379385589479040} a^{27} + \frac{3309738867044594790903570369317009292643484439914688413960263}{23180692207280362185760477327443472668828654509675877117895808} a^{26} + \frac{70214196943767472888157560582556545396804766407554139156748447}{115903461036401810928802386637217363344143272548379385589479040} a^{25} - \frac{81124274027970111110545149162261016194229893899288503638777277}{115903461036401810928802386637217363344143272548379385589479040} a^{24} - \frac{102301130410183469901664772124932461868921353519288916909927211}{23180692207280362185760477327443472668828654509675877117895808} a^{23} + \frac{147591019784582005442853605020284590818118566974682196982515649}{23180692207280362185760477327443472668828654509675877117895808} a^{22} + \frac{87364014943509442517944865125387597068531069478205545244717203}{5039280914626165692556625505965972319310577067320842851716480} a^{21} - \frac{946495057539009239743640810392289151986250614740775107991723289}{23180692207280362185760477327443472668828654509675877117895808} a^{20} - \frac{5817358183955966036324323004450384956735134047935838426768770967}{115903461036401810928802386637217363344143272548379385589479040} a^{19} + \frac{26454987229790633206532203406909439111748103453666545886414385027}{115903461036401810928802386637217363344143272548379385589479040} a^{18} + \frac{3719028091587074282714212485338341070894359441064948586275668171}{115903461036401810928802386637217363344143272548379385589479040} a^{17} - \frac{111785437064347871726877486513256468463298455202261632681173307361}{115903461036401810928802386637217363344143272548379385589479040} a^{16} + \frac{153608281348188150726314981609365634310592345532627352102105237403}{115903461036401810928802386637217363344143272548379385589479040} a^{15} + \frac{80612514941140175422109973976624849569898303028286360056842689041}{115903461036401810928802386637217363344143272548379385589479040} a^{14} - \frac{34600463354253990320031661681228727569961042401814980341046015891}{6817850649200106525223669802189256667302545444022316799381120} a^{13} + \frac{123431270881938354308381526994783713587629227092724213231804932685}{23180692207280362185760477327443472668828654509675877117895808} a^{12} + \frac{36959259259286210541651474822853272638715297507232298461142733587}{115903461036401810928802386637217363344143272548379385589479040} a^{11} - \frac{1238986033511898116737447230679560594422735747116885651149755275793}{115903461036401810928802386637217363344143272548379385589479040} a^{10} + \frac{1398986032039558894346289187777450754189835557005592966936067120327}{115903461036401810928802386637217363344143272548379385589479040} a^{9} - \frac{606410483461701213877456462996270890415500522693389005166402664943}{115903461036401810928802386637217363344143272548379385589479040} a^{8} - \frac{411768232483880793018590364900577551820566196453520257314014115259}{57951730518200905464401193318608681672071636274189692794739520} a^{7} + \frac{269359752296727155908272942912727812482729942437990792679637343811}{23180692207280362185760477327443472668828654509675877117895808} a^{6} - \frac{145090460152251241336397770344002692275709978425633697970184512833}{14487932629550226366100298329652170418017909068547423198684880} a^{5} + \frac{68431642116517052360064839836402682516042968818705767665130745017}{14487932629550226366100298329652170418017909068547423198684880} a^{4} - \frac{11193834138684197922724627275771201931704795754818869124313137641}{7243966314775113183050149164826085209008954534273711599342440} a^{3} + \frac{30439282668473099043992171347560518306675188759516371881121223}{157477528582067677892394547061436634978455533353776339116140} a^{2} - \frac{22977349369659350440206991126291976551492195095188619527266197}{905495789346889147881268645603260651126119316784213949917805} a + \frac{3011539238348273205376902118830814329187433326252113130034741}{1810991578693778295762537291206521302252238633568427899835610} \)
(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: |
$\frac{24\!\cdots\!93}{56\!\cdots\!40}a^{31}-\frac{11\!\cdots\!19}{64\!\cdots\!60}a^{30}-\frac{44\!\cdots\!41}{80\!\cdots\!20}a^{29}+\frac{22\!\cdots\!31}{16\!\cdots\!40}a^{28}+\frac{68\!\cdots\!97}{64\!\cdots\!60}a^{27}-\frac{14\!\cdots\!37}{80\!\cdots\!20}a^{26}-\frac{12\!\cdots\!59}{16\!\cdots\!40}a^{25}+\frac{29\!\cdots\!01}{32\!\cdots\!80}a^{24}+\frac{93\!\cdots\!17}{16\!\cdots\!40}a^{23}-\frac{13\!\cdots\!67}{16\!\cdots\!40}a^{22}-\frac{14\!\cdots\!51}{64\!\cdots\!60}a^{21}+\frac{42\!\cdots\!89}{80\!\cdots\!20}a^{20}+\frac{42\!\cdots\!29}{64\!\cdots\!60}a^{19}-\frac{19\!\cdots\!39}{64\!\cdots\!76}a^{18}-\frac{37\!\cdots\!11}{80\!\cdots\!20}a^{17}+\frac{17\!\cdots\!65}{14\!\cdots\!56}a^{16}-\frac{11\!\cdots\!49}{64\!\cdots\!60}a^{15}-\frac{75\!\cdots\!73}{80\!\cdots\!20}a^{14}+\frac{53\!\cdots\!63}{80\!\cdots\!20}a^{13}-\frac{22\!\cdots\!27}{32\!\cdots\!80}a^{12}-\frac{43\!\cdots\!13}{80\!\cdots\!20}a^{11}+\frac{44\!\cdots\!15}{32\!\cdots\!88}a^{10}-\frac{99\!\cdots\!81}{64\!\cdots\!60}a^{9}+\frac{26\!\cdots\!71}{40\!\cdots\!60}a^{8}+\frac{52\!\cdots\!41}{56\!\cdots\!40}a^{7}-\frac{96\!\cdots\!19}{64\!\cdots\!60}a^{6}+\frac{41\!\cdots\!49}{32\!\cdots\!88}a^{5}-\frac{23\!\cdots\!03}{40\!\cdots\!60}a^{4}+\frac{19\!\cdots\!93}{10\!\cdots\!90}a^{3}-\frac{42\!\cdots\!23}{17\!\cdots\!32}a^{2}+\frac{54\!\cdots\!97}{17\!\cdots\!32}a-\frac{20\!\cdots\!33}{10\!\cdots\!90}$, $\frac{71\!\cdots\!03}{64\!\cdots\!60}a^{31}-\frac{14\!\cdots\!29}{32\!\cdots\!80}a^{30}-\frac{86\!\cdots\!13}{64\!\cdots\!60}a^{29}+\frac{22\!\cdots\!11}{64\!\cdots\!60}a^{28}+\frac{17\!\cdots\!23}{64\!\cdots\!60}a^{27}-\frac{61\!\cdots\!07}{12\!\cdots\!52}a^{26}-\frac{12\!\cdots\!19}{64\!\cdots\!60}a^{25}+\frac{15\!\cdots\!61}{64\!\cdots\!60}a^{24}+\frac{91\!\cdots\!03}{64\!\cdots\!60}a^{23}-\frac{81\!\cdots\!13}{37\!\cdots\!80}a^{22}-\frac{35\!\cdots\!21}{64\!\cdots\!60}a^{21}+\frac{88\!\cdots\!53}{64\!\cdots\!60}a^{20}+\frac{10\!\cdots\!47}{64\!\cdots\!60}a^{19}-\frac{48\!\cdots\!47}{64\!\cdots\!60}a^{18}-\frac{35\!\cdots\!33}{56\!\cdots\!24}a^{17}+\frac{20\!\cdots\!13}{64\!\cdots\!60}a^{16}-\frac{28\!\cdots\!83}{64\!\cdots\!60}a^{15}-\frac{13\!\cdots\!53}{64\!\cdots\!60}a^{14}+\frac{10\!\cdots\!23}{64\!\cdots\!60}a^{13}-\frac{11\!\cdots\!41}{64\!\cdots\!60}a^{12}-\frac{21\!\cdots\!23}{64\!\cdots\!60}a^{11}+\frac{45\!\cdots\!29}{12\!\cdots\!52}a^{10}-\frac{26\!\cdots\!83}{64\!\cdots\!60}a^{9}+\frac{52\!\cdots\!13}{28\!\cdots\!20}a^{8}+\frac{14\!\cdots\!47}{64\!\cdots\!76}a^{7}-\frac{50\!\cdots\!83}{12\!\cdots\!52}a^{6}+\frac{55\!\cdots\!45}{16\!\cdots\!44}a^{5}-\frac{78\!\cdots\!69}{47\!\cdots\!60}a^{4}+\frac{93\!\cdots\!91}{17\!\cdots\!20}a^{3}-\frac{58\!\cdots\!47}{87\!\cdots\!60}a^{2}+\frac{45\!\cdots\!45}{10\!\cdots\!09}a-\frac{56\!\cdots\!23}{43\!\cdots\!30}$, $\frac{38\!\cdots\!71}{46\!\cdots\!80}a^{31}-\frac{75\!\cdots\!17}{23\!\cdots\!40}a^{30}-\frac{23\!\cdots\!99}{23\!\cdots\!40}a^{29}+\frac{35\!\cdots\!19}{13\!\cdots\!20}a^{28}+\frac{11\!\cdots\!39}{58\!\cdots\!60}a^{27}-\frac{16\!\cdots\!63}{46\!\cdots\!08}a^{26}-\frac{34\!\cdots\!53}{23\!\cdots\!40}a^{25}+\frac{23\!\cdots\!43}{13\!\cdots\!20}a^{24}+\frac{24\!\cdots\!97}{23\!\cdots\!40}a^{23}-\frac{35\!\cdots\!89}{23\!\cdots\!40}a^{22}-\frac{62\!\cdots\!07}{14\!\cdots\!60}a^{21}+\frac{22\!\cdots\!17}{23\!\cdots\!40}a^{20}+\frac{35\!\cdots\!79}{29\!\cdots\!88}a^{19}-\frac{25\!\cdots\!61}{46\!\cdots\!08}a^{18}-\frac{19\!\cdots\!61}{23\!\cdots\!40}a^{17}+\frac{54\!\cdots\!73}{23\!\cdots\!40}a^{16}-\frac{18\!\cdots\!97}{58\!\cdots\!76}a^{15}-\frac{39\!\cdots\!33}{23\!\cdots\!40}a^{14}+\frac{28\!\cdots\!89}{23\!\cdots\!40}a^{13}-\frac{29\!\cdots\!59}{23\!\cdots\!40}a^{12}-\frac{19\!\cdots\!49}{23\!\cdots\!40}a^{11}+\frac{60\!\cdots\!93}{23\!\cdots\!40}a^{10}-\frac{84\!\cdots\!83}{29\!\cdots\!80}a^{9}+\frac{29\!\cdots\!59}{23\!\cdots\!40}a^{8}+\frac{80\!\cdots\!81}{46\!\cdots\!80}a^{7}-\frac{65\!\cdots\!43}{23\!\cdots\!04}a^{6}+\frac{17\!\cdots\!17}{72\!\cdots\!20}a^{5}-\frac{16\!\cdots\!97}{14\!\cdots\!40}a^{4}+\frac{27\!\cdots\!89}{72\!\cdots\!20}a^{3}-\frac{36\!\cdots\!69}{78\!\cdots\!85}a^{2}+\frac{44\!\cdots\!83}{72\!\cdots\!20}a-\frac{17\!\cdots\!43}{10\!\cdots\!15}$, $\frac{10\!\cdots\!97}{12\!\cdots\!20}a^{31}-\frac{19\!\cdots\!89}{64\!\cdots\!60}a^{30}-\frac{36\!\cdots\!53}{37\!\cdots\!80}a^{29}+\frac{31\!\cdots\!95}{12\!\cdots\!52}a^{28}+\frac{30\!\cdots\!07}{16\!\cdots\!40}a^{27}-\frac{21\!\cdots\!31}{64\!\cdots\!60}a^{26}-\frac{53\!\cdots\!87}{37\!\cdots\!80}a^{25}+\frac{20\!\cdots\!43}{12\!\cdots\!52}a^{24}+\frac{65\!\cdots\!63}{64\!\cdots\!60}a^{23}-\frac{94\!\cdots\!73}{64\!\cdots\!60}a^{22}-\frac{80\!\cdots\!91}{20\!\cdots\!80}a^{21}+\frac{60\!\cdots\!17}{64\!\cdots\!60}a^{20}+\frac{18\!\cdots\!13}{16\!\cdots\!40}a^{19}-\frac{33\!\cdots\!53}{64\!\cdots\!60}a^{18}-\frac{22\!\cdots\!69}{28\!\cdots\!20}a^{17}+\frac{14\!\cdots\!09}{64\!\cdots\!60}a^{16}-\frac{24\!\cdots\!57}{80\!\cdots\!20}a^{15}-\frac{10\!\cdots\!13}{64\!\cdots\!60}a^{14}+\frac{75\!\cdots\!47}{64\!\cdots\!60}a^{13}-\frac{78\!\cdots\!79}{64\!\cdots\!60}a^{12}-\frac{55\!\cdots\!07}{64\!\cdots\!60}a^{11}+\frac{15\!\cdots\!17}{64\!\cdots\!60}a^{10}-\frac{44\!\cdots\!11}{16\!\cdots\!40}a^{9}+\frac{32\!\cdots\!61}{28\!\cdots\!20}a^{8}+\frac{21\!\cdots\!27}{12\!\cdots\!20}a^{7}-\frac{85\!\cdots\!57}{32\!\cdots\!80}a^{6}+\frac{36\!\cdots\!99}{16\!\cdots\!40}a^{5}-\frac{85\!\cdots\!79}{80\!\cdots\!20}a^{4}+\frac{60\!\cdots\!87}{17\!\cdots\!20}a^{3}-\frac{18\!\cdots\!51}{43\!\cdots\!30}a^{2}+\frac{11\!\cdots\!27}{20\!\cdots\!80}a-\frac{81\!\cdots\!47}{21\!\cdots\!15}$, $\frac{18\!\cdots\!45}{25\!\cdots\!04}a^{31}-\frac{36\!\cdots\!83}{12\!\cdots\!52}a^{30}-\frac{53\!\cdots\!33}{64\!\cdots\!60}a^{29}+\frac{14\!\cdots\!89}{64\!\cdots\!60}a^{28}+\frac{26\!\cdots\!19}{16\!\cdots\!40}a^{27}-\frac{20\!\cdots\!41}{64\!\cdots\!60}a^{26}-\frac{78\!\cdots\!23}{64\!\cdots\!60}a^{25}+\frac{10\!\cdots\!53}{64\!\cdots\!60}a^{24}+\frac{57\!\cdots\!99}{64\!\cdots\!60}a^{23}-\frac{10\!\cdots\!83}{75\!\cdots\!56}a^{22}-\frac{55\!\cdots\!39}{16\!\cdots\!40}a^{21}+\frac{57\!\cdots\!83}{64\!\cdots\!60}a^{20}+\frac{75\!\cdots\!09}{80\!\cdots\!20}a^{19}-\frac{31\!\cdots\!63}{64\!\cdots\!60}a^{18}-\frac{10\!\cdots\!83}{12\!\cdots\!52}a^{17}+\frac{12\!\cdots\!23}{64\!\cdots\!60}a^{16}-\frac{60\!\cdots\!13}{20\!\cdots\!80}a^{15}-\frac{71\!\cdots\!07}{64\!\cdots\!60}a^{14}+\frac{68\!\cdots\!39}{64\!\cdots\!60}a^{13}-\frac{79\!\cdots\!61}{64\!\cdots\!60}a^{12}+\frac{42\!\cdots\!41}{64\!\cdots\!60}a^{11}+\frac{14\!\cdots\!71}{64\!\cdots\!60}a^{10}-\frac{97\!\cdots\!19}{35\!\cdots\!40}a^{9}+\frac{17\!\cdots\!05}{12\!\cdots\!52}a^{8}+\frac{34\!\cdots\!55}{25\!\cdots\!04}a^{7}-\frac{83\!\cdots\!07}{32\!\cdots\!80}a^{6}+\frac{19\!\cdots\!91}{80\!\cdots\!20}a^{5}-\frac{28\!\cdots\!99}{23\!\cdots\!80}a^{4}+\frac{34\!\cdots\!27}{80\!\cdots\!72}a^{3}-\frac{12\!\cdots\!39}{17\!\cdots\!32}a^{2}+\frac{14\!\cdots\!51}{20\!\cdots\!80}a-\frac{92\!\cdots\!64}{50\!\cdots\!45}$, $\frac{96\!\cdots\!73}{64\!\cdots\!60}a^{31}-\frac{19\!\cdots\!93}{32\!\cdots\!80}a^{30}-\frac{51\!\cdots\!59}{28\!\cdots\!20}a^{29}+\frac{30\!\cdots\!11}{64\!\cdots\!60}a^{28}+\frac{23\!\cdots\!03}{64\!\cdots\!60}a^{27}-\frac{40\!\cdots\!19}{64\!\cdots\!60}a^{26}-\frac{17\!\cdots\!99}{64\!\cdots\!60}a^{25}+\frac{19\!\cdots\!93}{64\!\cdots\!60}a^{24}+\frac{12\!\cdots\!27}{64\!\cdots\!60}a^{23}-\frac{18\!\cdots\!37}{64\!\cdots\!60}a^{22}-\frac{49\!\cdots\!29}{64\!\cdots\!60}a^{21}+\frac{11\!\cdots\!33}{64\!\cdots\!60}a^{20}+\frac{14\!\cdots\!91}{64\!\cdots\!60}a^{19}-\frac{13\!\cdots\!03}{12\!\cdots\!52}a^{18}-\frac{97\!\cdots\!27}{64\!\cdots\!60}a^{17}+\frac{55\!\cdots\!05}{12\!\cdots\!52}a^{16}-\frac{22\!\cdots\!63}{37\!\cdots\!80}a^{15}-\frac{20\!\cdots\!41}{64\!\cdots\!60}a^{14}+\frac{14\!\cdots\!71}{64\!\cdots\!60}a^{13}-\frac{15\!\cdots\!21}{64\!\cdots\!60}a^{12}-\frac{11\!\cdots\!31}{64\!\cdots\!60}a^{11}+\frac{61\!\cdots\!97}{12\!\cdots\!52}a^{10}-\frac{34\!\cdots\!39}{64\!\cdots\!60}a^{9}+\frac{14\!\cdots\!99}{64\!\cdots\!60}a^{8}+\frac{51\!\cdots\!09}{16\!\cdots\!40}a^{7}-\frac{33\!\cdots\!71}{64\!\cdots\!60}a^{6}+\frac{70\!\cdots\!67}{16\!\cdots\!44}a^{5}-\frac{80\!\cdots\!07}{40\!\cdots\!60}a^{4}+\frac{24\!\cdots\!53}{40\!\cdots\!60}a^{3}-\frac{29\!\cdots\!29}{51\!\cdots\!98}a^{2}+\frac{28\!\cdots\!39}{10\!\cdots\!09}a+\frac{12\!\cdots\!13}{10\!\cdots\!90}$, $\frac{77\!\cdots\!11}{15\!\cdots\!20}a^{31}-\frac{31\!\cdots\!63}{15\!\cdots\!20}a^{30}-\frac{90\!\cdots\!77}{15\!\cdots\!20}a^{29}+\frac{25\!\cdots\!07}{15\!\cdots\!20}a^{28}+\frac{18\!\cdots\!51}{15\!\cdots\!20}a^{27}-\frac{35\!\cdots\!21}{15\!\cdots\!20}a^{26}-\frac{13\!\cdots\!03}{15\!\cdots\!20}a^{25}+\frac{36\!\cdots\!05}{31\!\cdots\!44}a^{24}+\frac{98\!\cdots\!07}{15\!\cdots\!20}a^{23}-\frac{16\!\cdots\!01}{15\!\cdots\!20}a^{22}-\frac{37\!\cdots\!89}{15\!\cdots\!20}a^{21}+\frac{99\!\cdots\!11}{15\!\cdots\!20}a^{20}+\frac{20\!\cdots\!95}{31\!\cdots\!44}a^{19}-\frac{31\!\cdots\!53}{92\!\cdots\!60}a^{18}+\frac{72\!\cdots\!97}{15\!\cdots\!20}a^{17}+\frac{22\!\cdots\!89}{15\!\cdots\!20}a^{16}-\frac{33\!\cdots\!23}{15\!\cdots\!20}a^{15}-\frac{11\!\cdots\!03}{15\!\cdots\!20}a^{14}+\frac{11\!\cdots\!83}{15\!\cdots\!20}a^{13}-\frac{13\!\cdots\!17}{15\!\cdots\!20}a^{12}+\frac{96\!\cdots\!17}{15\!\cdots\!20}a^{11}+\frac{49\!\cdots\!77}{31\!\cdots\!44}a^{10}-\frac{62\!\cdots\!79}{31\!\cdots\!44}a^{9}+\frac{15\!\cdots\!73}{15\!\cdots\!20}a^{8}+\frac{75\!\cdots\!87}{78\!\cdots\!60}a^{7}-\frac{73\!\cdots\!71}{39\!\cdots\!80}a^{6}+\frac{66\!\cdots\!97}{39\!\cdots\!68}a^{5}-\frac{34\!\cdots\!23}{39\!\cdots\!68}a^{4}+\frac{14\!\cdots\!23}{49\!\cdots\!60}a^{3}-\frac{95\!\cdots\!51}{21\!\cdots\!20}a^{2}+\frac{16\!\cdots\!23}{49\!\cdots\!46}a-\frac{16\!\cdots\!70}{24\!\cdots\!23}$, $\frac{95\!\cdots\!73}{34\!\cdots\!32}a^{31}-\frac{17\!\cdots\!55}{15\!\cdots\!72}a^{30}-\frac{51\!\cdots\!79}{15\!\cdots\!20}a^{29}+\frac{14\!\cdots\!67}{15\!\cdots\!20}a^{28}+\frac{10\!\cdots\!63}{15\!\cdots\!20}a^{27}-\frac{19\!\cdots\!43}{15\!\cdots\!20}a^{26}-\frac{76\!\cdots\!09}{15\!\cdots\!20}a^{25}+\frac{99\!\cdots\!89}{15\!\cdots\!20}a^{24}+\frac{55\!\cdots\!57}{15\!\cdots\!20}a^{23}-\frac{17\!\cdots\!57}{31\!\cdots\!44}a^{22}-\frac{21\!\cdots\!53}{15\!\cdots\!20}a^{21}+\frac{55\!\cdots\!49}{15\!\cdots\!20}a^{20}+\frac{58\!\cdots\!11}{15\!\cdots\!20}a^{19}-\frac{30\!\cdots\!39}{15\!\cdots\!20}a^{18}-\frac{17\!\cdots\!17}{31\!\cdots\!44}a^{17}+\frac{54\!\cdots\!03}{68\!\cdots\!40}a^{16}-\frac{18\!\cdots\!83}{15\!\cdots\!20}a^{15}-\frac{71\!\cdots\!41}{15\!\cdots\!20}a^{14}+\frac{39\!\cdots\!21}{92\!\cdots\!60}a^{13}-\frac{75\!\cdots\!13}{15\!\cdots\!20}a^{12}+\frac{27\!\cdots\!03}{15\!\cdots\!20}a^{11}+\frac{13\!\cdots\!93}{15\!\cdots\!20}a^{10}-\frac{17\!\cdots\!23}{15\!\cdots\!20}a^{9}+\frac{16\!\cdots\!59}{31\!\cdots\!44}a^{8}+\frac{75\!\cdots\!99}{13\!\cdots\!28}a^{7}-\frac{16\!\cdots\!87}{15\!\cdots\!20}a^{6}+\frac{36\!\cdots\!61}{39\!\cdots\!80}a^{5}-\frac{22\!\cdots\!37}{49\!\cdots\!60}a^{4}+\frac{31\!\cdots\!37}{19\!\cdots\!84}a^{3}-\frac{51\!\cdots\!41}{21\!\cdots\!02}a^{2}+\frac{99\!\cdots\!34}{53\!\cdots\!05}a-\frac{14\!\cdots\!99}{24\!\cdots\!30}$, $\frac{85\!\cdots\!23}{25\!\cdots\!04}a^{31}-\frac{10\!\cdots\!99}{75\!\cdots\!56}a^{30}-\frac{25\!\cdots\!51}{64\!\cdots\!60}a^{29}+\frac{68\!\cdots\!73}{64\!\cdots\!60}a^{28}+\frac{64\!\cdots\!09}{80\!\cdots\!20}a^{27}-\frac{92\!\cdots\!57}{64\!\cdots\!60}a^{26}-\frac{37\!\cdots\!61}{64\!\cdots\!60}a^{25}+\frac{46\!\cdots\!61}{64\!\cdots\!60}a^{24}+\frac{27\!\cdots\!93}{64\!\cdots\!60}a^{23}-\frac{83\!\cdots\!75}{12\!\cdots\!52}a^{22}-\frac{26\!\cdots\!23}{16\!\cdots\!40}a^{21}+\frac{26\!\cdots\!51}{64\!\cdots\!60}a^{20}+\frac{76\!\cdots\!71}{16\!\cdots\!40}a^{19}-\frac{14\!\cdots\!91}{64\!\cdots\!60}a^{18}-\frac{15\!\cdots\!53}{75\!\cdots\!56}a^{17}+\frac{61\!\cdots\!31}{64\!\cdots\!60}a^{16}-\frac{21\!\cdots\!23}{16\!\cdots\!40}a^{15}-\frac{40\!\cdots\!79}{64\!\cdots\!60}a^{14}+\frac{32\!\cdots\!73}{64\!\cdots\!60}a^{13}-\frac{35\!\cdots\!77}{64\!\cdots\!60}a^{12}-\frac{57\!\cdots\!73}{64\!\cdots\!60}a^{11}+\frac{67\!\cdots\!47}{64\!\cdots\!60}a^{10}-\frac{99\!\cdots\!09}{80\!\cdots\!20}a^{9}+\frac{72\!\cdots\!97}{12\!\cdots\!52}a^{8}+\frac{17\!\cdots\!85}{25\!\cdots\!04}a^{7}-\frac{38\!\cdots\!09}{32\!\cdots\!80}a^{6}+\frac{10\!\cdots\!29}{10\!\cdots\!90}a^{5}-\frac{20\!\cdots\!21}{40\!\cdots\!60}a^{4}+\frac{65\!\cdots\!11}{40\!\cdots\!36}a^{3}-\frac{88\!\cdots\!34}{43\!\cdots\!83}a^{2}+\frac{94\!\cdots\!67}{20\!\cdots\!80}a+\frac{64\!\cdots\!42}{50\!\cdots\!45}$, $\frac{12\!\cdots\!83}{64\!\cdots\!60}a^{31}-\frac{49\!\cdots\!11}{64\!\cdots\!60}a^{30}-\frac{71\!\cdots\!97}{32\!\cdots\!80}a^{29}+\frac{19\!\cdots\!11}{32\!\cdots\!80}a^{28}+\frac{71\!\cdots\!83}{16\!\cdots\!40}a^{27}-\frac{13\!\cdots\!47}{16\!\cdots\!40}a^{26}-\frac{10\!\cdots\!63}{32\!\cdots\!80}a^{25}+\frac{27\!\cdots\!65}{64\!\cdots\!76}a^{24}+\frac{77\!\cdots\!59}{32\!\cdots\!80}a^{23}-\frac{12\!\cdots\!81}{32\!\cdots\!80}a^{22}-\frac{14\!\cdots\!87}{16\!\cdots\!40}a^{21}+\frac{76\!\cdots\!31}{32\!\cdots\!88}a^{20}+\frac{40\!\cdots\!01}{16\!\cdots\!40}a^{19}-\frac{20\!\cdots\!53}{16\!\cdots\!40}a^{18}-\frac{92\!\cdots\!21}{32\!\cdots\!80}a^{17}+\frac{10\!\cdots\!93}{18\!\cdots\!40}a^{16}-\frac{25\!\cdots\!89}{32\!\cdots\!88}a^{15}-\frac{49\!\cdots\!91}{16\!\cdots\!40}a^{14}+\frac{92\!\cdots\!71}{32\!\cdots\!80}a^{13}-\frac{21\!\cdots\!29}{64\!\cdots\!76}a^{12}+\frac{40\!\cdots\!09}{32\!\cdots\!80}a^{11}+\frac{19\!\cdots\!97}{32\!\cdots\!80}a^{10}-\frac{11\!\cdots\!27}{16\!\cdots\!40}a^{9}+\frac{57\!\cdots\!51}{16\!\cdots\!40}a^{8}+\frac{23\!\cdots\!81}{64\!\cdots\!60}a^{7}-\frac{44\!\cdots\!97}{64\!\cdots\!60}a^{6}+\frac{43\!\cdots\!21}{70\!\cdots\!80}a^{5}-\frac{25\!\cdots\!83}{80\!\cdots\!20}a^{4}+\frac{42\!\cdots\!63}{40\!\cdots\!60}a^{3}-\frac{70\!\cdots\!95}{43\!\cdots\!83}a^{2}+\frac{12\!\cdots\!47}{10\!\cdots\!90}a-\frac{92\!\cdots\!89}{10\!\cdots\!90}$, $\frac{46\!\cdots\!43}{64\!\cdots\!60}a^{31}-\frac{18\!\cdots\!23}{64\!\cdots\!60}a^{30}-\frac{28\!\cdots\!93}{32\!\cdots\!80}a^{29}+\frac{14\!\cdots\!41}{64\!\cdots\!76}a^{28}+\frac{28\!\cdots\!51}{16\!\cdots\!40}a^{27}-\frac{24\!\cdots\!77}{80\!\cdots\!20}a^{26}-\frac{41\!\cdots\!71}{32\!\cdots\!80}a^{25}+\frac{46\!\cdots\!87}{32\!\cdots\!80}a^{24}+\frac{30\!\cdots\!91}{32\!\cdots\!80}a^{23}-\frac{18\!\cdots\!21}{14\!\cdots\!60}a^{22}-\frac{60\!\cdots\!93}{16\!\cdots\!40}a^{21}+\frac{68\!\cdots\!27}{80\!\cdots\!20}a^{20}+\frac{17\!\cdots\!91}{16\!\cdots\!40}a^{19}-\frac{77\!\cdots\!97}{16\!\cdots\!44}a^{18}-\frac{53\!\cdots\!97}{64\!\cdots\!76}a^{17}+\frac{65\!\cdots\!83}{32\!\cdots\!80}a^{16}-\frac{44\!\cdots\!81}{16\!\cdots\!40}a^{15}-\frac{91\!\cdots\!75}{59\!\cdots\!77}a^{14}+\frac{34\!\cdots\!91}{32\!\cdots\!80}a^{13}-\frac{35\!\cdots\!71}{32\!\cdots\!80}a^{12}-\frac{30\!\cdots\!11}{32\!\cdots\!80}a^{11}+\frac{72\!\cdots\!43}{32\!\cdots\!80}a^{10}-\frac{23\!\cdots\!73}{94\!\cdots\!20}a^{9}+\frac{42\!\cdots\!39}{40\!\cdots\!60}a^{8}+\frac{19\!\cdots\!53}{12\!\cdots\!52}a^{7}-\frac{15\!\cdots\!89}{64\!\cdots\!60}a^{6}+\frac{33\!\cdots\!27}{16\!\cdots\!40}a^{5}-\frac{77\!\cdots\!69}{80\!\cdots\!20}a^{4}+\frac{15\!\cdots\!06}{50\!\cdots\!45}a^{3}-\frac{79\!\cdots\!23}{21\!\cdots\!15}a^{2}+\frac{46\!\cdots\!73}{10\!\cdots\!90}a-\frac{20\!\cdots\!27}{10\!\cdots\!90}$, $\frac{11\!\cdots\!83}{64\!\cdots\!60}a^{31}-\frac{43\!\cdots\!33}{64\!\cdots\!60}a^{30}-\frac{14\!\cdots\!19}{64\!\cdots\!60}a^{29}+\frac{33\!\cdots\!49}{64\!\cdots\!60}a^{28}+\frac{54\!\cdots\!77}{12\!\cdots\!52}a^{27}-\frac{44\!\cdots\!07}{64\!\cdots\!60}a^{26}-\frac{40\!\cdots\!37}{12\!\cdots\!52}a^{25}+\frac{21\!\cdots\!87}{64\!\cdots\!60}a^{24}+\frac{14\!\cdots\!41}{64\!\cdots\!60}a^{23}-\frac{19\!\cdots\!71}{64\!\cdots\!60}a^{22}-\frac{59\!\cdots\!23}{64\!\cdots\!60}a^{21}+\frac{12\!\cdots\!89}{64\!\cdots\!60}a^{20}+\frac{17\!\cdots\!01}{64\!\cdots\!60}a^{19}-\frac{43\!\cdots\!59}{37\!\cdots\!80}a^{18}-\frac{36\!\cdots\!21}{12\!\cdots\!52}a^{17}+\frac{31\!\cdots\!59}{64\!\cdots\!60}a^{16}-\frac{81\!\cdots\!85}{12\!\cdots\!52}a^{15}-\frac{27\!\cdots\!97}{64\!\cdots\!60}a^{14}+\frac{16\!\cdots\!61}{64\!\cdots\!60}a^{13}-\frac{15\!\cdots\!63}{64\!\cdots\!60}a^{12}-\frac{13\!\cdots\!07}{28\!\cdots\!20}a^{11}+\frac{35\!\cdots\!47}{64\!\cdots\!60}a^{10}-\frac{72\!\cdots\!85}{12\!\cdots\!52}a^{9}+\frac{12\!\cdots\!59}{64\!\cdots\!60}a^{8}+\frac{12\!\cdots\!01}{32\!\cdots\!88}a^{7}-\frac{90\!\cdots\!27}{16\!\cdots\!40}a^{6}+\frac{72\!\cdots\!79}{16\!\cdots\!40}a^{5}-\frac{29\!\cdots\!77}{16\!\cdots\!44}a^{4}+\frac{19\!\cdots\!73}{40\!\cdots\!60}a^{3}+\frac{57\!\cdots\!23}{87\!\cdots\!60}a^{2}-\frac{25\!\cdots\!22}{50\!\cdots\!45}a+\frac{49\!\cdots\!69}{10\!\cdots\!09}$, $\frac{71\!\cdots\!63}{12\!\cdots\!20}a^{31}-\frac{72\!\cdots\!21}{32\!\cdots\!80}a^{30}-\frac{24\!\cdots\!51}{37\!\cdots\!28}a^{29}+\frac{72\!\cdots\!69}{40\!\cdots\!60}a^{28}+\frac{84\!\cdots\!23}{64\!\cdots\!60}a^{27}-\frac{79\!\cdots\!59}{32\!\cdots\!80}a^{26}-\frac{18\!\cdots\!91}{18\!\cdots\!40}a^{25}+\frac{40\!\cdots\!51}{32\!\cdots\!80}a^{24}+\frac{22\!\cdots\!47}{32\!\cdots\!80}a^{23}-\frac{18\!\cdots\!21}{16\!\cdots\!40}a^{22}-\frac{35\!\cdots\!65}{12\!\cdots\!52}a^{21}+\frac{22\!\cdots\!63}{32\!\cdots\!80}a^{20}+\frac{48\!\cdots\!83}{64\!\cdots\!60}a^{19}-\frac{15\!\cdots\!71}{40\!\cdots\!60}a^{18}-\frac{37\!\cdots\!41}{32\!\cdots\!80}a^{17}+\frac{63\!\cdots\!23}{40\!\cdots\!60}a^{16}-\frac{15\!\cdots\!91}{64\!\cdots\!60}a^{15}-\frac{59\!\cdots\!21}{64\!\cdots\!76}a^{14}+\frac{27\!\cdots\!09}{32\!\cdots\!80}a^{13}-\frac{30\!\cdots\!61}{32\!\cdots\!80}a^{12}+\frac{10\!\cdots\!11}{32\!\cdots\!80}a^{11}+\frac{28\!\cdots\!61}{16\!\cdots\!40}a^{10}-\frac{13\!\cdots\!79}{64\!\cdots\!60}a^{9}+\frac{33\!\cdots\!01}{32\!\cdots\!80}a^{8}+\frac{14\!\cdots\!47}{12\!\cdots\!20}a^{7}-\frac{65\!\cdots\!41}{32\!\cdots\!80}a^{6}+\frac{29\!\cdots\!33}{16\!\cdots\!40}a^{5}-\frac{74\!\cdots\!09}{80\!\cdots\!20}a^{4}+\frac{15\!\cdots\!98}{50\!\cdots\!45}a^{3}-\frac{41\!\cdots\!49}{87\!\cdots\!60}a^{2}+\frac{73\!\cdots\!59}{20\!\cdots\!80}a-\frac{13\!\cdots\!61}{50\!\cdots\!45}$, $\frac{18\!\cdots\!83}{64\!\cdots\!60}a^{31}-\frac{18\!\cdots\!41}{16\!\cdots\!40}a^{30}-\frac{12\!\cdots\!09}{37\!\cdots\!80}a^{29}+\frac{61\!\cdots\!47}{64\!\cdots\!60}a^{28}+\frac{42\!\cdots\!07}{64\!\cdots\!60}a^{27}-\frac{86\!\cdots\!67}{64\!\cdots\!60}a^{26}-\frac{17\!\cdots\!83}{37\!\cdots\!80}a^{25}+\frac{45\!\cdots\!77}{64\!\cdots\!60}a^{24}+\frac{22\!\cdots\!47}{64\!\cdots\!60}a^{23}-\frac{39\!\cdots\!93}{64\!\cdots\!60}a^{22}-\frac{84\!\cdots\!61}{64\!\cdots\!60}a^{21}+\frac{48\!\cdots\!93}{12\!\cdots\!52}a^{20}+\frac{21\!\cdots\!23}{64\!\cdots\!60}a^{19}-\frac{25\!\cdots\!39}{12\!\cdots\!52}a^{18}+\frac{11\!\cdots\!21}{64\!\cdots\!60}a^{17}+\frac{51\!\cdots\!29}{64\!\cdots\!60}a^{16}-\frac{83\!\cdots\!43}{64\!\cdots\!60}a^{15}-\frac{20\!\cdots\!37}{64\!\cdots\!60}a^{14}+\frac{56\!\cdots\!47}{12\!\cdots\!52}a^{13}-\frac{69\!\cdots\!33}{12\!\cdots\!52}a^{12}+\frac{41\!\cdots\!59}{56\!\cdots\!24}a^{11}+\frac{57\!\cdots\!09}{64\!\cdots\!60}a^{10}-\frac{78\!\cdots\!39}{64\!\cdots\!60}a^{9}+\frac{42\!\cdots\!99}{64\!\cdots\!60}a^{8}+\frac{15\!\cdots\!21}{32\!\cdots\!80}a^{7}-\frac{71\!\cdots\!33}{64\!\cdots\!60}a^{6}+\frac{84\!\cdots\!69}{80\!\cdots\!20}a^{5}-\frac{46\!\cdots\!89}{80\!\cdots\!20}a^{4}+\frac{10\!\cdots\!81}{50\!\cdots\!45}a^{3}-\frac{35\!\cdots\!43}{87\!\cdots\!60}a^{2}+\frac{36\!\cdots\!57}{10\!\cdots\!90}a-\frac{15\!\cdots\!07}{10\!\cdots\!90}$, $\frac{10\!\cdots\!97}{12\!\cdots\!20}a^{31}-\frac{24\!\cdots\!97}{64\!\cdots\!60}a^{30}-\frac{24\!\cdots\!37}{32\!\cdots\!80}a^{29}+\frac{26\!\cdots\!49}{80\!\cdots\!20}a^{28}+\frac{11\!\cdots\!81}{64\!\cdots\!60}a^{27}-\frac{78\!\cdots\!27}{16\!\cdots\!40}a^{26}-\frac{38\!\cdots\!49}{32\!\cdots\!80}a^{25}+\frac{88\!\cdots\!03}{32\!\cdots\!80}a^{24}+\frac{29\!\cdots\!17}{32\!\cdots\!80}a^{23}-\frac{37\!\cdots\!41}{16\!\cdots\!40}a^{22}-\frac{19\!\cdots\!83}{64\!\cdots\!60}a^{21}+\frac{10\!\cdots\!07}{80\!\cdots\!20}a^{20}+\frac{28\!\cdots\!37}{64\!\cdots\!60}a^{19}-\frac{40\!\cdots\!23}{64\!\cdots\!76}a^{18}+\frac{10\!\cdots\!83}{32\!\cdots\!80}a^{17}+\frac{37\!\cdots\!15}{16\!\cdots\!44}a^{16}-\frac{31\!\cdots\!97}{64\!\cdots\!60}a^{15}+\frac{12\!\cdots\!17}{16\!\cdots\!40}a^{14}+\frac{43\!\cdots\!51}{32\!\cdots\!80}a^{13}-\frac{71\!\cdots\!21}{32\!\cdots\!80}a^{12}+\frac{12\!\cdots\!23}{14\!\cdots\!60}a^{11}+\frac{86\!\cdots\!11}{32\!\cdots\!88}a^{10}-\frac{31\!\cdots\!33}{64\!\cdots\!60}a^{9}+\frac{28\!\cdots\!61}{80\!\cdots\!20}a^{8}+\frac{11\!\cdots\!69}{12\!\cdots\!20}a^{7}-\frac{26\!\cdots\!57}{64\!\cdots\!60}a^{6}+\frac{36\!\cdots\!65}{80\!\cdots\!72}a^{5}-\frac{11\!\cdots\!09}{40\!\cdots\!60}a^{4}+\frac{11\!\cdots\!79}{10\!\cdots\!90}a^{3}-\frac{22\!\cdots\!75}{87\!\cdots\!66}a^{2}+\frac{80\!\cdots\!45}{40\!\cdots\!36}a-\frac{13\!\cdots\!99}{10\!\cdots\!90}$
| 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: | \( 59356071958917.54 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 59356071958917.54 \cdot 1296}{6\cdot\sqrt{524243597623110466962067813334456972481513316626261344256}}\cr\approx \mathstrut & 3.30392537837143 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*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 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*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 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*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 - 4*x^31 - 12*x^30 + 32*x^29 + 238*x^28 - 436*x^27 - 1752*x^26 + 2180*x^25 + 12832*x^24 - 19600*x^23 - 49714*x^22 + 123516*x^21 + 139502*x^20 - 681160*x^19 - 49208*x^18 + 2842960*x^17 - 4085902*x^16 - 1791724*x^15 + 15066016*x^14 - 16647948*x^13 + 54648*x^12 + 31538976*x^11 - 37592014*x^10 + 17627156*x^9 + 19990921*x^8 - 35629052*x^7 + 31635244*x^6 - 15719312*x^5 + 5344480*x^4 - 815552*x^3 + 89408*x^2 - 5888*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 | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.8.0.1}{8} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(3\)
| 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 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(41\)
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(113\)
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.4.2.1 | $x^{4} + 18960 x^{3} + 90817911 x^{2} + 8982404280 x + 374946100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 113.4.2.1 | $x^{4} + 18960 x^{3} + 90817911 x^{2} + 8982404280 x + 374946100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |