Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496)
gp: K = bnfinit(y^32 - 2*y^31 - y^30 + 14*y^29 - 6*y^28 - 82*y^27 + 171*y^26 + 426*y^25 - 1103*y^24 + 612*y^23 + 1870*y^22 + 1692*y^21 - 15101*y^20 + 12498*y^19 + 60665*y^18 - 224090*y^17 + 470294*y^16 - 1196922*y^15 + 3481149*y^14 - 4522026*y^13 + 9950863*y^12 - 12441624*y^11 + 17394098*y^10 - 19804208*y^9 + 30506508*y^8 - 11192944*y^7 + 24538416*y^6 - 26063296*y^5 + 410816*y^4 - 22830080*y^3 + 11063808*y^2 + 9834496, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 2*x^31 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496)
\( x^{32} - 2 x^{31} - x^{30} + 14 x^{29} - 6 x^{28} - 82 x^{27} + 171 x^{26} + 426 x^{25} - 1103 x^{24} + \cdots + 9834496 \)
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: |
\(11108449995025704343652186204318263487217947546291798016\)
\(\medspace = 2^{48}\cdot 3^{16}\cdot 7^{16}\cdot 23^{8}\cdot 137^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.50\) | 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: | $2^{3/2}3^{1/2}7^{1/2}23^{1/2}137^{1/2}\approx 727.5768000699308$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(23\), \(137\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{6}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{7}$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{8}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{19}-\frac{1}{8}a^{17}-\frac{1}{4}a^{15}-\frac{1}{8}a^{9}-\frac{3}{8}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{22}-\frac{1}{8}a^{20}-\frac{1}{8}a^{18}-\frac{1}{8}a^{10}+\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{23}-\frac{1}{8}a^{17}-\frac{1}{4}a^{15}-\frac{1}{8}a^{11}-\frac{1}{2}a^{7}+\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{24}-\frac{1}{8}a^{20}-\frac{1}{8}a^{19}+\frac{1}{16}a^{18}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}-\frac{1}{4}a^{14}-\frac{1}{16}a^{12}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{3}{8}a^{5}-\frac{3}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{80}a^{25}+\frac{1}{40}a^{24}-\frac{1}{20}a^{23}+\frac{1}{40}a^{22}-\frac{1}{40}a^{21}-\frac{1}{20}a^{20}+\frac{1}{16}a^{19}-\frac{1}{40}a^{18}-\frac{3}{40}a^{17}+\frac{1}{20}a^{16}-\frac{1}{20}a^{15}+\frac{7}{80}a^{13}+\frac{3}{40}a^{12}-\frac{1}{4}a^{11}+\frac{1}{40}a^{10}-\frac{7}{40}a^{9}-\frac{3}{20}a^{8}-\frac{37}{80}a^{7}+\frac{17}{40}a^{6}-\frac{9}{40}a^{5}-\frac{1}{5}a^{4}-\frac{7}{20}a^{3}+\frac{3}{10}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{560}a^{26}-\frac{1}{280}a^{25}+\frac{13}{560}a^{24}+\frac{1}{20}a^{23}-\frac{1}{28}a^{22}-\frac{13}{280}a^{21}-\frac{39}{560}a^{20}-\frac{11}{280}a^{19}+\frac{17}{560}a^{18}+\frac{19}{280}a^{17}+\frac{5}{56}a^{16}+\frac{6}{35}a^{15}-\frac{93}{560}a^{14}-\frac{51}{280}a^{13}-\frac{109}{560}a^{12}-\frac{3}{35}a^{11}-\frac{23}{140}a^{10}+\frac{67}{280}a^{9}-\frac{7}{80}a^{8}+\frac{71}{280}a^{7}-\frac{169}{560}a^{6}+\frac{123}{280}a^{5}-\frac{17}{280}a^{4}-\frac{1}{140}a^{3}-\frac{41}{140}a^{2}+\frac{3}{10}a-\frac{2}{5}$, $\frac{1}{1120}a^{27}-\frac{1}{224}a^{25}+\frac{13}{560}a^{24}+\frac{11}{560}a^{23}+\frac{23}{560}a^{22}+\frac{1}{160}a^{21}+\frac{3}{35}a^{20}+\frac{113}{1120}a^{19}-\frac{1}{28}a^{18}-\frac{1}{16}a^{17}-\frac{1}{8}a^{16}+\frac{31}{224}a^{15}+\frac{17}{70}a^{14}-\frac{131}{1120}a^{13}-\frac{1}{16}a^{12}+\frac{81}{560}a^{11}-\frac{109}{560}a^{10}+\frac{13}{224}a^{9}-\frac{13}{70}a^{8}-\frac{417}{1120}a^{7}+\frac{33}{280}a^{6}-\frac{13}{112}a^{5}-\frac{51}{140}a^{4}+\frac{25}{56}a^{3}+\frac{2}{35}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{40423040}a^{28}+\frac{1979}{5052880}a^{27}+\frac{19907}{40423040}a^{26}+\frac{669}{206240}a^{25}-\frac{631333}{20211520}a^{24}-\frac{1173171}{20211520}a^{23}+\frac{888583}{40423040}a^{22}+\frac{105713}{5052880}a^{21}-\frac{819571}{40423040}a^{20}-\frac{2358729}{20211520}a^{19}+\frac{351111}{4042304}a^{18}+\frac{72861}{631610}a^{17}+\frac{1580843}{40423040}a^{16}+\frac{561251}{2526440}a^{15}+\frac{4058197}{40423040}a^{14}-\frac{193287}{2526440}a^{13}+\frac{794993}{20211520}a^{12}-\frac{924471}{4042304}a^{11}-\frac{251633}{5774720}a^{10}-\frac{161551}{5052880}a^{9}+\frac{739127}{8084608}a^{8}-\frac{9036401}{20211520}a^{7}-\frac{2883303}{20211520}a^{6}-\frac{339053}{10105760}a^{5}-\frac{113609}{10105760}a^{4}+\frac{22117}{144368}a^{3}-\frac{40599}{360920}a^{2}-\frac{162}{6445}a-\frac{31}{1289}$, $\frac{1}{80846080}a^{29}-\frac{9469}{80846080}a^{27}+\frac{2333}{4042304}a^{26}+\frac{227119}{40423040}a^{25}+\frac{865741}{40423040}a^{24}-\frac{10693}{329984}a^{23}-\frac{11675}{252644}a^{22}+\frac{704173}{80846080}a^{21}+\frac{2700727}{40423040}a^{20}+\frac{12091}{164992}a^{19}-\frac{4831}{144368}a^{18}+\frac{3814203}{80846080}a^{17}-\frac{607559}{10105760}a^{16}-\frac{6364363}{80846080}a^{15}-\frac{163}{1289}a^{14}+\frac{8814733}{40423040}a^{13}-\frac{859927}{8084608}a^{12}-\frac{189211}{16169216}a^{11}+\frac{1235739}{5052880}a^{10}-\frac{12924237}{80846080}a^{9}-\frac{1046813}{8084608}a^{8}-\frac{5285611}{40423040}a^{7}+\frac{2482791}{20211520}a^{6}-\frac{2607837}{20211520}a^{5}-\frac{4579133}{10105760}a^{4}-\frac{48357}{721840}a^{3}+\frac{72729}{180460}a^{2}+\frac{2097}{12890}a-\frac{2726}{6445}$, $\frac{1}{41\!\cdots\!00}a^{30}+\frac{21\!\cdots\!83}{52\!\cdots\!00}a^{29}-\frac{68\!\cdots\!57}{83\!\cdots\!00}a^{28}+\frac{64\!\cdots\!03}{14\!\cdots\!00}a^{27}-\frac{99\!\cdots\!69}{20\!\cdots\!00}a^{26}-\frac{25\!\cdots\!27}{20\!\cdots\!00}a^{25}+\frac{14\!\cdots\!67}{83\!\cdots\!00}a^{24}+\frac{49\!\cdots\!61}{13\!\cdots\!00}a^{23}-\frac{23\!\cdots\!59}{41\!\cdots\!00}a^{22}-\frac{95\!\cdots\!97}{20\!\cdots\!00}a^{21}-\frac{11\!\cdots\!01}{20\!\cdots\!00}a^{20}-\frac{16\!\cdots\!61}{26\!\cdots\!00}a^{19}-\frac{11\!\cdots\!33}{83\!\cdots\!00}a^{18}-\frac{60\!\cdots\!21}{52\!\cdots\!00}a^{17}-\frac{66\!\cdots\!31}{41\!\cdots\!00}a^{16}+\frac{39\!\cdots\!47}{52\!\cdots\!00}a^{15}+\frac{23\!\cdots\!49}{20\!\cdots\!00}a^{14}+\frac{33\!\cdots\!77}{20\!\cdots\!00}a^{13}+\frac{15\!\cdots\!47}{11\!\cdots\!00}a^{12}-\frac{21\!\cdots\!37}{26\!\cdots\!00}a^{11}+\frac{88\!\cdots\!19}{41\!\cdots\!00}a^{10}-\frac{38\!\cdots\!69}{20\!\cdots\!00}a^{9}-\frac{37\!\cdots\!07}{20\!\cdots\!00}a^{8}-\frac{33\!\cdots\!41}{10\!\cdots\!00}a^{7}+\frac{40\!\cdots\!79}{10\!\cdots\!00}a^{6}+\frac{19\!\cdots\!21}{15\!\cdots\!00}a^{5}-\frac{22\!\cdots\!21}{21\!\cdots\!80}a^{4}+\frac{40\!\cdots\!63}{13\!\cdots\!00}a^{3}-\frac{27\!\cdots\!99}{66\!\cdots\!00}a^{2}+\frac{70\!\cdots\!87}{23\!\cdots\!00}a+\frac{23\!\cdots\!27}{11\!\cdots\!00}$, $\frac{1}{17\!\cdots\!00}a^{31}+\frac{45\!\cdots\!79}{42\!\cdots\!00}a^{30}+\frac{66\!\cdots\!91}{34\!\cdots\!00}a^{29}+\frac{77\!\cdots\!03}{21\!\cdots\!00}a^{28}-\frac{33\!\cdots\!77}{85\!\cdots\!00}a^{27}-\frac{29\!\cdots\!71}{85\!\cdots\!00}a^{26}-\frac{45\!\cdots\!69}{17\!\cdots\!00}a^{25}-\frac{36\!\cdots\!89}{18\!\cdots\!00}a^{24}+\frac{40\!\cdots\!93}{17\!\cdots\!00}a^{23}-\frac{11\!\cdots\!13}{41\!\cdots\!00}a^{22}-\frac{38\!\cdots\!73}{85\!\cdots\!00}a^{21}+\frac{15\!\cdots\!59}{21\!\cdots\!00}a^{20}+\frac{84\!\cdots\!91}{34\!\cdots\!00}a^{19}+\frac{35\!\cdots\!23}{42\!\cdots\!00}a^{18}+\frac{17\!\cdots\!43}{24\!\cdots\!00}a^{17}-\frac{39\!\cdots\!49}{42\!\cdots\!00}a^{16}+\frac{98\!\cdots\!29}{85\!\cdots\!00}a^{15}-\frac{20\!\cdots\!99}{85\!\cdots\!00}a^{14}+\frac{34\!\cdots\!49}{17\!\cdots\!00}a^{13}+\frac{31\!\cdots\!07}{42\!\cdots\!00}a^{12}+\frac{21\!\cdots\!27}{17\!\cdots\!00}a^{11}+\frac{54\!\cdots\!53}{85\!\cdots\!00}a^{10}+\frac{19\!\cdots\!07}{12\!\cdots\!00}a^{9}-\frac{34\!\cdots\!07}{42\!\cdots\!00}a^{8}-\frac{13\!\cdots\!37}{42\!\cdots\!00}a^{7}-\frac{23\!\cdots\!19}{42\!\cdots\!00}a^{6}+\frac{66\!\cdots\!27}{15\!\cdots\!00}a^{5}+\frac{39\!\cdots\!61}{67\!\cdots\!00}a^{4}+\frac{97\!\cdots\!57}{77\!\cdots\!00}a^{3}-\frac{57\!\cdots\!87}{12\!\cdots\!00}a^{2}-\frac{49\!\cdots\!67}{48\!\cdots\!00}a+\frac{25\!\cdots\!13}{12\!\cdots\!00}$
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
$C_{2}\times C_{4}\times C_{48}$, which has order $384$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $384$ (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{688015877785208730788504150522650665105291527049307}{8965993924830536794926252156063681459147162231969139060480} a^{31} + \frac{28050507770270434111320676869846056618924354310769743}{89659939248305367949262521560636814591471622319691390604800} a^{30} - \frac{356544587871833678184785715109942413246079006533293}{1120749240603817099365781519507960182393395278996142382560} a^{29} - \frac{3173652855729077250496730659292855038357167438255909}{2561712549951581941407500616018194702613474923419754017280} a^{28} + \frac{142920012835290179768637386542764337920287520542574371}{44829969624152683974631260780318407295735811159845695302400} a^{27} + \frac{190387916709158431586956373948115738357644375774761633}{44829969624152683974631260780318407295735811159845695302400} a^{26} - \frac{158129752425890828716763601652094827940932663533822607}{5603746203019085496828907597539800911966976394980711912800} a^{25} + \frac{61710399573425771654041853398782662194169896642152129}{17931987849661073589852504312127362918294324463938278120960} a^{24} + \frac{1706982672871201892046254340335688328554636460901980027}{11207492406038170993657815195079601823933952789961423825600} a^{23} - \frac{904775275325437768354070380321192977876206330020389473}{3091722043045012687905604191746097054878331804127289331200} a^{22} - \frac{17350698895401663133244275594141430990454713181676203}{1400936550754771374207226899384950227991744098745177978200} a^{21} + \frac{15898911312062870825309662807180896265268589083178283757}{44829969624152683974631260780318407295735811159845695302400} a^{20} + \frac{47698434189713902075052400535300977911414804227604147341}{44829969624152683974631260780318407295735811159845695302400} a^{19} - \frac{68197399117989047278988741607640873497000791103787782263}{17931987849661073589852504312127362918294324463938278120960} a^{18} - \frac{7980214041612763101430450748405832268917210087613532249}{5603746203019085496828907597539800911966976394980711912800} a^{17} + \frac{2548298122459538467939068128122832041991711898737765921727}{89659939248305367949262521560636814591471622319691390604800} a^{16} - \frac{718775674253091829978843286716091191589185909167386881793}{8965993924830536794926252156063681459147162231969139060480} a^{15} + \frac{7851093876882328071554542604577701368362908531298211694907}{44829969624152683974631260780318407295735811159845695302400} a^{14} - \frac{1441012963668496693609898029524165500765094252435446445271}{3202140687439477426759375770022743378266843654274692521600} a^{13} + \frac{16053907856498562921832883854009132082267704957235352113719}{17931987849661073589852504312127362918294324463938278120960} a^{12} - \frac{17087807789168096798203879609835514490718038667466660450887}{11207492406038170993657815195079601823933952789961423825600} a^{11} + \frac{201994360946300598165128672882600997774697091113015849054677}{89659939248305367949262521560636814591471622319691390604800} a^{10} - \frac{69300931275072010946556002539497682431168397514124039030951}{22414984812076341987315630390159203647867905579922847651200} a^{9} + \frac{154189229326664307304688962379906755845018589111212218977999}{44829969624152683974631260780318407295735811159845695302400} a^{8} - \frac{48616114046313679912081295026030246905834633703834770712969}{11207492406038170993657815195079601823933952789961423825600} a^{7} + \frac{12751537361848079551171443492477881596806123747359960580771}{3202140687439477426759375770022743378266843654274692521600} a^{6} - \frac{283829597559115989899525251477577316316196752065199079637}{114362167408552765241406277500812263509530130509810447200} a^{5} + \frac{53201020426480758159638668235366118812559841663910048321}{32021406874394774267593757700227433782668436542746925216} a^{4} - \frac{13385697665910720582524299581715098833686017934606375103}{4084363121734027330050224196457580839626076089636087400} a^{3} + \frac{4263643320086452498488384958340511586465319712201013123}{4084363121734027330050224196457580839626076089636087400} a^{2} - \frac{456350524210946270996296346874179662456749712898852369}{510545390216753416256278024557197604953259511204510925} a + \frac{1133097598858785504557409687704500332751823764852726982}{510545390216753416256278024557197604953259511204510925} \)
(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: |
$\frac{23\!\cdots\!09}{29\!\cdots\!00}a^{31}-\frac{19\!\cdots\!07}{47\!\cdots\!00}a^{30}+\frac{30\!\cdots\!23}{47\!\cdots\!00}a^{29}+\frac{57\!\cdots\!13}{68\!\cdots\!00}a^{28}-\frac{40\!\cdots\!37}{95\!\cdots\!80}a^{27}-\frac{37\!\cdots\!43}{47\!\cdots\!00}a^{26}+\frac{71\!\cdots\!79}{23\!\cdots\!00}a^{25}-\frac{14\!\cdots\!57}{47\!\cdots\!00}a^{24}-\frac{65\!\cdots\!23}{47\!\cdots\!00}a^{23}+\frac{66\!\cdots\!73}{16\!\cdots\!00}a^{22}-\frac{89\!\cdots\!67}{23\!\cdots\!00}a^{21}+\frac{18\!\cdots\!19}{47\!\cdots\!00}a^{20}-\frac{11\!\cdots\!17}{11\!\cdots\!00}a^{19}+\frac{17\!\cdots\!63}{47\!\cdots\!00}a^{18}-\frac{11\!\cdots\!79}{23\!\cdots\!00}a^{17}-\frac{13\!\cdots\!83}{47\!\cdots\!00}a^{16}+\frac{25\!\cdots\!71}{23\!\cdots\!00}a^{15}-\frac{13\!\cdots\!29}{47\!\cdots\!00}a^{14}+\frac{24\!\cdots\!33}{34\!\cdots\!00}a^{13}-\frac{70\!\cdots\!63}{47\!\cdots\!00}a^{12}+\frac{23\!\cdots\!67}{95\!\cdots\!80}a^{11}-\frac{38\!\cdots\!89}{95\!\cdots\!00}a^{10}+\frac{63\!\cdots\!67}{11\!\cdots\!00}a^{9}-\frac{15\!\cdots\!39}{23\!\cdots\!00}a^{8}+\frac{20\!\cdots\!37}{29\!\cdots\!00}a^{7}-\frac{16\!\cdots\!21}{24\!\cdots\!00}a^{6}+\frac{15\!\cdots\!91}{42\!\cdots\!00}a^{5}-\frac{16\!\cdots\!59}{42\!\cdots\!00}a^{4}+\frac{51\!\cdots\!67}{15\!\cdots\!00}a^{3}-\frac{37\!\cdots\!13}{21\!\cdots\!00}a^{2}+\frac{70\!\cdots\!47}{27\!\cdots\!75}a-\frac{39\!\cdots\!68}{27\!\cdots\!75}$, $\frac{48\!\cdots\!19}{57\!\cdots\!00}a^{31}-\frac{21\!\cdots\!03}{14\!\cdots\!00}a^{30}-\frac{44\!\cdots\!39}{11\!\cdots\!00}a^{29}+\frac{13\!\cdots\!27}{71\!\cdots\!00}a^{28}-\frac{30\!\cdots\!47}{57\!\cdots\!00}a^{27}-\frac{66\!\cdots\!49}{57\!\cdots\!00}a^{26}+\frac{11\!\cdots\!53}{57\!\cdots\!00}a^{25}+\frac{82\!\cdots\!67}{14\!\cdots\!00}a^{24}-\frac{18\!\cdots\!13}{11\!\cdots\!00}a^{23}-\frac{68\!\cdots\!61}{99\!\cdots\!00}a^{22}+\frac{35\!\cdots\!93}{57\!\cdots\!00}a^{21}-\frac{16\!\cdots\!03}{28\!\cdots\!80}a^{20}-\frac{92\!\cdots\!63}{57\!\cdots\!00}a^{19}+\frac{34\!\cdots\!97}{14\!\cdots\!00}a^{18}+\frac{43\!\cdots\!07}{57\!\cdots\!00}a^{17}-\frac{39\!\cdots\!07}{14\!\cdots\!00}a^{16}+\frac{82\!\cdots\!71}{28\!\cdots\!00}a^{15}-\frac{31\!\cdots\!73}{57\!\cdots\!00}a^{14}+\frac{29\!\cdots\!67}{57\!\cdots\!00}a^{13}+\frac{14\!\cdots\!53}{14\!\cdots\!00}a^{12}-\frac{24\!\cdots\!83}{11\!\cdots\!00}a^{11}+\frac{38\!\cdots\!51}{57\!\cdots\!00}a^{10}-\frac{98\!\cdots\!53}{57\!\cdots\!00}a^{9}+\frac{23\!\cdots\!23}{14\!\cdots\!00}a^{8}-\frac{92\!\cdots\!47}{14\!\cdots\!00}a^{7}+\frac{18\!\cdots\!63}{71\!\cdots\!00}a^{6}-\frac{43\!\cdots\!33}{35\!\cdots\!00}a^{5}+\frac{36\!\cdots\!07}{22\!\cdots\!00}a^{4}-\frac{10\!\cdots\!51}{44\!\cdots\!00}a^{3}+\frac{84\!\cdots\!09}{14\!\cdots\!00}a^{2}-\frac{17\!\cdots\!67}{56\!\cdots\!00}a+\frac{72\!\cdots\!77}{14\!\cdots\!00}$, $\frac{16\!\cdots\!27}{17\!\cdots\!00}a^{31}-\frac{97\!\cdots\!67}{42\!\cdots\!00}a^{30}-\frac{17\!\cdots\!23}{34\!\cdots\!00}a^{29}+\frac{44\!\cdots\!83}{30\!\cdots\!00}a^{28}-\frac{94\!\cdots\!79}{85\!\cdots\!00}a^{27}-\frac{69\!\cdots\!17}{85\!\cdots\!00}a^{26}+\frac{34\!\cdots\!37}{17\!\cdots\!00}a^{25}+\frac{15\!\cdots\!31}{42\!\cdots\!00}a^{24}-\frac{22\!\cdots\!89}{17\!\cdots\!00}a^{23}+\frac{25\!\cdots\!43}{29\!\cdots\!00}a^{22}+\frac{18\!\cdots\!29}{85\!\cdots\!00}a^{21}+\frac{93\!\cdots\!93}{21\!\cdots\!00}a^{20}-\frac{28\!\cdots\!07}{17\!\cdots\!00}a^{19}+\frac{34\!\cdots\!27}{18\!\cdots\!00}a^{18}+\frac{10\!\cdots\!27}{17\!\cdots\!00}a^{17}-\frac{10\!\cdots\!23}{42\!\cdots\!00}a^{16}+\frac{18\!\cdots\!21}{37\!\cdots\!00}a^{15}-\frac{10\!\cdots\!73}{85\!\cdots\!00}a^{14}+\frac{86\!\cdots\!89}{24\!\cdots\!00}a^{13}-\frac{22\!\cdots\!11}{42\!\cdots\!00}a^{12}+\frac{16\!\cdots\!29}{17\!\cdots\!00}a^{11}-\frac{11\!\cdots\!69}{85\!\cdots\!00}a^{10}+\frac{14\!\cdots\!23}{85\!\cdots\!00}a^{9}-\frac{83\!\cdots\!89}{42\!\cdots\!00}a^{8}+\frac{12\!\cdots\!01}{42\!\cdots\!00}a^{7}-\frac{15\!\cdots\!91}{12\!\cdots\!00}a^{6}+\frac{26\!\cdots\!47}{21\!\cdots\!00}a^{5}-\frac{26\!\cdots\!49}{11\!\cdots\!00}a^{4}+\frac{28\!\cdots\!59}{77\!\cdots\!00}a^{3}-\frac{10\!\cdots\!99}{12\!\cdots\!00}a^{2}+\frac{45\!\cdots\!91}{48\!\cdots\!00}a+\frac{32\!\cdots\!51}{12\!\cdots\!00}$, $\frac{13\!\cdots\!73}{17\!\cdots\!00}a^{31}-\frac{43\!\cdots\!01}{37\!\cdots\!00}a^{30}-\frac{78\!\cdots\!09}{34\!\cdots\!00}a^{29}+\frac{61\!\cdots\!61}{52\!\cdots\!00}a^{28}+\frac{24\!\cdots\!91}{85\!\cdots\!00}a^{27}-\frac{66\!\cdots\!17}{85\!\cdots\!00}a^{26}+\frac{16\!\cdots\!19}{17\!\cdots\!00}a^{25}+\frac{40\!\cdots\!43}{85\!\cdots\!00}a^{24}-\frac{13\!\cdots\!39}{17\!\cdots\!00}a^{23}-\frac{16\!\cdots\!87}{36\!\cdots\!00}a^{22}+\frac{21\!\cdots\!29}{85\!\cdots\!00}a^{21}+\frac{10\!\cdots\!71}{42\!\cdots\!00}a^{20}-\frac{24\!\cdots\!29}{17\!\cdots\!00}a^{19}+\frac{18\!\cdots\!63}{85\!\cdots\!00}a^{18}+\frac{45\!\cdots\!81}{68\!\cdots\!80}a^{17}-\frac{13\!\cdots\!87}{85\!\cdots\!00}a^{16}+\frac{17\!\cdots\!97}{85\!\cdots\!00}a^{15}-\frac{48\!\cdots\!13}{85\!\cdots\!00}a^{14}+\frac{48\!\cdots\!03}{24\!\cdots\!00}a^{13}-\frac{14\!\cdots\!43}{85\!\cdots\!00}a^{12}+\frac{66\!\cdots\!59}{17\!\cdots\!00}a^{11}-\frac{18\!\cdots\!57}{42\!\cdots\!00}a^{10}+\frac{38\!\cdots\!43}{85\!\cdots\!00}a^{9}-\frac{53\!\cdots\!03}{10\!\cdots\!00}a^{8}+\frac{45\!\cdots\!73}{42\!\cdots\!00}a^{7}+\frac{10\!\cdots\!33}{15\!\cdots\!00}a^{6}+\frac{71\!\cdots\!09}{76\!\cdots\!00}a^{5}-\frac{14\!\cdots\!89}{76\!\cdots\!00}a^{4}-\frac{24\!\cdots\!41}{16\!\cdots\!00}a^{3}-\frac{91\!\cdots\!67}{19\!\cdots\!00}a^{2}+\frac{66\!\cdots\!01}{97\!\cdots\!20}a+\frac{10\!\cdots\!59}{24\!\cdots\!00}$, $\frac{15\!\cdots\!23}{17\!\cdots\!00}a^{31}-\frac{25\!\cdots\!07}{12\!\cdots\!00}a^{30}-\frac{35\!\cdots\!53}{24\!\cdots\!00}a^{29}+\frac{20\!\cdots\!19}{12\!\cdots\!00}a^{28}-\frac{77\!\cdots\!83}{60\!\cdots\!00}a^{27}-\frac{13\!\cdots\!51}{15\!\cdots\!00}a^{26}+\frac{27\!\cdots\!73}{12\!\cdots\!00}a^{25}+\frac{68\!\cdots\!61}{17\!\cdots\!00}a^{24}-\frac{18\!\cdots\!43}{12\!\cdots\!00}a^{23}+\frac{37\!\cdots\!91}{41\!\cdots\!00}a^{22}+\frac{10\!\cdots\!99}{30\!\cdots\!00}a^{21}-\frac{18\!\cdots\!21}{60\!\cdots\!00}a^{20}-\frac{16\!\cdots\!93}{12\!\cdots\!00}a^{19}+\frac{37\!\cdots\!01}{17\!\cdots\!00}a^{18}+\frac{59\!\cdots\!27}{10\!\cdots\!00}a^{17}-\frac{31\!\cdots\!83}{12\!\cdots\!00}a^{16}+\frac{29\!\cdots\!29}{60\!\cdots\!00}a^{15}-\frac{27\!\cdots\!03}{30\!\cdots\!00}a^{14}+\frac{42\!\cdots\!01}{17\!\cdots\!00}a^{13}-\frac{33\!\cdots\!27}{12\!\cdots\!00}a^{12}+\frac{59\!\cdots\!83}{12\!\cdots\!00}a^{11}-\frac{94\!\cdots\!73}{17\!\cdots\!00}a^{10}+\frac{68\!\cdots\!79}{15\!\cdots\!00}a^{9}-\frac{19\!\cdots\!73}{60\!\cdots\!00}a^{8}+\frac{15\!\cdots\!73}{15\!\cdots\!00}a^{7}+\frac{74\!\cdots\!53}{30\!\cdots\!00}a^{6}-\frac{19\!\cdots\!91}{15\!\cdots\!00}a^{5}-\frac{10\!\cdots\!97}{76\!\cdots\!00}a^{4}-\frac{28\!\cdots\!73}{27\!\cdots\!00}a^{3}-\frac{24\!\cdots\!51}{19\!\cdots\!00}a^{2}+\frac{56\!\cdots\!07}{97\!\cdots\!00}a+\frac{15\!\cdots\!17}{24\!\cdots\!00}$, $\frac{64\!\cdots\!31}{17\!\cdots\!00}a^{31}-\frac{67\!\cdots\!69}{17\!\cdots\!00}a^{30}-\frac{73\!\cdots\!41}{48\!\cdots\!00}a^{29}+\frac{47\!\cdots\!27}{85\!\cdots\!00}a^{28}+\frac{33\!\cdots\!01}{85\!\cdots\!00}a^{27}-\frac{33\!\cdots\!67}{85\!\cdots\!00}a^{26}+\frac{11\!\cdots\!21}{34\!\cdots\!00}a^{25}+\frac{22\!\cdots\!81}{85\!\cdots\!00}a^{24}-\frac{11\!\cdots\!61}{34\!\cdots\!00}a^{23}-\frac{63\!\cdots\!77}{15\!\cdots\!00}a^{22}+\frac{11\!\cdots\!79}{85\!\cdots\!00}a^{21}+\frac{60\!\cdots\!01}{42\!\cdots\!00}a^{20}-\frac{21\!\cdots\!67}{34\!\cdots\!00}a^{19}-\frac{13\!\cdots\!79}{85\!\cdots\!00}a^{18}+\frac{58\!\cdots\!79}{17\!\cdots\!00}a^{17}-\frac{10\!\cdots\!01}{17\!\cdots\!00}a^{16}+\frac{54\!\cdots\!19}{85\!\cdots\!00}a^{15}-\frac{16\!\cdots\!83}{85\!\cdots\!00}a^{14}+\frac{10\!\cdots\!91}{13\!\cdots\!96}a^{13}-\frac{97\!\cdots\!01}{85\!\cdots\!00}a^{12}+\frac{77\!\cdots\!63}{74\!\cdots\!00}a^{11}-\frac{92\!\cdots\!87}{42\!\cdots\!00}a^{10}+\frac{51\!\cdots\!53}{85\!\cdots\!00}a^{9}+\frac{10\!\cdots\!71}{10\!\cdots\!00}a^{8}+\frac{98\!\cdots\!79}{42\!\cdots\!00}a^{7}+\frac{95\!\cdots\!29}{10\!\cdots\!00}a^{6}+\frac{11\!\cdots\!79}{15\!\cdots\!00}a^{5}-\frac{40\!\cdots\!83}{76\!\cdots\!00}a^{4}-\frac{37\!\cdots\!83}{33\!\cdots\!00}a^{3}-\frac{14\!\cdots\!07}{13\!\cdots\!00}a^{2}+\frac{63\!\cdots\!91}{24\!\cdots\!00}a+\frac{26\!\cdots\!69}{48\!\cdots\!00}$, $\frac{13\!\cdots\!09}{37\!\cdots\!00}a^{31}-\frac{23\!\cdots\!99}{37\!\cdots\!00}a^{30}-\frac{53\!\cdots\!79}{10\!\cdots\!00}a^{29}+\frac{94\!\cdots\!53}{18\!\cdots\!00}a^{28}-\frac{32\!\cdots\!21}{18\!\cdots\!00}a^{27}-\frac{54\!\cdots\!93}{18\!\cdots\!00}a^{26}+\frac{43\!\cdots\!23}{74\!\cdots\!00}a^{25}+\frac{29\!\cdots\!59}{18\!\cdots\!00}a^{24}-\frac{20\!\cdots\!33}{53\!\cdots\!00}a^{23}+\frac{29\!\cdots\!47}{16\!\cdots\!00}a^{22}+\frac{14\!\cdots\!41}{18\!\cdots\!00}a^{21}+\frac{36\!\cdots\!79}{93\!\cdots\!00}a^{20}-\frac{36\!\cdots\!77}{74\!\cdots\!00}a^{19}+\frac{67\!\cdots\!19}{18\!\cdots\!00}a^{18}+\frac{81\!\cdots\!21}{37\!\cdots\!00}a^{17}-\frac{28\!\cdots\!71}{37\!\cdots\!00}a^{16}+\frac{28\!\cdots\!41}{18\!\cdots\!00}a^{15}-\frac{70\!\cdots\!57}{18\!\cdots\!00}a^{14}+\frac{80\!\cdots\!41}{74\!\cdots\!00}a^{13}-\frac{22\!\cdots\!39}{18\!\cdots\!00}a^{12}+\frac{10\!\cdots\!71}{37\!\cdots\!00}a^{11}-\frac{27\!\cdots\!23}{93\!\cdots\!00}a^{10}+\frac{75\!\cdots\!87}{18\!\cdots\!00}a^{9}-\frac{45\!\cdots\!93}{11\!\cdots\!00}a^{8}+\frac{63\!\cdots\!61}{93\!\cdots\!00}a^{7}+\frac{10\!\cdots\!01}{23\!\cdots\!00}a^{6}+\frac{32\!\cdots\!31}{66\!\cdots\!60}a^{5}-\frac{86\!\cdots\!37}{16\!\cdots\!00}a^{4}-\frac{43\!\cdots\!79}{14\!\cdots\!00}a^{3}-\frac{30\!\cdots\!59}{42\!\cdots\!00}a^{2}-\frac{99\!\cdots\!41}{53\!\cdots\!00}a-\frac{16\!\cdots\!61}{21\!\cdots\!60}$, $\frac{59\!\cdots\!41}{37\!\cdots\!00}a^{31}-\frac{11\!\cdots\!83}{26\!\cdots\!00}a^{30}+\frac{44\!\cdots\!07}{74\!\cdots\!00}a^{29}+\frac{38\!\cdots\!07}{18\!\cdots\!00}a^{28}-\frac{28\!\cdots\!13}{18\!\cdots\!00}a^{27}-\frac{24\!\cdots\!69}{18\!\cdots\!00}a^{26}+\frac{16\!\cdots\!49}{53\!\cdots\!00}a^{25}+\frac{10\!\cdots\!81}{18\!\cdots\!00}a^{24}-\frac{32\!\cdots\!01}{16\!\cdots\!00}a^{23}+\frac{45\!\cdots\!29}{32\!\cdots\!00}a^{22}+\frac{62\!\cdots\!79}{26\!\cdots\!00}a^{21}+\frac{65\!\cdots\!71}{13\!\cdots\!00}a^{20}-\frac{12\!\cdots\!13}{37\!\cdots\!00}a^{19}+\frac{67\!\cdots\!21}{18\!\cdots\!00}a^{18}+\frac{69\!\cdots\!93}{74\!\cdots\!00}a^{17}-\frac{10\!\cdots\!27}{26\!\cdots\!00}a^{16}+\frac{16\!\cdots\!49}{18\!\cdots\!00}a^{15}-\frac{45\!\cdots\!41}{18\!\cdots\!00}a^{14}+\frac{28\!\cdots\!37}{37\!\cdots\!00}a^{13}-\frac{24\!\cdots\!81}{18\!\cdots\!00}a^{12}+\frac{96\!\cdots\!63}{37\!\cdots\!00}a^{11}-\frac{19\!\cdots\!37}{46\!\cdots\!00}a^{10}+\frac{11\!\cdots\!51}{18\!\cdots\!00}a^{9}-\frac{38\!\cdots\!37}{46\!\cdots\!00}a^{8}+\frac{10\!\cdots\!21}{93\!\cdots\!00}a^{7}-\frac{11\!\cdots\!89}{11\!\cdots\!00}a^{6}+\frac{84\!\cdots\!49}{83\!\cdots\!00}a^{5}-\frac{18\!\cdots\!63}{16\!\cdots\!00}a^{4}+\frac{17\!\cdots\!39}{29\!\cdots\!00}a^{3}-\frac{17\!\cdots\!09}{42\!\cdots\!00}a^{2}+\frac{19\!\cdots\!53}{57\!\cdots\!75}a-\frac{65\!\cdots\!27}{53\!\cdots\!00}$, $\frac{16\!\cdots\!17}{17\!\cdots\!00}a^{31}-\frac{49\!\cdots\!73}{17\!\cdots\!00}a^{30}+\frac{14\!\cdots\!71}{34\!\cdots\!00}a^{29}+\frac{34\!\cdots\!41}{21\!\cdots\!00}a^{28}-\frac{17\!\cdots\!13}{85\!\cdots\!00}a^{27}-\frac{69\!\cdots\!79}{85\!\cdots\!00}a^{26}+\frac{51\!\cdots\!49}{19\!\cdots\!60}a^{25}+\frac{12\!\cdots\!21}{42\!\cdots\!00}a^{24}-\frac{27\!\cdots\!43}{17\!\cdots\!00}a^{23}+\frac{48\!\cdots\!29}{29\!\cdots\!00}a^{22}+\frac{35\!\cdots\!27}{17\!\cdots\!00}a^{21}-\frac{50\!\cdots\!67}{30\!\cdots\!00}a^{20}-\frac{54\!\cdots\!37}{34\!\cdots\!00}a^{19}+\frac{11\!\cdots\!11}{42\!\cdots\!00}a^{18}+\frac{38\!\cdots\!71}{74\!\cdots\!00}a^{17}-\frac{72\!\cdots\!31}{24\!\cdots\!00}a^{16}+\frac{56\!\cdots\!33}{85\!\cdots\!00}a^{15}-\frac{12\!\cdots\!71}{85\!\cdots\!00}a^{14}+\frac{13\!\cdots\!17}{34\!\cdots\!00}a^{13}-\frac{27\!\cdots\!41}{42\!\cdots\!00}a^{12}+\frac{19\!\cdots\!63}{17\!\cdots\!00}a^{11}-\frac{13\!\cdots\!63}{85\!\cdots\!00}a^{10}+\frac{17\!\cdots\!61}{85\!\cdots\!00}a^{9}-\frac{95\!\cdots\!27}{42\!\cdots\!00}a^{8}+\frac{13\!\cdots\!63}{42\!\cdots\!00}a^{7}-\frac{40\!\cdots\!39}{21\!\cdots\!00}a^{6}+\frac{33\!\cdots\!47}{30\!\cdots\!00}a^{5}-\frac{22\!\cdots\!07}{95\!\cdots\!00}a^{4}+\frac{25\!\cdots\!87}{27\!\cdots\!00}a^{3}-\frac{84\!\cdots\!19}{84\!\cdots\!00}a^{2}+\frac{67\!\cdots\!89}{48\!\cdots\!00}a-\frac{54\!\cdots\!47}{24\!\cdots\!00}$, $\frac{49\!\cdots\!01}{68\!\cdots\!80}a^{31}+\frac{24\!\cdots\!51}{85\!\cdots\!00}a^{30}-\frac{82\!\cdots\!49}{13\!\cdots\!96}a^{29}-\frac{41\!\cdots\!07}{85\!\cdots\!60}a^{28}+\frac{83\!\cdots\!83}{17\!\cdots\!00}a^{27}-\frac{32\!\cdots\!71}{17\!\cdots\!00}a^{26}-\frac{98\!\cdots\!81}{34\!\cdots\!00}a^{25}+\frac{19\!\cdots\!61}{34\!\cdots\!24}a^{24}+\frac{51\!\cdots\!13}{34\!\cdots\!00}a^{23}-\frac{23\!\cdots\!47}{58\!\cdots\!00}a^{22}+\frac{23\!\cdots\!23}{32\!\cdots\!00}a^{21}+\frac{45\!\cdots\!39}{42\!\cdots\!00}a^{20}+\frac{59\!\cdots\!51}{34\!\cdots\!00}a^{19}-\frac{95\!\cdots\!33}{17\!\cdots\!20}a^{18}+\frac{16\!\cdots\!73}{34\!\cdots\!00}a^{17}+\frac{18\!\cdots\!59}{85\!\cdots\!00}a^{16}-\frac{49\!\cdots\!63}{69\!\cdots\!60}a^{15}+\frac{27\!\cdots\!03}{24\!\cdots\!00}a^{14}-\frac{87\!\cdots\!39}{34\!\cdots\!00}a^{13}+\frac{14\!\cdots\!27}{17\!\cdots\!20}a^{12}-\frac{34\!\cdots\!73}{34\!\cdots\!00}a^{11}+\frac{52\!\cdots\!19}{24\!\cdots\!00}a^{10}-\frac{43\!\cdots\!71}{17\!\cdots\!00}a^{9}+\frac{28\!\cdots\!41}{85\!\cdots\!00}a^{8}-\frac{42\!\cdots\!07}{12\!\cdots\!00}a^{7}+\frac{29\!\cdots\!53}{42\!\cdots\!00}a^{6}-\frac{11\!\cdots\!19}{13\!\cdots\!00}a^{5}+\frac{16\!\cdots\!13}{54\!\cdots\!20}a^{4}-\frac{34\!\cdots\!69}{54\!\cdots\!00}a^{3}+\frac{17\!\cdots\!51}{24\!\cdots\!00}a^{2}-\frac{23\!\cdots\!11}{97\!\cdots\!00}a+\frac{70\!\cdots\!17}{24\!\cdots\!00}$, $\frac{52\!\cdots\!33}{10\!\cdots\!00}a^{31}-\frac{40\!\cdots\!01}{42\!\cdots\!00}a^{30}-\frac{20\!\cdots\!73}{21\!\cdots\!00}a^{29}+\frac{34\!\cdots\!73}{42\!\cdots\!00}a^{28}-\frac{66\!\cdots\!51}{21\!\cdots\!00}a^{27}-\frac{39\!\cdots\!83}{85\!\cdots\!60}a^{26}+\frac{19\!\cdots\!97}{21\!\cdots\!00}a^{25}+\frac{10\!\cdots\!29}{42\!\cdots\!00}a^{24}-\frac{26\!\cdots\!83}{42\!\cdots\!80}a^{23}+\frac{30\!\cdots\!99}{14\!\cdots\!00}a^{22}+\frac{12\!\cdots\!33}{85\!\cdots\!60}a^{21}-\frac{23\!\cdots\!07}{42\!\cdots\!00}a^{20}-\frac{76\!\cdots\!81}{10\!\cdots\!00}a^{19}+\frac{24\!\cdots\!89}{42\!\cdots\!00}a^{18}+\frac{35\!\cdots\!79}{10\!\cdots\!00}a^{17}-\frac{49\!\cdots\!69}{42\!\cdots\!00}a^{16}+\frac{15\!\cdots\!71}{76\!\cdots\!00}a^{15}-\frac{28\!\cdots\!63}{60\!\cdots\!00}a^{14}+\frac{29\!\cdots\!33}{21\!\cdots\!00}a^{13}-\frac{57\!\cdots\!89}{42\!\cdots\!00}a^{12}+\frac{60\!\cdots\!43}{21\!\cdots\!00}a^{11}-\frac{34\!\cdots\!53}{12\!\cdots\!00}a^{10}+\frac{12\!\cdots\!53}{42\!\cdots\!00}a^{9}-\frac{46\!\cdots\!57}{21\!\cdots\!00}a^{8}+\frac{88\!\cdots\!07}{15\!\cdots\!00}a^{7}+\frac{38\!\cdots\!93}{10\!\cdots\!00}a^{6}+\frac{70\!\cdots\!51}{76\!\cdots\!00}a^{5}-\frac{20\!\cdots\!57}{23\!\cdots\!00}a^{4}-\frac{78\!\cdots\!39}{13\!\cdots\!00}a^{3}-\frac{80\!\cdots\!47}{97\!\cdots\!00}a^{2}+\frac{10\!\cdots\!81}{24\!\cdots\!00}a+\frac{30\!\cdots\!13}{12\!\cdots\!00}$, $\frac{32\!\cdots\!01}{37\!\cdots\!00}a^{31}-\frac{11\!\cdots\!27}{72\!\cdots\!00}a^{30}-\frac{93\!\cdots\!33}{74\!\cdots\!00}a^{29}+\frac{14\!\cdots\!97}{11\!\cdots\!00}a^{28}-\frac{27\!\cdots\!11}{81\!\cdots\!00}a^{27}-\frac{13\!\cdots\!39}{18\!\cdots\!00}a^{26}+\frac{51\!\cdots\!43}{37\!\cdots\!00}a^{25}+\frac{36\!\cdots\!83}{93\!\cdots\!00}a^{24}-\frac{33\!\cdots\!63}{37\!\cdots\!00}a^{23}+\frac{24\!\cdots\!53}{64\!\cdots\!00}a^{22}+\frac{34\!\cdots\!43}{18\!\cdots\!00}a^{21}+\frac{66\!\cdots\!41}{46\!\cdots\!00}a^{20}-\frac{46\!\cdots\!13}{37\!\cdots\!00}a^{19}+\frac{77\!\cdots\!53}{93\!\cdots\!00}a^{18}+\frac{41\!\cdots\!61}{74\!\cdots\!00}a^{17}-\frac{17\!\cdots\!07}{93\!\cdots\!00}a^{16}+\frac{68\!\cdots\!89}{18\!\cdots\!00}a^{15}-\frac{17\!\cdots\!71}{18\!\cdots\!00}a^{14}+\frac{44\!\cdots\!19}{16\!\cdots\!00}a^{13}-\frac{28\!\cdots\!83}{93\!\cdots\!00}a^{12}+\frac{38\!\cdots\!29}{53\!\cdots\!00}a^{11}-\frac{14\!\cdots\!63}{18\!\cdots\!00}a^{10}+\frac{20\!\cdots\!81}{18\!\cdots\!00}a^{9}-\frac{15\!\cdots\!37}{13\!\cdots\!00}a^{8}+\frac{17\!\cdots\!11}{93\!\cdots\!00}a^{7}-\frac{23\!\cdots\!41}{22\!\cdots\!00}a^{6}+\frac{48\!\cdots\!71}{33\!\cdots\!00}a^{5}-\frac{62\!\cdots\!17}{41\!\cdots\!00}a^{4}-\frac{35\!\cdots\!37}{59\!\cdots\!00}a^{3}-\frac{14\!\cdots\!77}{74\!\cdots\!00}a^{2}-\frac{87\!\cdots\!47}{21\!\cdots\!00}a-\frac{15\!\cdots\!01}{26\!\cdots\!00}$, $\frac{93\!\cdots\!97}{21\!\cdots\!00}a^{31}-\frac{60\!\cdots\!63}{42\!\cdots\!00}a^{30}+\frac{42\!\cdots\!99}{42\!\cdots\!00}a^{29}+\frac{26\!\cdots\!51}{42\!\cdots\!00}a^{28}-\frac{32\!\cdots\!69}{26\!\cdots\!00}a^{27}-\frac{74\!\cdots\!93}{30\!\cdots\!00}a^{26}+\frac{16\!\cdots\!51}{13\!\cdots\!00}a^{25}+\frac{25\!\cdots\!69}{60\!\cdots\!00}a^{24}-\frac{13\!\cdots\!71}{21\!\cdots\!00}a^{23}+\frac{16\!\cdots\!49}{14\!\cdots\!00}a^{22}-\frac{27\!\cdots\!63}{21\!\cdots\!00}a^{21}-\frac{11\!\cdots\!49}{21\!\cdots\!00}a^{20}-\frac{94\!\cdots\!81}{21\!\cdots\!00}a^{19}+\frac{68\!\cdots\!29}{60\!\cdots\!00}a^{18}+\frac{11\!\cdots\!77}{85\!\cdots\!60}a^{17}-\frac{52\!\cdots\!47}{42\!\cdots\!00}a^{16}+\frac{38\!\cdots\!33}{10\!\cdots\!00}a^{15}-\frac{18\!\cdots\!89}{21\!\cdots\!00}a^{14}+\frac{23\!\cdots\!97}{10\!\cdots\!00}a^{13}-\frac{68\!\cdots\!21}{18\!\cdots\!00}a^{12}+\frac{13\!\cdots\!51}{21\!\cdots\!00}a^{11}-\frac{38\!\cdots\!09}{42\!\cdots\!00}a^{10}+\frac{25\!\cdots\!29}{21\!\cdots\!00}a^{9}-\frac{26\!\cdots\!47}{21\!\cdots\!00}a^{8}+\frac{16\!\cdots\!69}{10\!\cdots\!00}a^{7}-\frac{11\!\cdots\!53}{10\!\cdots\!00}a^{6}+\frac{74\!\cdots\!33}{76\!\cdots\!00}a^{5}-\frac{43\!\cdots\!59}{38\!\cdots\!00}a^{4}+\frac{11\!\cdots\!39}{13\!\cdots\!00}a^{3}-\frac{37\!\cdots\!39}{67\!\cdots\!00}a^{2}+\frac{39\!\cdots\!23}{97\!\cdots\!20}a-\frac{59\!\cdots\!21}{12\!\cdots\!00}$, $\frac{21\!\cdots\!99}{17\!\cdots\!00}a^{31}-\frac{11\!\cdots\!77}{42\!\cdots\!00}a^{30}-\frac{24\!\cdots\!43}{34\!\cdots\!40}a^{29}+\frac{18\!\cdots\!61}{10\!\cdots\!00}a^{28}-\frac{72\!\cdots\!87}{85\!\cdots\!00}a^{27}-\frac{17\!\cdots\!09}{17\!\cdots\!00}a^{26}+\frac{37\!\cdots\!57}{17\!\cdots\!00}a^{25}+\frac{32\!\cdots\!31}{60\!\cdots\!00}a^{24}-\frac{24\!\cdots\!57}{17\!\cdots\!00}a^{23}+\frac{24\!\cdots\!47}{29\!\cdots\!00}a^{22}+\frac{21\!\cdots\!17}{85\!\cdots\!00}a^{21}+\frac{71\!\cdots\!79}{21\!\cdots\!00}a^{20}-\frac{37\!\cdots\!87}{17\!\cdots\!00}a^{19}+\frac{21\!\cdots\!69}{12\!\cdots\!00}a^{18}+\frac{30\!\cdots\!43}{34\!\cdots\!40}a^{17}-\frac{12\!\cdots\!33}{42\!\cdots\!00}a^{16}+\frac{51\!\cdots\!71}{85\!\cdots\!00}a^{15}-\frac{13\!\cdots\!89}{85\!\cdots\!00}a^{14}+\frac{82\!\cdots\!43}{17\!\cdots\!00}a^{13}-\frac{29\!\cdots\!97}{42\!\cdots\!00}a^{12}+\frac{24\!\cdots\!17}{17\!\cdots\!00}a^{11}-\frac{14\!\cdots\!97}{85\!\cdots\!00}a^{10}+\frac{22\!\cdots\!39}{85\!\cdots\!00}a^{9}-\frac{10\!\cdots\!21}{42\!\cdots\!00}a^{8}+\frac{18\!\cdots\!69}{42\!\cdots\!00}a^{7}-\frac{20\!\cdots\!29}{21\!\cdots\!00}a^{6}+\frac{45\!\cdots\!69}{15\!\cdots\!00}a^{5}-\frac{36\!\cdots\!13}{19\!\cdots\!00}a^{4}+\frac{91\!\cdots\!57}{27\!\cdots\!00}a^{3}-\frac{90\!\cdots\!93}{33\!\cdots\!00}a^{2}+\frac{25\!\cdots\!59}{97\!\cdots\!20}a-\frac{19\!\cdots\!49}{12\!\cdots\!50}$, $\frac{58\!\cdots\!19}{68\!\cdots\!48}a^{31}-\frac{17\!\cdots\!23}{60\!\cdots\!00}a^{30}-\frac{20\!\cdots\!63}{34\!\cdots\!40}a^{29}+\frac{38\!\cdots\!19}{21\!\cdots\!64}a^{28}-\frac{17\!\cdots\!43}{85\!\cdots\!00}a^{27}-\frac{78\!\cdots\!49}{85\!\cdots\!00}a^{26}+\frac{62\!\cdots\!63}{24\!\cdots\!00}a^{25}+\frac{28\!\cdots\!13}{85\!\cdots\!60}a^{24}-\frac{30\!\cdots\!53}{17\!\cdots\!00}a^{23}+\frac{27\!\cdots\!27}{29\!\cdots\!00}a^{22}+\frac{40\!\cdots\!39}{12\!\cdots\!00}a^{21}-\frac{55\!\cdots\!67}{30\!\cdots\!00}a^{20}-\frac{31\!\cdots\!71}{17\!\cdots\!00}a^{19}+\frac{18\!\cdots\!63}{85\!\cdots\!60}a^{18}+\frac{13\!\cdots\!87}{17\!\cdots\!00}a^{17}-\frac{10\!\cdots\!07}{37\!\cdots\!00}a^{16}+\frac{86\!\cdots\!83}{17\!\cdots\!20}a^{15}-\frac{91\!\cdots\!41}{85\!\cdots\!00}a^{14}+\frac{58\!\cdots\!79}{17\!\cdots\!00}a^{13}-\frac{89\!\cdots\!41}{17\!\cdots\!12}a^{12}+\frac{89\!\cdots\!33}{17\!\cdots\!00}a^{11}-\frac{76\!\cdots\!13}{85\!\cdots\!00}a^{10}+\frac{38\!\cdots\!51}{85\!\cdots\!00}a^{9}-\frac{18\!\cdots\!01}{42\!\cdots\!00}a^{8}+\frac{12\!\cdots\!29}{42\!\cdots\!00}a^{7}+\frac{62\!\cdots\!09}{92\!\cdots\!00}a^{6}-\frac{53\!\cdots\!63}{15\!\cdots\!00}a^{5}-\frac{22\!\cdots\!35}{76\!\cdots\!88}a^{4}+\frac{52\!\cdots\!27}{38\!\cdots\!00}a^{3}+\frac{88\!\cdots\!97}{33\!\cdots\!00}a^{2}+\frac{23\!\cdots\!39}{37\!\cdots\!00}a+\frac{15\!\cdots\!03}{12\!\cdots\!50}$
| 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: | \( 17203826154694.41 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 17203826154694.41 \cdot 384}{6\cdot\sqrt{11108449995025704343652186204318263487217947546291798016}}\cr\approx \mathstrut & 1.94919900432321 \end{aligned}\] (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 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496)
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 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496, 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 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496);
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 - x^30 + 14*x^29 - 6*x^28 - 82*x^27 + 171*x^26 + 426*x^25 - 1103*x^24 + 612*x^23 + 1870*x^22 + 1692*x^21 - 15101*x^20 + 12498*x^19 + 60665*x^18 - 224090*x^17 + 470294*x^16 - 1196922*x^15 + 3481149*x^14 - 4522026*x^13 + 9950863*x^12 - 12441624*x^11 + 17394098*x^10 - 19804208*x^9 + 30506508*x^8 - 11192944*x^7 + 24538416*x^6 - 26063296*x^5 + 410816*x^4 - 22830080*x^3 + 11063808*x^2 + 9834496);
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 | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(23\)
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(137\)
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 137.4.2.1 | $x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 137.4.2.1 | $x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |