Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 256)
gp: K = bnfinit(y^32 - 2*y^31 - 7*y^30 + 6*y^29 + 41*y^28 + 140*y^27 + 178*y^26 - 2824*y^25 + 684*y^24 + 10324*y^23 + 9232*y^22 - 2516*y^21 - 12401*y^20 - 32042*y^19 - 31601*y^18 + 6402*y^17 + 66991*y^16 + 65484*y^15 - 7418*y^14 - 55672*y^13 - 53412*y^12 - 80560*y^11 + 544*y^10 + 118368*y^9 + 65792*y^8 - 23616*y^7 + 10688*y^6 + 2560*y^5 - 3520*y^4 + 2048*y^3 - 384*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 2*x^31 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 256)
\( x^{32} - 2 x^{31} - 7 x^{30} + 6 x^{29} + 41 x^{28} + 140 x^{27} + 178 x^{26} - 2824 x^{25} + 684 x^{24} + \cdots + 256 \)
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: |
\(2407046967087022733336044808386337832960000000000000000\)
\(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 41^{8}\cdot 113^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.05\) | 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: | $2^{3/2}3^{1/2}5^{1/2}41^{1/2}113^{1/2}\approx 745.6272527208216$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(41\), \(113\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{5}$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{20}-\frac{1}{8}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{9}-\frac{1}{2}a^{6}$, $\frac{1}{16}a^{22}-\frac{1}{16}a^{20}-\frac{1}{16}a^{18}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{16}a^{10}-\frac{1}{4}a^{9}+\frac{1}{16}a^{8}-\frac{1}{2}a^{7}-\frac{7}{16}a^{6}-\frac{1}{2}$, $\frac{1}{16}a^{23}-\frac{1}{16}a^{21}-\frac{1}{16}a^{19}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{16}a^{11}-\frac{1}{4}a^{10}+\frac{1}{16}a^{9}-\frac{7}{16}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{24}-\frac{1}{32}a^{22}-\frac{1}{32}a^{20}-\frac{1}{16}a^{19}+\frac{1}{8}a^{15}+\frac{1}{8}a^{13}+\frac{7}{32}a^{12}-\frac{1}{8}a^{11}+\frac{1}{32}a^{10}-\frac{7}{32}a^{8}-\frac{3}{16}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{25}-\frac{1}{32}a^{23}-\frac{1}{32}a^{21}-\frac{1}{16}a^{20}-\frac{1}{8}a^{16}+\frac{1}{8}a^{14}+\frac{7}{32}a^{13}-\frac{1}{8}a^{12}+\frac{1}{32}a^{11}-\frac{7}{32}a^{9}-\frac{3}{16}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{26}-\frac{1}{64}a^{24}-\frac{1}{64}a^{22}-\frac{1}{32}a^{21}-\frac{1}{16}a^{20}+\frac{1}{16}a^{17}+\frac{1}{16}a^{15}-\frac{9}{64}a^{14}+\frac{3}{16}a^{13}+\frac{1}{64}a^{12}+\frac{9}{64}a^{10}-\frac{3}{32}a^{9}-\frac{1}{16}a^{8}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}-\frac{3}{8}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{27}-\frac{1}{64}a^{25}-\frac{1}{64}a^{23}-\frac{1}{32}a^{22}-\frac{1}{16}a^{21}-\frac{1}{16}a^{18}+\frac{1}{16}a^{16}-\frac{9}{64}a^{15}+\frac{3}{16}a^{14}+\frac{1}{64}a^{13}+\frac{9}{64}a^{11}-\frac{3}{32}a^{10}-\frac{1}{16}a^{9}+\frac{3}{8}a^{7}-\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2178560}a^{28}-\frac{3449}{1089280}a^{27}+\frac{3599}{2178560}a^{26}-\frac{13023}{1089280}a^{25}-\frac{3733}{435712}a^{24}+\frac{7793}{272320}a^{23}+\frac{513}{272320}a^{22}+\frac{1247}{54464}a^{21}-\frac{195}{54464}a^{20}+\frac{29561}{544640}a^{19}-\frac{258}{4255}a^{18}+\frac{31}{23680}a^{17}+\frac{256783}{2178560}a^{16}+\frac{269143}{1089280}a^{15}-\frac{9883}{435712}a^{14}+\frac{110303}{1089280}a^{13}-\frac{94767}{2178560}a^{12}+\frac{6177}{34040}a^{11}+\frac{321}{34040}a^{10}-\frac{6633}{54464}a^{9}-\frac{5227}{27232}a^{8}-\frac{11571}{34040}a^{7}-\frac{26003}{68080}a^{6}+\frac{2897}{8510}a^{5}+\frac{9167}{34040}a^{4}+\frac{1637}{6808}a^{3}+\frac{273}{6808}a^{2}-\frac{7243}{17020}a-\frac{1561}{34040}$, $\frac{1}{4357120}a^{29}+\frac{1823}{871424}a^{27}+\frac{1171}{272320}a^{26}+\frac{47227}{4357120}a^{25}-\frac{8813}{2178560}a^{24}-\frac{9553}{544640}a^{23}+\frac{4749}{544640}a^{22}-\frac{1999}{108928}a^{21}-\frac{9779}{1089280}a^{20}+\frac{23617}{544640}a^{19}+\frac{64921}{1089280}a^{18}-\frac{221801}{4357120}a^{17}+\frac{5675}{217856}a^{16}-\frac{1056987}{4357120}a^{15}-\frac{33157}{136160}a^{14}+\frac{1010541}{4357120}a^{13}+\frac{198361}{2178560}a^{12}-\frac{21881}{136160}a^{11}-\frac{87981}{544640}a^{10}+\frac{5201}{54464}a^{9}-\frac{16327}{136160}a^{8}+\frac{5101}{136160}a^{7}-\frac{28419}{68080}a^{6}-\frac{8259}{68080}a^{5}-\frac{4169}{68080}a^{4}-\frac{3875}{13616}a^{3}-\frac{6949}{17020}a^{2}-\frac{18589}{68080}a-\frac{5569}{34040}$, $\frac{1}{46\!\cdots\!00}a^{30}+\frac{51\!\cdots\!17}{92\!\cdots\!80}a^{29}+\frac{48\!\cdots\!49}{46\!\cdots\!00}a^{28}+\frac{15\!\cdots\!59}{46\!\cdots\!00}a^{27}+\frac{37\!\cdots\!13}{46\!\cdots\!00}a^{26}+\frac{14\!\cdots\!09}{92\!\cdots\!80}a^{25}+\frac{69\!\cdots\!19}{72\!\cdots\!00}a^{24}-\frac{33\!\cdots\!11}{14\!\cdots\!00}a^{23}-\frac{90\!\cdots\!71}{72\!\cdots\!00}a^{22}-\frac{38\!\cdots\!49}{11\!\cdots\!00}a^{21}-\frac{61\!\cdots\!61}{11\!\cdots\!00}a^{20}-\frac{20\!\cdots\!59}{46\!\cdots\!44}a^{19}+\frac{19\!\cdots\!47}{92\!\cdots\!80}a^{18}+\frac{53\!\cdots\!83}{46\!\cdots\!00}a^{17}-\frac{14\!\cdots\!61}{18\!\cdots\!76}a^{16}-\frac{14\!\cdots\!63}{92\!\cdots\!80}a^{15}+\frac{93\!\cdots\!11}{46\!\cdots\!00}a^{14}-\frac{10\!\cdots\!09}{46\!\cdots\!00}a^{13}-\frac{76\!\cdots\!29}{58\!\cdots\!68}a^{12}+\frac{13\!\cdots\!73}{58\!\cdots\!00}a^{11}-\frac{13\!\cdots\!43}{58\!\cdots\!00}a^{10}-\frac{22\!\cdots\!07}{14\!\cdots\!00}a^{9}-\frac{85\!\cdots\!37}{72\!\cdots\!00}a^{8}-\frac{60\!\cdots\!91}{14\!\cdots\!00}a^{7}-\frac{60\!\cdots\!19}{36\!\cdots\!80}a^{6}+\frac{72\!\cdots\!49}{36\!\cdots\!00}a^{5}+\frac{15\!\cdots\!59}{36\!\cdots\!00}a^{4}+\frac{22\!\cdots\!39}{72\!\cdots\!00}a^{3}-\frac{31\!\cdots\!59}{72\!\cdots\!00}a^{2}-\frac{59\!\cdots\!89}{31\!\cdots\!00}a+\frac{94\!\cdots\!97}{90\!\cdots\!50}$, $\frac{1}{10\!\cdots\!00}a^{31}+\frac{750290800782691}{46\!\cdots\!40}a^{30}+\frac{11\!\cdots\!19}{10\!\cdots\!00}a^{29}+\frac{10\!\cdots\!27}{53\!\cdots\!00}a^{28}+\frac{32\!\cdots\!23}{10\!\cdots\!00}a^{27}-\frac{25\!\cdots\!57}{53\!\cdots\!60}a^{26}+\frac{72\!\cdots\!23}{11\!\cdots\!00}a^{25}+\frac{11\!\cdots\!31}{13\!\cdots\!00}a^{24}+\frac{47\!\cdots\!87}{13\!\cdots\!00}a^{23}+\frac{69\!\cdots\!81}{26\!\cdots\!00}a^{22}+\frac{11\!\cdots\!71}{66\!\cdots\!00}a^{21}+\frac{36\!\cdots\!69}{10\!\cdots\!52}a^{20}+\frac{95\!\cdots\!23}{21\!\cdots\!40}a^{19}-\frac{27\!\cdots\!51}{53\!\cdots\!00}a^{18}-\frac{14\!\cdots\!91}{21\!\cdots\!40}a^{17}+\frac{48\!\cdots\!33}{10\!\cdots\!20}a^{16}-\frac{11\!\cdots\!39}{10\!\cdots\!00}a^{15}+\frac{32\!\cdots\!39}{26\!\cdots\!00}a^{14}+\frac{78\!\cdots\!29}{53\!\cdots\!60}a^{13}+\frac{29\!\cdots\!43}{13\!\cdots\!00}a^{12}-\frac{62\!\cdots\!57}{20\!\cdots\!00}a^{11}+\frac{31\!\cdots\!59}{36\!\cdots\!00}a^{10}+\frac{25\!\cdots\!07}{14\!\cdots\!00}a^{9}+\frac{85\!\cdots\!41}{83\!\cdots\!00}a^{8}+\frac{14\!\cdots\!61}{33\!\cdots\!60}a^{7}-\frac{22\!\cdots\!17}{16\!\cdots\!00}a^{6}-\frac{49\!\cdots\!57}{16\!\cdots\!00}a^{5}+\frac{40\!\cdots\!87}{83\!\cdots\!00}a^{4}-\frac{13\!\cdots\!03}{38\!\cdots\!00}a^{3}-\frac{80\!\cdots\!43}{41\!\cdots\!00}a^{2}-\frac{12\!\cdots\!21}{41\!\cdots\!00}a-\frac{73\!\cdots\!21}{20\!\cdots\!10}$
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
$C_{2}\times C_{6}\times C_{24}$, which has order $288$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $288$ (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{162634508511431153864851004851}{37530532616396663825214071684096} a^{31} - \frac{198885165936462469345152133323}{18765266308198331912607035842048} a^{30} - \frac{119809346302913040484045319447}{4691316577049582978151758960512} a^{29} + \frac{695972557180426606206123258485}{18765266308198331912607035842048} a^{28} + \frac{3018722185888677092614874315475}{18765266308198331912607035842048} a^{27} + \frac{5035019663787533673349674951757}{9382633154099165956303517921024} a^{26} + \frac{20068075790220384108020467785401}{37530532616396663825214071684096} a^{25} - \frac{234055626040092590276852613233733}{18765266308198331912607035842048} a^{24} + \frac{19964759671211094267911271293847}{2345658288524791489075879480256} a^{23} + \frac{381911534615092260098527814588989}{9382633154099165956303517921024} a^{22} + \frac{104762562031513048139774638371327}{4691316577049582978151758960512} a^{21} - \frac{93805329454035621457064244649859}{4691316577049582978151758960512} a^{20} - \frac{1718706218379858515253605331463131}{37530532616396663825214071684096} a^{19} - \frac{2269962042196941597825562278268501}{18765266308198331912607035842048} a^{18} - \frac{396233832102907222376046381439563}{4691316577049582978151758960512} a^{17} + \frac{1222053985898160093111236550020741}{18765266308198331912607035842048} a^{16} + \frac{4975460352694821472598674751576129}{18765266308198331912607035842048} a^{15} + \frac{1631255355259766200302116384906477}{9382633154099165956303517921024} a^{14} - \frac{3855037920134173675143862946581545}{37530532616396663825214071684096} a^{13} - \frac{3778165509956202438374708853830303}{18765266308198331912607035842048} a^{12} - \frac{92839581638117785567088974191751}{586414572131197872268969870064} a^{11} - \frac{1351330015665040915337075197344501}{4691316577049582978151758960512} a^{10} + \frac{317628726226036840528892952727629}{2345658288524791489075879480256} a^{9} + \frac{269799912939675499518531908543639}{586414572131197872268969870064} a^{8} + \frac{115000658340096174706691532650167}{1172829144262395744537939740128} a^{7} - \frac{76289919837651117729298077597907}{586414572131197872268969870064} a^{6} + \frac{13497958017207702746424165060569}{146603643032799468067242467516} a^{5} - \frac{30506079473998954764691161239511}{586414572131197872268969870064} a^{4} + \frac{177996299821160330612215028840}{36650910758199867016810616879} a^{3} + \frac{968734577951648090013859062461}{146603643032799468067242467516} a^{2} - \frac{3012649335272807243164960897855}{586414572131197872268969870064} a + \frac{858075443717918190916830032927}{293207286065598936134484935032} \)
(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: |
$\frac{16\!\cdots\!83}{62\!\cdots\!60}a^{31}-\frac{94\!\cdots\!37}{14\!\cdots\!80}a^{30}-\frac{11\!\cdots\!47}{71\!\cdots\!40}a^{29}+\frac{32\!\cdots\!63}{14\!\cdots\!80}a^{28}+\frac{35\!\cdots\!51}{35\!\cdots\!20}a^{27}+\frac{47\!\cdots\!89}{14\!\cdots\!80}a^{26}+\frac{47\!\cdots\!99}{14\!\cdots\!80}a^{25}-\frac{24\!\cdots\!25}{31\!\cdots\!88}a^{24}+\frac{94\!\cdots\!81}{17\!\cdots\!60}a^{23}+\frac{90\!\cdots\!31}{35\!\cdots\!20}a^{22}+\frac{49\!\cdots\!59}{35\!\cdots\!20}a^{21}-\frac{22\!\cdots\!09}{17\!\cdots\!60}a^{20}-\frac{81\!\cdots\!01}{28\!\cdots\!96}a^{19}-\frac{25\!\cdots\!81}{33\!\cdots\!60}a^{18}-\frac{16\!\cdots\!81}{31\!\cdots\!80}a^{17}+\frac{11\!\cdots\!05}{28\!\cdots\!96}a^{16}+\frac{58\!\cdots\!21}{35\!\cdots\!20}a^{15}+\frac{15\!\cdots\!87}{14\!\cdots\!80}a^{14}-\frac{91\!\cdots\!27}{14\!\cdots\!80}a^{13}-\frac{44\!\cdots\!61}{35\!\cdots\!20}a^{12}-\frac{17\!\cdots\!63}{17\!\cdots\!60}a^{11}-\frac{37\!\cdots\!11}{20\!\cdots\!60}a^{10}+\frac{18\!\cdots\!59}{22\!\cdots\!20}a^{9}+\frac{12\!\cdots\!79}{44\!\cdots\!40}a^{8}+\frac{67\!\cdots\!53}{11\!\cdots\!60}a^{7}-\frac{18\!\cdots\!33}{22\!\cdots\!20}a^{6}+\frac{12\!\cdots\!91}{22\!\cdots\!20}a^{5}-\frac{14\!\cdots\!83}{44\!\cdots\!64}a^{4}+\frac{16\!\cdots\!39}{56\!\cdots\!30}a^{3}+\frac{18\!\cdots\!65}{44\!\cdots\!64}a^{2}-\frac{71\!\cdots\!57}{22\!\cdots\!20}a+\frac{66\!\cdots\!17}{56\!\cdots\!30}$, $\frac{43\!\cdots\!43}{11\!\cdots\!00}a^{31}-\frac{39\!\cdots\!41}{56\!\cdots\!00}a^{30}-\frac{59\!\cdots\!21}{22\!\cdots\!00}a^{29}+\frac{16\!\cdots\!79}{11\!\cdots\!00}a^{28}+\frac{32\!\cdots\!97}{22\!\cdots\!00}a^{27}+\frac{65\!\cdots\!39}{11\!\cdots\!00}a^{26}+\frac{19\!\cdots\!71}{22\!\cdots\!00}a^{25}-\frac{59\!\cdots\!09}{56\!\cdots\!00}a^{24}+\frac{33\!\cdots\!23}{28\!\cdots\!00}a^{23}+\frac{19\!\cdots\!63}{56\!\cdots\!92}a^{22}+\frac{23\!\cdots\!73}{56\!\cdots\!20}a^{21}+\frac{11\!\cdots\!19}{56\!\cdots\!00}a^{20}-\frac{52\!\cdots\!27}{22\!\cdots\!80}a^{19}-\frac{19\!\cdots\!37}{14\!\cdots\!00}a^{18}-\frac{40\!\cdots\!27}{22\!\cdots\!00}a^{17}-\frac{16\!\cdots\!01}{22\!\cdots\!80}a^{16}+\frac{41\!\cdots\!31}{22\!\cdots\!00}a^{15}+\frac{35\!\cdots\!21}{11\!\cdots\!00}a^{14}+\frac{41\!\cdots\!81}{22\!\cdots\!00}a^{13}-\frac{22\!\cdots\!09}{56\!\cdots\!00}a^{12}-\frac{36\!\cdots\!43}{14\!\cdots\!80}a^{11}-\frac{14\!\cdots\!01}{28\!\cdots\!00}a^{10}-\frac{28\!\cdots\!67}{14\!\cdots\!00}a^{9}+\frac{40\!\cdots\!83}{14\!\cdots\!80}a^{8}+\frac{22\!\cdots\!09}{70\!\cdots\!00}a^{7}+\frac{36\!\cdots\!27}{14\!\cdots\!00}a^{6}+\frac{90\!\cdots\!31}{35\!\cdots\!00}a^{5}-\frac{58\!\cdots\!19}{94\!\cdots\!00}a^{4}-\frac{39\!\cdots\!37}{70\!\cdots\!40}a^{3}+\frac{31\!\cdots\!97}{17\!\cdots\!50}a^{2}-\frac{45\!\cdots\!41}{35\!\cdots\!00}a-\frac{11\!\cdots\!87}{17\!\cdots\!50}$, $\frac{46\!\cdots\!47}{24\!\cdots\!00}a^{31}-\frac{12\!\cdots\!53}{24\!\cdots\!00}a^{30}-\frac{25\!\cdots\!17}{24\!\cdots\!00}a^{29}+\frac{51\!\cdots\!91}{24\!\cdots\!00}a^{28}+\frac{16\!\cdots\!19}{24\!\cdots\!00}a^{27}+\frac{51\!\cdots\!81}{24\!\cdots\!00}a^{26}+\frac{25\!\cdots\!87}{15\!\cdots\!00}a^{25}-\frac{67\!\cdots\!11}{12\!\cdots\!00}a^{24}+\frac{76\!\cdots\!73}{15\!\cdots\!00}a^{23}+\frac{44\!\cdots\!85}{24\!\cdots\!36}a^{22}+\frac{19\!\cdots\!37}{12\!\cdots\!80}a^{21}-\frac{71\!\cdots\!37}{60\!\cdots\!00}a^{20}-\frac{78\!\cdots\!63}{48\!\cdots\!20}a^{19}-\frac{87\!\cdots\!59}{24\!\cdots\!00}a^{18}-\frac{39\!\cdots\!79}{24\!\cdots\!00}a^{17}+\frac{20\!\cdots\!41}{48\!\cdots\!20}a^{16}+\frac{18\!\cdots\!37}{24\!\cdots\!00}a^{15}+\frac{73\!\cdots\!59}{24\!\cdots\!00}a^{14}-\frac{15\!\cdots\!43}{15\!\cdots\!00}a^{13}-\frac{46\!\cdots\!61}{12\!\cdots\!00}a^{12}-\frac{87\!\cdots\!09}{60\!\cdots\!40}a^{11}-\frac{13\!\cdots\!97}{18\!\cdots\!00}a^{10}+\frac{96\!\cdots\!41}{15\!\cdots\!00}a^{9}+\frac{28\!\cdots\!01}{15\!\cdots\!60}a^{8}-\frac{10\!\cdots\!83}{18\!\cdots\!00}a^{7}+\frac{15\!\cdots\!13}{18\!\cdots\!00}a^{6}+\frac{28\!\cdots\!31}{18\!\cdots\!00}a^{5}-\frac{11\!\cdots\!13}{10\!\cdots\!00}a^{4}+\frac{33\!\cdots\!41}{75\!\cdots\!80}a^{3}-\frac{11\!\cdots\!87}{37\!\cdots\!00}a^{2}-\frac{21\!\cdots\!29}{47\!\cdots\!75}a+\frac{45\!\cdots\!01}{18\!\cdots\!00}$, $\frac{20\!\cdots\!71}{10\!\cdots\!00}a^{31}+\frac{12\!\cdots\!67}{32\!\cdots\!00}a^{30}-\frac{23\!\cdots\!01}{10\!\cdots\!00}a^{29}-\frac{77\!\cdots\!29}{51\!\cdots\!20}a^{28}+\frac{10\!\cdots\!59}{10\!\cdots\!00}a^{27}+\frac{22\!\cdots\!51}{51\!\cdots\!00}a^{26}+\frac{49\!\cdots\!93}{51\!\cdots\!00}a^{25}-\frac{30\!\cdots\!83}{64\!\cdots\!00}a^{24}-\frac{13\!\cdots\!19}{12\!\cdots\!00}a^{23}+\frac{27\!\cdots\!49}{11\!\cdots\!00}a^{22}+\frac{30\!\cdots\!43}{56\!\cdots\!00}a^{21}+\frac{92\!\cdots\!81}{25\!\cdots\!00}a^{20}-\frac{28\!\cdots\!31}{41\!\cdots\!36}a^{19}-\frac{23\!\cdots\!53}{25\!\cdots\!00}a^{18}-\frac{21\!\cdots\!03}{10\!\cdots\!00}a^{17}-\frac{79\!\cdots\!81}{51\!\cdots\!20}a^{16}+\frac{73\!\cdots\!01}{10\!\cdots\!00}a^{15}+\frac{33\!\cdots\!49}{10\!\cdots\!40}a^{14}+\frac{14\!\cdots\!07}{51\!\cdots\!00}a^{13}+\frac{13\!\cdots\!17}{32\!\cdots\!00}a^{12}-\frac{10\!\cdots\!13}{64\!\cdots\!00}a^{11}-\frac{52\!\cdots\!51}{12\!\cdots\!80}a^{10}-\frac{68\!\cdots\!29}{14\!\cdots\!00}a^{9}+\frac{19\!\cdots\!43}{20\!\cdots\!00}a^{8}+\frac{80\!\cdots\!31}{20\!\cdots\!00}a^{7}+\frac{15\!\cdots\!21}{70\!\cdots\!00}a^{6}+\frac{81\!\cdots\!97}{32\!\cdots\!20}a^{5}+\frac{18\!\cdots\!83}{80\!\cdots\!00}a^{4}-\frac{99\!\cdots\!13}{16\!\cdots\!00}a^{3}+\frac{15\!\cdots\!51}{80\!\cdots\!00}a^{2}+\frac{11\!\cdots\!69}{80\!\cdots\!00}a-\frac{22\!\cdots\!37}{20\!\cdots\!00}$, $\frac{52\!\cdots\!49}{53\!\cdots\!00}a^{31}-\frac{98\!\cdots\!43}{53\!\cdots\!00}a^{30}-\frac{34\!\cdots\!59}{53\!\cdots\!00}a^{29}+\frac{21\!\cdots\!19}{53\!\cdots\!00}a^{28}+\frac{38\!\cdots\!61}{10\!\cdots\!20}a^{27}+\frac{76\!\cdots\!01}{53\!\cdots\!00}a^{26}+\frac{14\!\cdots\!93}{66\!\cdots\!00}a^{25}-\frac{47\!\cdots\!51}{17\!\cdots\!00}a^{24}+\frac{15\!\cdots\!31}{33\!\cdots\!60}a^{23}+\frac{11\!\cdots\!67}{13\!\cdots\!00}a^{22}+\frac{59\!\cdots\!61}{57\!\cdots\!00}a^{21}+\frac{64\!\cdots\!33}{13\!\cdots\!00}a^{20}-\frac{61\!\cdots\!17}{10\!\cdots\!20}a^{19}-\frac{78\!\cdots\!11}{23\!\cdots\!00}a^{18}-\frac{23\!\cdots\!69}{53\!\cdots\!00}a^{17}-\frac{37\!\cdots\!31}{21\!\cdots\!04}a^{16}+\frac{24\!\cdots\!39}{53\!\cdots\!00}a^{15}+\frac{40\!\cdots\!91}{53\!\cdots\!00}a^{14}+\frac{29\!\cdots\!39}{66\!\cdots\!00}a^{13}-\frac{18\!\cdots\!39}{20\!\cdots\!00}a^{12}-\frac{41\!\cdots\!51}{66\!\cdots\!00}a^{11}-\frac{26\!\cdots\!59}{20\!\cdots\!00}a^{10}-\frac{15\!\cdots\!89}{33\!\cdots\!60}a^{9}+\frac{11\!\cdots\!93}{16\!\cdots\!00}a^{8}+\frac{32\!\cdots\!17}{41\!\cdots\!00}a^{7}+\frac{51\!\cdots\!07}{83\!\cdots\!00}a^{6}+\frac{25\!\cdots\!89}{41\!\cdots\!00}a^{5}-\frac{12\!\cdots\!73}{83\!\cdots\!00}a^{4}-\frac{11\!\cdots\!03}{83\!\cdots\!00}a^{3}+\frac{36\!\cdots\!49}{83\!\cdots\!00}a^{2}+\frac{37\!\cdots\!15}{48\!\cdots\!97}a-\frac{81\!\cdots\!93}{51\!\cdots\!75}$, $\frac{48\!\cdots\!39}{23\!\cdots\!00}a^{31}-\frac{80\!\cdots\!57}{13\!\cdots\!00}a^{30}-\frac{27\!\cdots\!71}{26\!\cdots\!00}a^{29}+\frac{16\!\cdots\!27}{66\!\cdots\!00}a^{28}+\frac{46\!\cdots\!43}{66\!\cdots\!00}a^{27}+\frac{58\!\cdots\!03}{26\!\cdots\!00}a^{26}+\frac{65\!\cdots\!17}{53\!\cdots\!00}a^{25}-\frac{71\!\cdots\!07}{11\!\cdots\!00}a^{24}+\frac{11\!\cdots\!49}{16\!\cdots\!00}a^{23}+\frac{10\!\cdots\!47}{53\!\cdots\!76}a^{22}+\frac{26\!\cdots\!33}{66\!\cdots\!20}a^{21}-\frac{56\!\cdots\!53}{33\!\cdots\!00}a^{20}-\frac{23\!\cdots\!33}{10\!\cdots\!20}a^{19}-\frac{51\!\cdots\!21}{13\!\cdots\!00}a^{18}-\frac{20\!\cdots\!77}{26\!\cdots\!00}a^{17}+\frac{15\!\cdots\!39}{26\!\cdots\!80}a^{16}+\frac{15\!\cdots\!91}{16\!\cdots\!00}a^{15}+\frac{23\!\cdots\!79}{11\!\cdots\!00}a^{14}-\frac{62\!\cdots\!13}{53\!\cdots\!00}a^{13}-\frac{13\!\cdots\!11}{26\!\cdots\!00}a^{12}-\frac{69\!\cdots\!71}{33\!\cdots\!60}a^{11}-\frac{39\!\cdots\!27}{66\!\cdots\!00}a^{10}+\frac{35\!\cdots\!41}{33\!\cdots\!00}a^{9}+\frac{65\!\cdots\!71}{33\!\cdots\!60}a^{8}-\frac{44\!\cdots\!09}{72\!\cdots\!00}a^{7}+\frac{10\!\cdots\!51}{83\!\cdots\!00}a^{6}+\frac{61\!\cdots\!81}{41\!\cdots\!00}a^{5}-\frac{25\!\cdots\!63}{22\!\cdots\!00}a^{4}+\frac{41\!\cdots\!93}{83\!\cdots\!40}a^{3}-\frac{17\!\cdots\!31}{41\!\cdots\!00}a^{2}-\frac{33\!\cdots\!07}{83\!\cdots\!00}a-\frac{82\!\cdots\!49}{41\!\cdots\!00}$, $\frac{65\!\cdots\!13}{10\!\cdots\!00}a^{31}-\frac{29\!\cdots\!83}{26\!\cdots\!00}a^{30}-\frac{44\!\cdots\!73}{10\!\cdots\!00}a^{29}+\frac{14\!\cdots\!49}{66\!\cdots\!00}a^{28}+\frac{24\!\cdots\!71}{10\!\cdots\!00}a^{27}+\frac{49\!\cdots\!67}{53\!\cdots\!00}a^{26}+\frac{81\!\cdots\!87}{57\!\cdots\!00}a^{25}-\frac{44\!\cdots\!07}{26\!\cdots\!00}a^{24}+\frac{20\!\cdots\!29}{13\!\cdots\!00}a^{23}+\frac{28\!\cdots\!61}{53\!\cdots\!60}a^{22}+\frac{18\!\cdots\!87}{26\!\cdots\!80}a^{21}+\frac{91\!\cdots\!17}{26\!\cdots\!00}a^{20}-\frac{77\!\cdots\!53}{21\!\cdots\!40}a^{19}-\frac{29\!\cdots\!07}{13\!\cdots\!00}a^{18}-\frac{30\!\cdots\!71}{10\!\cdots\!00}a^{17}-\frac{20\!\cdots\!57}{16\!\cdots\!80}a^{16}+\frac{31\!\cdots\!33}{10\!\cdots\!00}a^{15}+\frac{26\!\cdots\!93}{53\!\cdots\!00}a^{14}+\frac{20\!\cdots\!53}{66\!\cdots\!00}a^{13}-\frac{71\!\cdots\!19}{11\!\cdots\!00}a^{12}-\frac{55\!\cdots\!77}{13\!\cdots\!40}a^{11}-\frac{17\!\cdots\!37}{20\!\cdots\!00}a^{10}-\frac{27\!\cdots\!07}{83\!\cdots\!00}a^{9}+\frac{74\!\cdots\!23}{16\!\cdots\!80}a^{8}+\frac{87\!\cdots\!81}{16\!\cdots\!00}a^{7}+\frac{69\!\cdots\!99}{16\!\cdots\!00}a^{6}+\frac{69\!\cdots\!33}{16\!\cdots\!00}a^{5}-\frac{10\!\cdots\!79}{10\!\cdots\!50}a^{4}-\frac{30\!\cdots\!77}{33\!\cdots\!60}a^{3}+\frac{24\!\cdots\!31}{83\!\cdots\!00}a^{2}-\frac{21\!\cdots\!23}{10\!\cdots\!50}a-\frac{27\!\cdots\!91}{18\!\cdots\!00}$, $\frac{32\!\cdots\!57}{53\!\cdots\!00}a^{31}-\frac{34\!\cdots\!43}{26\!\cdots\!00}a^{30}-\frac{22\!\cdots\!57}{53\!\cdots\!00}a^{29}+\frac{47\!\cdots\!79}{11\!\cdots\!00}a^{28}+\frac{13\!\cdots\!67}{53\!\cdots\!00}a^{27}+\frac{55\!\cdots\!79}{66\!\cdots\!00}a^{26}+\frac{57\!\cdots\!41}{57\!\cdots\!00}a^{25}-\frac{46\!\cdots\!01}{26\!\cdots\!80}a^{24}+\frac{39\!\cdots\!03}{66\!\cdots\!00}a^{23}+\frac{19\!\cdots\!51}{30\!\cdots\!00}a^{22}+\frac{17\!\cdots\!73}{33\!\cdots\!00}a^{21}-\frac{31\!\cdots\!99}{13\!\cdots\!00}a^{20}-\frac{86\!\cdots\!57}{10\!\cdots\!20}a^{19}-\frac{50\!\cdots\!27}{26\!\cdots\!00}a^{18}-\frac{89\!\cdots\!03}{53\!\cdots\!00}a^{17}+\frac{37\!\cdots\!39}{53\!\cdots\!60}a^{16}+\frac{22\!\cdots\!37}{53\!\cdots\!00}a^{15}+\frac{12\!\cdots\!41}{33\!\cdots\!00}a^{14}-\frac{14\!\cdots\!49}{13\!\cdots\!00}a^{13}-\frac{50\!\cdots\!43}{13\!\cdots\!00}a^{12}-\frac{26\!\cdots\!33}{83\!\cdots\!00}a^{11}-\frac{36\!\cdots\!91}{83\!\cdots\!00}a^{10}+\frac{35\!\cdots\!43}{38\!\cdots\!00}a^{9}+\frac{17\!\cdots\!63}{22\!\cdots\!00}a^{8}+\frac{35\!\cdots\!53}{97\!\cdots\!00}a^{7}-\frac{91\!\cdots\!17}{41\!\cdots\!00}a^{6}+\frac{93\!\cdots\!43}{83\!\cdots\!00}a^{5}-\frac{73\!\cdots\!67}{83\!\cdots\!40}a^{4}-\frac{81\!\cdots\!97}{83\!\cdots\!00}a^{3}+\frac{19\!\cdots\!31}{20\!\cdots\!10}a^{2}+\frac{10\!\cdots\!21}{20\!\cdots\!00}a+\frac{50\!\cdots\!31}{20\!\cdots\!00}$, $\frac{36\!\cdots\!13}{36\!\cdots\!40}a^{31}-\frac{46\!\cdots\!27}{23\!\cdots\!00}a^{30}-\frac{52\!\cdots\!25}{73\!\cdots\!28}a^{29}+\frac{30\!\cdots\!29}{46\!\cdots\!00}a^{28}+\frac{18\!\cdots\!17}{42\!\cdots\!00}a^{27}+\frac{12\!\cdots\!01}{92\!\cdots\!00}a^{26}+\frac{92\!\cdots\!29}{57\!\cdots\!60}a^{25}-\frac{13\!\cdots\!09}{46\!\cdots\!00}a^{24}+\frac{16\!\cdots\!29}{23\!\cdots\!00}a^{23}+\frac{50\!\cdots\!81}{46\!\cdots\!00}a^{22}+\frac{20\!\cdots\!97}{23\!\cdots\!00}a^{21}-\frac{25\!\cdots\!99}{46\!\cdots\!00}a^{20}-\frac{55\!\cdots\!73}{36\!\cdots\!40}a^{19}-\frac{28\!\cdots\!39}{92\!\cdots\!60}a^{18}-\frac{48\!\cdots\!23}{18\!\cdots\!00}a^{17}+\frac{14\!\cdots\!31}{92\!\cdots\!60}a^{16}+\frac{11\!\cdots\!23}{16\!\cdots\!80}a^{15}+\frac{55\!\cdots\!07}{92\!\cdots\!00}a^{14}-\frac{28\!\cdots\!99}{10\!\cdots\!00}a^{13}-\frac{68\!\cdots\!31}{92\!\cdots\!60}a^{12}-\frac{56\!\cdots\!99}{11\!\cdots\!00}a^{11}-\frac{34\!\cdots\!03}{57\!\cdots\!00}a^{10}+\frac{28\!\cdots\!63}{14\!\cdots\!00}a^{9}+\frac{97\!\cdots\!43}{72\!\cdots\!00}a^{8}+\frac{17\!\cdots\!13}{28\!\cdots\!00}a^{7}-\frac{34\!\cdots\!97}{57\!\cdots\!60}a^{6}-\frac{31\!\cdots\!03}{28\!\cdots\!00}a^{5}+\frac{17\!\cdots\!97}{18\!\cdots\!50}a^{4}+\frac{15\!\cdots\!91}{28\!\cdots\!00}a^{3}+\frac{15\!\cdots\!57}{14\!\cdots\!00}a^{2}-\frac{30\!\cdots\!31}{36\!\cdots\!00}a+\frac{46\!\cdots\!61}{72\!\cdots\!00}$, $\frac{47\!\cdots\!07}{83\!\cdots\!00}a^{31}-\frac{57\!\cdots\!17}{53\!\cdots\!00}a^{30}-\frac{10\!\cdots\!69}{26\!\cdots\!00}a^{29}+\frac{12\!\cdots\!59}{53\!\cdots\!00}a^{28}+\frac{24\!\cdots\!41}{11\!\cdots\!00}a^{27}+\frac{44\!\cdots\!99}{53\!\cdots\!00}a^{26}+\frac{82\!\cdots\!51}{66\!\cdots\!00}a^{25}-\frac{41\!\cdots\!39}{26\!\cdots\!00}a^{24}+\frac{18\!\cdots\!79}{66\!\cdots\!00}a^{23}+\frac{66\!\cdots\!71}{13\!\cdots\!40}a^{22}+\frac{31\!\cdots\!75}{53\!\cdots\!76}a^{21}+\frac{11\!\cdots\!21}{41\!\cdots\!00}a^{20}-\frac{95\!\cdots\!91}{26\!\cdots\!80}a^{19}-\frac{10\!\cdots\!71}{53\!\cdots\!00}a^{18}-\frac{67\!\cdots\!73}{26\!\cdots\!00}a^{17}-\frac{10\!\cdots\!71}{10\!\cdots\!20}a^{16}+\frac{31\!\cdots\!43}{11\!\cdots\!00}a^{15}+\frac{23\!\cdots\!81}{53\!\cdots\!00}a^{14}+\frac{42\!\cdots\!69}{16\!\cdots\!00}a^{13}-\frac{15\!\cdots\!09}{26\!\cdots\!00}a^{12}-\frac{49\!\cdots\!27}{13\!\cdots\!40}a^{11}-\frac{30\!\cdots\!43}{41\!\cdots\!00}a^{10}-\frac{92\!\cdots\!21}{33\!\cdots\!00}a^{9}+\frac{27\!\cdots\!37}{66\!\cdots\!72}a^{8}+\frac{37\!\cdots\!71}{83\!\cdots\!00}a^{7}+\frac{75\!\cdots\!31}{20\!\cdots\!00}a^{6}+\frac{30\!\cdots\!03}{83\!\cdots\!00}a^{5}-\frac{72\!\cdots\!09}{83\!\cdots\!00}a^{4}-\frac{32\!\cdots\!67}{41\!\cdots\!20}a^{3}+\frac{21\!\cdots\!87}{83\!\cdots\!00}a^{2}-\frac{19\!\cdots\!71}{10\!\cdots\!50}a-\frac{57\!\cdots\!71}{41\!\cdots\!00}$, $\frac{33\!\cdots\!79}{26\!\cdots\!00}a^{31}-\frac{78\!\cdots\!49}{53\!\cdots\!00}a^{30}-\frac{61\!\cdots\!43}{53\!\cdots\!00}a^{29}+\frac{45\!\cdots\!61}{53\!\cdots\!00}a^{28}+\frac{33\!\cdots\!63}{53\!\cdots\!00}a^{27}+\frac{11\!\cdots\!53}{53\!\cdots\!00}a^{26}+\frac{18\!\cdots\!23}{53\!\cdots\!00}a^{25}-\frac{18\!\cdots\!21}{53\!\cdots\!60}a^{24}-\frac{32\!\cdots\!93}{14\!\cdots\!00}a^{23}+\frac{10\!\cdots\!51}{66\!\cdots\!00}a^{22}+\frac{30\!\cdots\!33}{13\!\cdots\!00}a^{21}+\frac{51\!\cdots\!27}{35\!\cdots\!00}a^{20}-\frac{25\!\cdots\!97}{10\!\cdots\!52}a^{19}-\frac{28\!\cdots\!91}{53\!\cdots\!00}a^{18}-\frac{35\!\cdots\!17}{53\!\cdots\!00}a^{17}-\frac{10\!\cdots\!33}{10\!\cdots\!20}a^{16}+\frac{58\!\cdots\!73}{53\!\cdots\!00}a^{15}+\frac{81\!\cdots\!39}{53\!\cdots\!00}a^{14}+\frac{36\!\cdots\!73}{14\!\cdots\!00}a^{13}-\frac{31\!\cdots\!19}{26\!\cdots\!00}a^{12}-\frac{83\!\cdots\!71}{66\!\cdots\!00}a^{11}-\frac{82\!\cdots\!27}{66\!\cdots\!00}a^{10}-\frac{18\!\cdots\!53}{33\!\cdots\!00}a^{9}+\frac{31\!\cdots\!33}{16\!\cdots\!00}a^{8}+\frac{36\!\cdots\!69}{16\!\cdots\!00}a^{7}-\frac{94\!\cdots\!53}{41\!\cdots\!00}a^{6}-\frac{25\!\cdots\!69}{41\!\cdots\!00}a^{5}+\frac{26\!\cdots\!79}{83\!\cdots\!40}a^{4}+\frac{62\!\cdots\!97}{83\!\cdots\!00}a^{3}-\frac{16\!\cdots\!39}{16\!\cdots\!80}a^{2}+\frac{17\!\cdots\!91}{83\!\cdots\!00}a+\frac{49\!\cdots\!23}{41\!\cdots\!00}$, $\frac{14\!\cdots\!79}{53\!\cdots\!00}a^{31}-\frac{33\!\cdots\!27}{53\!\cdots\!60}a^{30}-\frac{57\!\cdots\!99}{33\!\cdots\!00}a^{29}+\frac{29\!\cdots\!49}{13\!\cdots\!00}a^{28}+\frac{28\!\cdots\!81}{26\!\cdots\!00}a^{27}+\frac{18\!\cdots\!57}{53\!\cdots\!60}a^{26}+\frac{49\!\cdots\!17}{14\!\cdots\!00}a^{25}-\frac{10\!\cdots\!47}{13\!\cdots\!00}a^{24}+\frac{13\!\cdots\!29}{33\!\cdots\!00}a^{23}+\frac{36\!\cdots\!49}{13\!\cdots\!00}a^{22}+\frac{11\!\cdots\!73}{66\!\cdots\!00}a^{21}-\frac{63\!\cdots\!41}{35\!\cdots\!20}a^{20}-\frac{41\!\cdots\!23}{10\!\cdots\!20}a^{19}-\frac{58\!\cdots\!37}{71\!\cdots\!00}a^{18}-\frac{14\!\cdots\!23}{26\!\cdots\!88}a^{17}+\frac{49\!\cdots\!31}{83\!\cdots\!40}a^{16}+\frac{14\!\cdots\!71}{71\!\cdots\!00}a^{15}+\frac{36\!\cdots\!87}{26\!\cdots\!00}a^{14}-\frac{96\!\cdots\!93}{10\!\cdots\!20}a^{13}-\frac{25\!\cdots\!31}{13\!\cdots\!00}a^{12}-\frac{45\!\cdots\!41}{33\!\cdots\!00}a^{11}-\frac{11\!\cdots\!77}{66\!\cdots\!00}a^{10}+\frac{91\!\cdots\!87}{83\!\cdots\!00}a^{9}+\frac{72\!\cdots\!01}{19\!\cdots\!00}a^{8}+\frac{43\!\cdots\!93}{33\!\cdots\!60}a^{7}-\frac{14\!\cdots\!11}{10\!\cdots\!50}a^{6}-\frac{10\!\cdots\!69}{41\!\cdots\!00}a^{5}-\frac{38\!\cdots\!29}{83\!\cdots\!00}a^{4}-\frac{91\!\cdots\!63}{41\!\cdots\!00}a^{3}-\frac{16\!\cdots\!59}{41\!\cdots\!00}a^{2}-\frac{40\!\cdots\!31}{83\!\cdots\!00}a-\frac{29\!\cdots\!01}{41\!\cdots\!20}$, $\frac{19\!\cdots\!69}{10\!\cdots\!00}a^{31}-\frac{10\!\cdots\!67}{53\!\cdots\!00}a^{30}+\frac{19\!\cdots\!41}{10\!\cdots\!00}a^{29}+\frac{25\!\cdots\!21}{21\!\cdots\!04}a^{28}-\frac{24\!\cdots\!59}{10\!\cdots\!00}a^{27}-\frac{51\!\cdots\!89}{13\!\cdots\!00}a^{26}-\frac{95\!\cdots\!03}{53\!\cdots\!00}a^{25}-\frac{51\!\cdots\!17}{66\!\cdots\!00}a^{24}+\frac{59\!\cdots\!29}{13\!\cdots\!00}a^{23}+\frac{22\!\cdots\!13}{26\!\cdots\!00}a^{22}-\frac{49\!\cdots\!01}{33\!\cdots\!00}a^{21}-\frac{37\!\cdots\!61}{26\!\cdots\!00}a^{20}+\frac{82\!\cdots\!07}{21\!\cdots\!40}a^{19}+\frac{80\!\cdots\!41}{53\!\cdots\!00}a^{18}+\frac{45\!\cdots\!43}{10\!\cdots\!00}a^{17}+\frac{49\!\cdots\!63}{10\!\cdots\!20}a^{16}-\frac{51\!\cdots\!01}{10\!\cdots\!00}a^{15}-\frac{34\!\cdots\!81}{35\!\cdots\!20}a^{14}-\frac{52\!\cdots\!77}{53\!\cdots\!00}a^{13}+\frac{11\!\cdots\!21}{66\!\cdots\!00}a^{12}+\frac{29\!\cdots\!59}{33\!\cdots\!00}a^{11}+\frac{85\!\cdots\!49}{13\!\cdots\!40}a^{10}+\frac{36\!\cdots\!87}{33\!\cdots\!00}a^{9}+\frac{66\!\cdots\!93}{83\!\cdots\!00}a^{8}-\frac{75\!\cdots\!59}{40\!\cdots\!00}a^{7}-\frac{17\!\cdots\!13}{16\!\cdots\!00}a^{6}+\frac{21\!\cdots\!01}{33\!\cdots\!60}a^{5}-\frac{57\!\cdots\!89}{41\!\cdots\!00}a^{4}-\frac{15\!\cdots\!07}{16\!\cdots\!00}a^{3}+\frac{83\!\cdots\!53}{10\!\cdots\!50}a^{2}-\frac{12\!\cdots\!79}{83\!\cdots\!00}a-\frac{33\!\cdots\!13}{20\!\cdots\!00}$, $\frac{13\!\cdots\!89}{66\!\cdots\!00}a^{31}-\frac{98\!\cdots\!27}{26\!\cdots\!00}a^{30}-\frac{41\!\cdots\!41}{26\!\cdots\!00}a^{29}+\frac{25\!\cdots\!31}{26\!\cdots\!00}a^{28}+\frac{94\!\cdots\!45}{10\!\cdots\!52}a^{27}+\frac{82\!\cdots\!59}{26\!\cdots\!00}a^{26}+\frac{11\!\cdots\!11}{26\!\cdots\!00}a^{25}-\frac{77\!\cdots\!07}{13\!\cdots\!00}a^{24}+\frac{10\!\cdots\!37}{66\!\cdots\!20}a^{23}+\frac{22\!\cdots\!97}{10\!\cdots\!50}a^{22}+\frac{15\!\cdots\!47}{66\!\cdots\!00}a^{21}-\frac{16\!\cdots\!83}{66\!\cdots\!00}a^{20}-\frac{18\!\cdots\!33}{66\!\cdots\!20}a^{19}-\frac{82\!\cdots\!39}{11\!\cdots\!00}a^{18}-\frac{20\!\cdots\!51}{26\!\cdots\!00}a^{17}+\frac{21\!\cdots\!21}{53\!\cdots\!60}a^{16}+\frac{16\!\cdots\!17}{11\!\cdots\!00}a^{15}+\frac{43\!\cdots\!49}{26\!\cdots\!00}a^{14}+\frac{67\!\cdots\!33}{26\!\cdots\!00}a^{13}-\frac{17\!\cdots\!53}{13\!\cdots\!00}a^{12}-\frac{56\!\cdots\!73}{41\!\cdots\!00}a^{11}-\frac{59\!\cdots\!67}{33\!\cdots\!00}a^{10}-\frac{82\!\cdots\!79}{33\!\cdots\!60}a^{9}+\frac{18\!\cdots\!03}{72\!\cdots\!00}a^{8}+\frac{15\!\cdots\!87}{83\!\cdots\!00}a^{7}-\frac{29\!\cdots\!29}{83\!\cdots\!00}a^{6}-\frac{13\!\cdots\!63}{41\!\cdots\!00}a^{5}+\frac{72\!\cdots\!03}{48\!\cdots\!00}a^{4}-\frac{57\!\cdots\!39}{18\!\cdots\!00}a^{3}-\frac{35\!\cdots\!59}{41\!\cdots\!00}a^{2}-\frac{13\!\cdots\!17}{83\!\cdots\!40}a+\frac{31\!\cdots\!59}{20\!\cdots\!00}$, $\frac{15\!\cdots\!99}{53\!\cdots\!00}a^{31}-\frac{89\!\cdots\!43}{14\!\cdots\!00}a^{30}-\frac{75\!\cdots\!79}{53\!\cdots\!00}a^{29}+\frac{40\!\cdots\!17}{53\!\cdots\!00}a^{28}+\frac{38\!\cdots\!73}{53\!\cdots\!00}a^{27}+\frac{23\!\cdots\!87}{53\!\cdots\!00}a^{26}+\frac{21\!\cdots\!77}{26\!\cdots\!00}a^{25}-\frac{20\!\cdots\!87}{26\!\cdots\!00}a^{24}+\frac{12\!\cdots\!51}{33\!\cdots\!00}a^{23}+\frac{66\!\cdots\!41}{53\!\cdots\!76}a^{22}+\frac{70\!\cdots\!91}{26\!\cdots\!80}a^{21}+\frac{81\!\cdots\!21}{13\!\cdots\!00}a^{20}+\frac{81\!\cdots\!49}{21\!\cdots\!04}a^{19}-\frac{57\!\cdots\!93}{53\!\cdots\!00}a^{18}-\frac{99\!\cdots\!13}{53\!\cdots\!00}a^{17}-\frac{42\!\cdots\!89}{21\!\cdots\!04}a^{16}-\frac{13\!\cdots\!61}{53\!\cdots\!00}a^{15}+\frac{12\!\cdots\!73}{53\!\cdots\!00}a^{14}+\frac{12\!\cdots\!27}{26\!\cdots\!00}a^{13}+\frac{90\!\cdots\!43}{26\!\cdots\!00}a^{12}-\frac{29\!\cdots\!39}{13\!\cdots\!40}a^{11}-\frac{10\!\cdots\!79}{14\!\cdots\!00}a^{10}-\frac{14\!\cdots\!43}{33\!\cdots\!00}a^{9}-\frac{83\!\cdots\!03}{66\!\cdots\!72}a^{8}+\frac{17\!\cdots\!23}{83\!\cdots\!00}a^{7}+\frac{66\!\cdots\!07}{83\!\cdots\!00}a^{6}+\frac{24\!\cdots\!17}{41\!\cdots\!00}a^{5}-\frac{23\!\cdots\!77}{83\!\cdots\!00}a^{4}-\frac{57\!\cdots\!53}{33\!\cdots\!36}a^{3}+\frac{38\!\cdots\!71}{83\!\cdots\!00}a^{2}+\frac{86\!\cdots\!33}{41\!\cdots\!00}a+\frac{10\!\cdots\!17}{41\!\cdots\!00}$
| 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: | \( 9642731609916.342 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 9642731609916.342 \cdot 288}{6\cdot\sqrt{2407046967087022733336044808386337832960000000000000000}}\cr\approx \mathstrut & 1.76026043651467 \end{aligned}\] (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 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 256)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 - 2*x^31 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 256, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 2*x^31 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 256);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 - 7*x^30 + 6*x^29 + 41*x^28 + 140*x^27 + 178*x^26 - 2824*x^25 + 684*x^24 + 10324*x^23 + 9232*x^22 - 2516*x^21 - 12401*x^20 - 32042*x^19 - 31601*x^18 + 6402*x^17 + 66991*x^16 + 65484*x^15 - 7418*x^14 - 55672*x^13 - 53412*x^12 - 80560*x^11 + 544*x^10 + 118368*x^9 + 65792*x^8 - 23616*x^7 + 10688*x^6 + 2560*x^5 - 3520*x^4 + 2048*x^3 - 384*x^2 + 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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.8.0.1}{8} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{8}$ | 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 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(41\)
| 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 | $x^{4} + 18960 x^{3} + 90817911 x^{2} + 8982404280 x + 374946100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |