Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1)
gp: K = bnfinit(y^40 - 20*y^38 + 231*y^36 - 1812*y^34 + 10709*y^32 - 49280*y^30 + 181674*y^28 - 540148*y^26 + 1304886*y^24 - 2544812*y^22 + 3994121*y^20 - 4954644*y^18 + 4817692*y^16 - 3548680*y^14 + 1959963*y^12 - 753104*y^10 + 202622*y^8 - 30356*y^6 + 3025*y^4 - 60*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1)
\( x^{40} - 20 x^{38} + 231 x^{36} - 1812 x^{34} + 10709 x^{32} - 49280 x^{30} + 181674 x^{28} - 540148 x^{26} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(130305099804548492884220428175380349368393046678311823693003457545895936\)
\(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.96\) | 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: | $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(264=2^{3}\cdot 3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(263,·)$, $\chi_{264}(137,·)$, $\chi_{264}(13,·)$, $\chi_{264}(149,·)$, $\chi_{264}(151,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(29,·)$, $\chi_{264}(163,·)$, $\chi_{264}(167,·)$, $\chi_{264}(169,·)$, $\chi_{264}(7,·)$, $\chi_{264}(257,·)$, $\chi_{264}(173,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(185,·)$, $\chi_{264}(59,·)$, $\chi_{264}(61,·)$, $\chi_{264}(67,·)$, $\chi_{264}(197,·)$, $\chi_{264}(203,·)$, $\chi_{264}(205,·)$, $\chi_{264}(79,·)$, $\chi_{264}(85,·)$, $\chi_{264}(215,·)$, $\chi_{264}(89,·)$, $\chi_{264}(91,·)$, $\chi_{264}(95,·)$, $\chi_{264}(97,·)$, $\chi_{264}(101,·)$, $\chi_{264}(235,·)$, $\chi_{264}(109,·)$, $\chi_{264}(239,·)$, $\chi_{264}(113,·)$, $\chi_{264}(115,·)$, $\chi_{264}(251,·)$, $\chi_{264}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{43}a^{32}-\frac{7}{43}a^{30}+\frac{11}{43}a^{28}-\frac{14}{43}a^{26}+\frac{2}{43}a^{24}+\frac{16}{43}a^{22}+\frac{6}{43}a^{20}+\frac{4}{43}a^{18}-\frac{21}{43}a^{16}+\frac{14}{43}a^{14}-\frac{16}{43}a^{12}+\frac{10}{43}a^{10}-\frac{20}{43}a^{8}-\frac{8}{43}a^{6}-\frac{18}{43}a^{4}+\frac{11}{43}a^{2}-\frac{20}{43}$, $\frac{1}{43}a^{33}-\frac{7}{43}a^{31}+\frac{11}{43}a^{29}-\frac{14}{43}a^{27}+\frac{2}{43}a^{25}+\frac{16}{43}a^{23}+\frac{6}{43}a^{21}+\frac{4}{43}a^{19}-\frac{21}{43}a^{17}+\frac{14}{43}a^{15}-\frac{16}{43}a^{13}+\frac{10}{43}a^{11}-\frac{20}{43}a^{9}-\frac{8}{43}a^{7}-\frac{18}{43}a^{5}+\frac{11}{43}a^{3}-\frac{20}{43}a$, $\frac{1}{43}a^{34}+\frac{5}{43}a^{30}+\frac{20}{43}a^{28}-\frac{10}{43}a^{26}-\frac{13}{43}a^{24}-\frac{11}{43}a^{22}+\frac{3}{43}a^{20}+\frac{7}{43}a^{18}-\frac{4}{43}a^{16}-\frac{4}{43}a^{14}-\frac{16}{43}a^{12}+\frac{7}{43}a^{10}-\frac{19}{43}a^{8}+\frac{12}{43}a^{6}+\frac{14}{43}a^{4}+\frac{14}{43}a^{2}-\frac{11}{43}$, $\frac{1}{43}a^{35}+\frac{5}{43}a^{31}+\frac{20}{43}a^{29}-\frac{10}{43}a^{27}-\frac{13}{43}a^{25}-\frac{11}{43}a^{23}+\frac{3}{43}a^{21}+\frac{7}{43}a^{19}-\frac{4}{43}a^{17}-\frac{4}{43}a^{15}-\frac{16}{43}a^{13}+\frac{7}{43}a^{11}-\frac{19}{43}a^{9}+\frac{12}{43}a^{7}+\frac{14}{43}a^{5}+\frac{14}{43}a^{3}-\frac{11}{43}a$, $\frac{1}{43}a^{36}+\frac{12}{43}a^{30}+\frac{21}{43}a^{28}+\frac{14}{43}a^{26}-\frac{21}{43}a^{24}+\frac{9}{43}a^{22}+\frac{20}{43}a^{20}+\frac{19}{43}a^{18}+\frac{15}{43}a^{16}+\frac{1}{43}a^{12}+\frac{17}{43}a^{10}-\frac{17}{43}a^{8}+\frac{11}{43}a^{6}+\frac{18}{43}a^{4}+\frac{20}{43}a^{2}+\frac{14}{43}$, $\frac{1}{43}a^{37}+\frac{12}{43}a^{31}+\frac{21}{43}a^{29}+\frac{14}{43}a^{27}-\frac{21}{43}a^{25}+\frac{9}{43}a^{23}+\frac{20}{43}a^{21}+\frac{19}{43}a^{19}+\frac{15}{43}a^{17}+\frac{1}{43}a^{13}+\frac{17}{43}a^{11}-\frac{17}{43}a^{9}+\frac{11}{43}a^{7}+\frac{18}{43}a^{5}+\frac{20}{43}a^{3}+\frac{14}{43}a$, $\frac{1}{22\!\cdots\!29}a^{38}+\frac{23\!\cdots\!28}{22\!\cdots\!29}a^{36}-\frac{28\!\cdots\!32}{22\!\cdots\!29}a^{34}+\frac{47\!\cdots\!21}{22\!\cdots\!29}a^{32}-\frac{10\!\cdots\!91}{22\!\cdots\!29}a^{30}-\frac{24\!\cdots\!38}{97\!\cdots\!23}a^{28}-\frac{30\!\cdots\!91}{22\!\cdots\!29}a^{26}-\frac{11\!\cdots\!15}{22\!\cdots\!29}a^{24}-\frac{91\!\cdots\!99}{22\!\cdots\!29}a^{22}+\frac{77\!\cdots\!31}{22\!\cdots\!29}a^{20}+\frac{85\!\cdots\!52}{22\!\cdots\!29}a^{18}+\frac{29\!\cdots\!33}{22\!\cdots\!29}a^{16}+\frac{10\!\cdots\!59}{22\!\cdots\!29}a^{14}+\frac{23\!\cdots\!62}{22\!\cdots\!29}a^{12}-\frac{78\!\cdots\!19}{22\!\cdots\!29}a^{10}+\frac{78\!\cdots\!30}{22\!\cdots\!29}a^{8}+\frac{11\!\cdots\!43}{22\!\cdots\!29}a^{6}+\frac{31\!\cdots\!92}{22\!\cdots\!29}a^{4}+\frac{31\!\cdots\!25}{22\!\cdots\!29}a^{2}+\frac{47\!\cdots\!98}{22\!\cdots\!29}$, $\frac{1}{22\!\cdots\!29}a^{39}+\frac{23\!\cdots\!28}{22\!\cdots\!29}a^{37}-\frac{28\!\cdots\!32}{22\!\cdots\!29}a^{35}+\frac{47\!\cdots\!21}{22\!\cdots\!29}a^{33}-\frac{10\!\cdots\!91}{22\!\cdots\!29}a^{31}-\frac{24\!\cdots\!38}{97\!\cdots\!23}a^{29}-\frac{30\!\cdots\!91}{22\!\cdots\!29}a^{27}-\frac{11\!\cdots\!15}{22\!\cdots\!29}a^{25}-\frac{91\!\cdots\!99}{22\!\cdots\!29}a^{23}+\frac{77\!\cdots\!31}{22\!\cdots\!29}a^{21}+\frac{85\!\cdots\!52}{22\!\cdots\!29}a^{19}+\frac{29\!\cdots\!33}{22\!\cdots\!29}a^{17}+\frac{10\!\cdots\!59}{22\!\cdots\!29}a^{15}+\frac{23\!\cdots\!62}{22\!\cdots\!29}a^{13}-\frac{78\!\cdots\!19}{22\!\cdots\!29}a^{11}+\frac{78\!\cdots\!30}{22\!\cdots\!29}a^{9}+\frac{11\!\cdots\!43}{22\!\cdots\!29}a^{7}+\frac{31\!\cdots\!92}{22\!\cdots\!29}a^{5}+\frac{31\!\cdots\!25}{22\!\cdots\!29}a^{3}+\frac{47\!\cdots\!98}{22\!\cdots\!29}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{22}\times C_{110}$, which has order $2420$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{5080791259918300299740496912046655776}{224658185921597823206597305721673242429} a^{38} + \frac{101451924046779396192754697460220609903}{224658185921597823206597305721673242429} a^{36} - \frac{1170410703975544732317290899165648530944}{224658185921597823206597305721673242429} a^{34} + \frac{9169045072830155735773051351456043006888}{224658185921597823206597305721673242429} a^{32} - \frac{54119072247195440536245166968075232034124}{224658185921597823206597305721673242429} a^{30} + \frac{10811815539627020670491264589715384316209}{9767747213982514052460752422681445323} a^{28} - \frac{915237434801245657191272986145370561091636}{224658185921597823206597305721673242429} a^{26} + \frac{2715817121311291661864357525611831054438697}{224658185921597823206597305721673242429} a^{24} - \frac{6545746692282391260349052949038788511323256}{224658185921597823206597305721673242429} a^{22} + \frac{12728755983245911553739024764537008612303686}{224658185921597823206597305721673242429} a^{20} - \frac{19907024450807145541181119779646685184616180}{224658185921597823206597305721673242429} a^{18} + \frac{24577788814115491124269186355572593226390486}{224658185921597823206597305721673242429} a^{16} - \frac{23755874145055474657407286129028929607548560}{224658185921597823206597305721673242429} a^{14} + \frac{17348934078521056249858340015679985489178873}{224658185921597823206597305721673242429} a^{12} - \frac{9478004883568827701515967247945978499664712}{224658185921597823206597305721673242429} a^{10} + \frac{3575881750535688428642484243944196077931123}{224658185921597823206597305721673242429} a^{8} - \frac{942779388565629517685965251399746338891980}{224658185921597823206597305721673242429} a^{6} + \frac{133670352136312976626545402469902327786412}{224658185921597823206597305721673242429} a^{4} - \frac{13728457126047526199356492463843048242376}{224658185921597823206597305721673242429} a^{2} + \frac{272131199688104057554246519781892536581}{224658185921597823206597305721673242429} \)
(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: |
$\frac{22\!\cdots\!72}{22\!\cdots\!29}a^{38}-\frac{44\!\cdots\!34}{22\!\cdots\!29}a^{36}+\frac{52\!\cdots\!76}{22\!\cdots\!29}a^{34}-\frac{41\!\cdots\!00}{22\!\cdots\!29}a^{32}+\frac{24\!\cdots\!88}{22\!\cdots\!29}a^{30}-\frac{49\!\cdots\!14}{97\!\cdots\!23}a^{28}+\frac{98\!\cdots\!32}{52\!\cdots\!03}a^{26}-\frac{12\!\cdots\!50}{22\!\cdots\!29}a^{24}+\frac{31\!\cdots\!12}{22\!\cdots\!29}a^{22}-\frac{62\!\cdots\!96}{22\!\cdots\!29}a^{20}+\frac{99\!\cdots\!32}{22\!\cdots\!29}a^{18}-\frac{12\!\cdots\!03}{22\!\cdots\!29}a^{16}+\frac{12\!\cdots\!16}{22\!\cdots\!29}a^{14}-\frac{98\!\cdots\!34}{22\!\cdots\!29}a^{12}+\frac{57\!\cdots\!24}{22\!\cdots\!29}a^{10}-\frac{24\!\cdots\!30}{22\!\cdots\!29}a^{8}+\frac{71\!\cdots\!56}{22\!\cdots\!29}a^{6}-\frac{12\!\cdots\!28}{22\!\cdots\!29}a^{4}+\frac{11\!\cdots\!16}{22\!\cdots\!29}a^{2}-\frac{23\!\cdots\!22}{22\!\cdots\!29}$, $\frac{31\!\cdots\!12}{22\!\cdots\!29}a^{38}-\frac{63\!\cdots\!40}{22\!\cdots\!29}a^{36}+\frac{73\!\cdots\!00}{22\!\cdots\!29}a^{34}-\frac{57\!\cdots\!28}{22\!\cdots\!29}a^{32}+\frac{34\!\cdots\!88}{22\!\cdots\!29}a^{30}-\frac{68\!\cdots\!23}{97\!\cdots\!23}a^{28}+\frac{58\!\cdots\!72}{22\!\cdots\!29}a^{26}-\frac{17\!\cdots\!32}{22\!\cdots\!29}a^{24}+\frac{41\!\cdots\!28}{22\!\cdots\!29}a^{22}-\frac{81\!\cdots\!48}{22\!\cdots\!29}a^{20}+\frac{12\!\cdots\!68}{22\!\cdots\!29}a^{18}-\frac{16\!\cdots\!44}{22\!\cdots\!29}a^{16}+\frac{15\!\cdots\!04}{22\!\cdots\!29}a^{14}-\frac{11\!\cdots\!56}{22\!\cdots\!29}a^{12}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{10}-\frac{25\!\cdots\!20}{22\!\cdots\!29}a^{8}+\frac{69\!\cdots\!92}{22\!\cdots\!29}a^{6}-\frac{11\!\cdots\!59}{22\!\cdots\!29}a^{4}+\frac{10\!\cdots\!92}{22\!\cdots\!29}a^{2}-\frac{21\!\cdots\!36}{22\!\cdots\!29}$, $\frac{16\!\cdots\!16}{17\!\cdots\!13}a^{38}-\frac{31\!\cdots\!71}{17\!\cdots\!13}a^{36}+\frac{36\!\cdots\!76}{17\!\cdots\!13}a^{34}-\frac{28\!\cdots\!80}{17\!\cdots\!13}a^{32}+\frac{16\!\cdots\!44}{17\!\cdots\!13}a^{30}-\frac{76\!\cdots\!66}{17\!\cdots\!13}a^{28}+\frac{28\!\cdots\!64}{17\!\cdots\!13}a^{26}-\frac{82\!\cdots\!25}{17\!\cdots\!13}a^{24}+\frac{19\!\cdots\!96}{17\!\cdots\!13}a^{22}-\frac{38\!\cdots\!06}{17\!\cdots\!13}a^{20}+\frac{58\!\cdots\!28}{17\!\cdots\!13}a^{18}-\frac{71\!\cdots\!02}{17\!\cdots\!13}a^{16}+\frac{67\!\cdots\!88}{17\!\cdots\!13}a^{14}-\frac{47\!\cdots\!25}{17\!\cdots\!13}a^{12}+\frac{24\!\cdots\!52}{17\!\cdots\!13}a^{10}-\frac{86\!\cdots\!88}{17\!\cdots\!13}a^{8}+\frac{20\!\cdots\!64}{17\!\cdots\!13}a^{6}-\frac{16\!\cdots\!82}{17\!\cdots\!13}a^{4}+\frac{32\!\cdots\!72}{17\!\cdots\!13}a^{2}+\frac{28\!\cdots\!25}{17\!\cdots\!13}$, $\frac{25\!\cdots\!16}{17\!\cdots\!13}a^{38}-\frac{50\!\cdots\!79}{17\!\cdots\!13}a^{36}+\frac{57\!\cdots\!52}{17\!\cdots\!13}a^{34}-\frac{45\!\cdots\!80}{17\!\cdots\!13}a^{32}+\frac{26\!\cdots\!28}{17\!\cdots\!13}a^{30}-\frac{12\!\cdots\!29}{17\!\cdots\!13}a^{28}+\frac{44\!\cdots\!48}{17\!\cdots\!13}a^{26}-\frac{13\!\cdots\!74}{17\!\cdots\!13}a^{24}+\frac{31\!\cdots\!88}{17\!\cdots\!13}a^{22}-\frac{60\!\cdots\!13}{17\!\cdots\!13}a^{20}+\frac{94\!\cdots\!44}{17\!\cdots\!13}a^{18}-\frac{11\!\cdots\!71}{17\!\cdots\!13}a^{16}+\frac{10\!\cdots\!96}{17\!\cdots\!13}a^{14}-\frac{77\!\cdots\!19}{17\!\cdots\!13}a^{12}+\frac{40\!\cdots\!08}{17\!\cdots\!13}a^{10}-\frac{14\!\cdots\!06}{17\!\cdots\!13}a^{8}+\frac{32\!\cdots\!16}{17\!\cdots\!13}a^{6}-\frac{26\!\cdots\!07}{17\!\cdots\!13}a^{4}+\frac{53\!\cdots\!32}{17\!\cdots\!13}a^{2}+\frac{44\!\cdots\!24}{17\!\cdots\!13}$, $\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{39}-\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{37}+\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{35}-\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{33}+\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{31}-\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{29}+\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{27}-\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{25}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{23}-\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{21}+\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{19}-\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{17}+\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{15}-\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{13}+\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{11}-\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{9}+\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{7}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{5}+\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{3}-\frac{47\!\cdots\!52}{22\!\cdots\!29}a$, $\frac{58\!\cdots\!29}{22\!\cdots\!29}a^{39}-\frac{11\!\cdots\!61}{22\!\cdots\!29}a^{37}+\frac{13\!\cdots\!36}{22\!\cdots\!29}a^{35}-\frac{10\!\cdots\!71}{22\!\cdots\!29}a^{33}+\frac{63\!\cdots\!21}{22\!\cdots\!29}a^{31}-\frac{12\!\cdots\!38}{97\!\cdots\!23}a^{29}+\frac{10\!\cdots\!60}{22\!\cdots\!29}a^{27}-\frac{32\!\cdots\!54}{22\!\cdots\!29}a^{25}+\frac{77\!\cdots\!64}{22\!\cdots\!29}a^{23}-\frac{15\!\cdots\!79}{22\!\cdots\!29}a^{21}+\frac{23\!\cdots\!07}{22\!\cdots\!29}a^{19}-\frac{29\!\cdots\!42}{22\!\cdots\!29}a^{17}+\frac{28\!\cdots\!62}{22\!\cdots\!29}a^{15}-\frac{21\!\cdots\!33}{22\!\cdots\!29}a^{13}+\frac{11\!\cdots\!24}{22\!\cdots\!29}a^{11}-\frac{46\!\cdots\!58}{22\!\cdots\!29}a^{9}+\frac{12\!\cdots\!38}{22\!\cdots\!29}a^{7}-\frac{19\!\cdots\!02}{22\!\cdots\!29}a^{5}+\frac{19\!\cdots\!79}{22\!\cdots\!29}a^{3}-\frac{37\!\cdots\!63}{22\!\cdots\!29}a$, $\frac{42\!\cdots\!70}{22\!\cdots\!29}a^{39}-\frac{83\!\cdots\!90}{22\!\cdots\!29}a^{37}+\frac{96\!\cdots\!36}{22\!\cdots\!29}a^{35}-\frac{75\!\cdots\!20}{22\!\cdots\!29}a^{33}+\frac{44\!\cdots\!70}{22\!\cdots\!29}a^{31}-\frac{87\!\cdots\!07}{97\!\cdots\!23}a^{29}+\frac{73\!\cdots\!74}{22\!\cdots\!29}a^{27}-\frac{21\!\cdots\!93}{22\!\cdots\!29}a^{25}+\frac{51\!\cdots\!16}{22\!\cdots\!29}a^{23}-\frac{10\!\cdots\!29}{22\!\cdots\!29}a^{21}+\frac{15\!\cdots\!24}{22\!\cdots\!29}a^{19}-\frac{18\!\cdots\!15}{22\!\cdots\!29}a^{17}+\frac{17\!\cdots\!14}{22\!\cdots\!29}a^{15}-\frac{12\!\cdots\!59}{22\!\cdots\!29}a^{13}+\frac{65\!\cdots\!18}{22\!\cdots\!29}a^{11}-\frac{22\!\cdots\!82}{22\!\cdots\!29}a^{9}+\frac{52\!\cdots\!61}{22\!\cdots\!29}a^{7}-\frac{42\!\cdots\!23}{22\!\cdots\!29}a^{5}+\frac{85\!\cdots\!74}{22\!\cdots\!29}a^{3}+\frac{18\!\cdots\!83}{22\!\cdots\!29}a$, $\frac{57\!\cdots\!09}{22\!\cdots\!29}a^{39}-\frac{11\!\cdots\!39}{22\!\cdots\!29}a^{37}+\frac{13\!\cdots\!51}{22\!\cdots\!29}a^{35}-\frac{10\!\cdots\!45}{22\!\cdots\!29}a^{33}+\frac{61\!\cdots\!81}{22\!\cdots\!29}a^{31}-\frac{12\!\cdots\!46}{97\!\cdots\!23}a^{29}+\frac{24\!\cdots\!32}{52\!\cdots\!03}a^{27}-\frac{31\!\cdots\!10}{22\!\cdots\!29}a^{25}+\frac{75\!\cdots\!05}{22\!\cdots\!29}a^{23}-\frac{14\!\cdots\!29}{22\!\cdots\!29}a^{21}+\frac{23\!\cdots\!87}{22\!\cdots\!29}a^{19}-\frac{28\!\cdots\!74}{22\!\cdots\!29}a^{17}+\frac{28\!\cdots\!82}{22\!\cdots\!29}a^{15}-\frac{20\!\cdots\!15}{22\!\cdots\!29}a^{13}+\frac{11\!\cdots\!86}{22\!\cdots\!29}a^{11}-\frac{44\!\cdots\!90}{22\!\cdots\!29}a^{9}+\frac{28\!\cdots\!10}{52\!\cdots\!03}a^{7}-\frac{18\!\cdots\!90}{22\!\cdots\!29}a^{5}+\frac{18\!\cdots\!75}{22\!\cdots\!29}a^{3}-\frac{36\!\cdots\!69}{22\!\cdots\!29}a$, $\frac{55\!\cdots\!37}{22\!\cdots\!29}a^{39}-\frac{11\!\cdots\!05}{22\!\cdots\!29}a^{37}+\frac{12\!\cdots\!75}{22\!\cdots\!29}a^{35}-\frac{10\!\cdots\!45}{22\!\cdots\!29}a^{33}+\frac{59\!\cdots\!93}{22\!\cdots\!29}a^{31}-\frac{11\!\cdots\!32}{97\!\cdots\!23}a^{29}+\frac{23\!\cdots\!00}{52\!\cdots\!03}a^{27}-\frac{29\!\cdots\!60}{22\!\cdots\!29}a^{25}+\frac{72\!\cdots\!93}{22\!\cdots\!29}a^{23}-\frac{14\!\cdots\!33}{22\!\cdots\!29}a^{21}+\frac{22\!\cdots\!55}{22\!\cdots\!29}a^{19}-\frac{27\!\cdots\!71}{22\!\cdots\!29}a^{17}+\frac{26\!\cdots\!66}{22\!\cdots\!29}a^{15}-\frac{19\!\cdots\!81}{22\!\cdots\!29}a^{13}+\frac{10\!\cdots\!62}{22\!\cdots\!29}a^{11}-\frac{42\!\cdots\!60}{22\!\cdots\!29}a^{9}+\frac{11\!\cdots\!74}{22\!\cdots\!29}a^{7}-\frac{17\!\cdots\!62}{22\!\cdots\!29}a^{5}+\frac{17\!\cdots\!59}{22\!\cdots\!29}a^{3}-\frac{33\!\cdots\!47}{22\!\cdots\!29}a$, $\frac{18\!\cdots\!93}{75\!\cdots\!43}a^{39}-\frac{10\!\cdots\!52}{22\!\cdots\!29}a^{38}-\frac{36\!\cdots\!22}{75\!\cdots\!43}a^{37}+\frac{20\!\cdots\!06}{22\!\cdots\!29}a^{36}+\frac{41\!\cdots\!16}{75\!\cdots\!43}a^{35}-\frac{23\!\cdots\!88}{22\!\cdots\!29}a^{34}-\frac{32\!\cdots\!36}{75\!\cdots\!43}a^{33}+\frac{18\!\cdots\!76}{22\!\cdots\!29}a^{32}+\frac{19\!\cdots\!34}{75\!\cdots\!43}a^{31}-\frac{10\!\cdots\!48}{22\!\cdots\!29}a^{30}-\frac{38\!\cdots\!66}{32\!\cdots\!41}a^{29}+\frac{21\!\cdots\!18}{97\!\cdots\!23}a^{28}+\frac{32\!\cdots\!91}{75\!\cdots\!43}a^{27}-\frac{18\!\cdots\!72}{22\!\cdots\!29}a^{26}-\frac{97\!\cdots\!37}{75\!\cdots\!43}a^{25}+\frac{54\!\cdots\!94}{22\!\cdots\!29}a^{24}+\frac{23\!\cdots\!66}{75\!\cdots\!43}a^{23}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{22}-\frac{45\!\cdots\!57}{75\!\cdots\!43}a^{21}+\frac{25\!\cdots\!72}{22\!\cdots\!29}a^{20}+\frac{71\!\cdots\!46}{75\!\cdots\!43}a^{19}-\frac{39\!\cdots\!60}{22\!\cdots\!29}a^{18}-\frac{20\!\cdots\!45}{17\!\cdots\!01}a^{17}+\frac{49\!\cdots\!72}{22\!\cdots\!29}a^{16}+\frac{19\!\cdots\!49}{17\!\cdots\!01}a^{15}-\frac{47\!\cdots\!20}{22\!\cdots\!29}a^{14}-\frac{62\!\cdots\!59}{75\!\cdots\!43}a^{13}+\frac{34\!\cdots\!46}{22\!\cdots\!29}a^{12}+\frac{34\!\cdots\!52}{75\!\cdots\!43}a^{11}-\frac{18\!\cdots\!24}{22\!\cdots\!29}a^{10}-\frac{12\!\cdots\!06}{75\!\cdots\!43}a^{9}+\frac{71\!\cdots\!46}{22\!\cdots\!29}a^{8}+\frac{34\!\cdots\!78}{75\!\cdots\!43}a^{7}-\frac{18\!\cdots\!60}{22\!\cdots\!29}a^{6}-\frac{48\!\cdots\!82}{75\!\cdots\!43}a^{5}+\frac{26\!\cdots\!24}{22\!\cdots\!29}a^{4}+\frac{46\!\cdots\!74}{75\!\cdots\!43}a^{3}-\frac{27\!\cdots\!52}{22\!\cdots\!29}a^{2}-\frac{17\!\cdots\!93}{75\!\cdots\!43}a+\frac{31\!\cdots\!33}{22\!\cdots\!29}$, $\frac{15\!\cdots\!11}{22\!\cdots\!29}a^{39}+\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{30\!\cdots\!97}{22\!\cdots\!29}a^{37}-\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{34\!\cdots\!08}{22\!\cdots\!29}a^{35}+\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{27\!\cdots\!04}{22\!\cdots\!29}a^{33}-\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{16\!\cdots\!99}{22\!\cdots\!29}a^{31}+\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{32\!\cdots\!71}{97\!\cdots\!23}a^{29}-\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{27\!\cdots\!57}{22\!\cdots\!29}a^{27}+\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{80\!\cdots\!11}{22\!\cdots\!29}a^{25}-\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{19\!\cdots\!42}{22\!\cdots\!29}a^{23}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{37\!\cdots\!42}{22\!\cdots\!29}a^{21}-\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{59\!\cdots\!82}{22\!\cdots\!29}a^{19}+\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{73\!\cdots\!26}{22\!\cdots\!29}a^{17}-\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{70\!\cdots\!73}{22\!\cdots\!29}a^{15}+\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{51\!\cdots\!43}{22\!\cdots\!29}a^{13}-\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{28\!\cdots\!55}{22\!\cdots\!29}a^{11}+\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{10\!\cdots\!77}{22\!\cdots\!29}a^{9}-\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{28\!\cdots\!96}{22\!\cdots\!29}a^{7}+\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{39\!\cdots\!12}{22\!\cdots\!29}a^{5}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{40\!\cdots\!20}{22\!\cdots\!29}a^{3}+\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}-\frac{81\!\cdots\!67}{22\!\cdots\!29}a-\frac{49\!\cdots\!10}{22\!\cdots\!29}$, $\frac{10\!\cdots\!35}{39\!\cdots\!91}a^{39}-\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{90\!\cdots\!00}{17\!\cdots\!13}a^{37}+\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{10\!\cdots\!68}{17\!\cdots\!13}a^{35}-\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{82\!\cdots\!04}{17\!\cdots\!13}a^{33}+\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{48\!\cdots\!25}{17\!\cdots\!13}a^{31}-\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{22\!\cdots\!28}{17\!\cdots\!13}a^{29}+\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{82\!\cdots\!84}{17\!\cdots\!13}a^{27}-\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{24\!\cdots\!36}{17\!\cdots\!13}a^{25}+\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{59\!\cdots\!68}{17\!\cdots\!13}a^{23}-\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{11\!\cdots\!84}{17\!\cdots\!13}a^{21}+\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{18\!\cdots\!91}{17\!\cdots\!13}a^{19}-\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{22\!\cdots\!48}{17\!\cdots\!13}a^{17}+\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{22\!\cdots\!50}{17\!\cdots\!13}a^{15}-\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{16\!\cdots\!76}{17\!\cdots\!13}a^{13}+\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{91\!\cdots\!88}{17\!\cdots\!13}a^{11}-\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{35\!\cdots\!52}{17\!\cdots\!13}a^{9}+\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{97\!\cdots\!18}{17\!\cdots\!13}a^{7}-\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{15\!\cdots\!16}{17\!\cdots\!13}a^{5}+\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{14\!\cdots\!67}{17\!\cdots\!13}a^{3}-\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}-\frac{68\!\cdots\!12}{39\!\cdots\!91}a+\frac{49\!\cdots\!10}{22\!\cdots\!29}$, $\frac{79\!\cdots\!57}{22\!\cdots\!29}a^{39}+\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{15\!\cdots\!57}{22\!\cdots\!29}a^{37}-\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{18\!\cdots\!76}{22\!\cdots\!29}a^{35}+\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{14\!\cdots\!20}{22\!\cdots\!29}a^{33}-\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{82\!\cdots\!93}{22\!\cdots\!29}a^{31}+\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{16\!\cdots\!71}{97\!\cdots\!23}a^{29}-\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{13\!\cdots\!19}{22\!\cdots\!29}a^{27}+\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{40\!\cdots\!64}{22\!\cdots\!29}a^{25}-\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{97\!\cdots\!42}{22\!\cdots\!29}a^{23}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{18\!\cdots\!81}{22\!\cdots\!29}a^{21}-\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{28\!\cdots\!74}{22\!\cdots\!29}a^{19}+\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{35\!\cdots\!03}{22\!\cdots\!29}a^{17}-\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{76\!\cdots\!57}{52\!\cdots\!03}a^{15}+\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{23\!\cdots\!90}{22\!\cdots\!29}a^{13}-\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{12\!\cdots\!29}{22\!\cdots\!29}a^{11}+\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{42\!\cdots\!30}{22\!\cdots\!29}a^{9}-\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{99\!\cdots\!92}{22\!\cdots\!29}a^{7}+\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{79\!\cdots\!03}{22\!\cdots\!29}a^{5}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{15\!\cdots\!51}{22\!\cdots\!29}a^{3}+\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}+\frac{10\!\cdots\!82}{22\!\cdots\!29}a+\frac{17\!\cdots\!77}{22\!\cdots\!29}$, $\frac{40\!\cdots\!71}{22\!\cdots\!29}a^{39}-\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{81\!\cdots\!72}{22\!\cdots\!29}a^{37}+\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{94\!\cdots\!76}{22\!\cdots\!29}a^{35}-\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{73\!\cdots\!32}{22\!\cdots\!29}a^{33}+\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{43\!\cdots\!99}{22\!\cdots\!29}a^{31}-\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{87\!\cdots\!19}{97\!\cdots\!23}a^{29}+\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{73\!\cdots\!50}{22\!\cdots\!29}a^{27}-\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{21\!\cdots\!88}{22\!\cdots\!29}a^{25}+\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{52\!\cdots\!45}{22\!\cdots\!29}a^{23}-\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{10\!\cdots\!20}{22\!\cdots\!29}a^{21}+\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{16\!\cdots\!39}{22\!\cdots\!29}a^{19}-\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{19\!\cdots\!48}{22\!\cdots\!29}a^{17}+\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{19\!\cdots\!08}{22\!\cdots\!29}a^{15}-\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{14\!\cdots\!60}{22\!\cdots\!29}a^{13}+\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{77\!\cdots\!76}{22\!\cdots\!29}a^{11}-\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{29\!\cdots\!36}{22\!\cdots\!29}a^{9}+\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{78\!\cdots\!62}{22\!\cdots\!29}a^{7}-\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{11\!\cdots\!47}{22\!\cdots\!29}a^{5}+\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{11\!\cdots\!19}{22\!\cdots\!29}a^{3}-\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}-\frac{22\!\cdots\!44}{22\!\cdots\!29}a+\frac{49\!\cdots\!10}{22\!\cdots\!29}$, $\frac{33\!\cdots\!51}{22\!\cdots\!29}a^{39}-\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{66\!\cdots\!77}{22\!\cdots\!29}a^{37}+\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{76\!\cdots\!28}{22\!\cdots\!29}a^{35}-\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{60\!\cdots\!56}{22\!\cdots\!29}a^{33}+\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{35\!\cdots\!79}{22\!\cdots\!29}a^{31}-\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{70\!\cdots\!11}{97\!\cdots\!23}a^{29}+\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{60\!\cdots\!46}{22\!\cdots\!29}a^{27}-\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{17\!\cdots\!79}{22\!\cdots\!29}a^{25}+\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{43\!\cdots\!65}{22\!\cdots\!29}a^{23}-\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{83\!\cdots\!50}{22\!\cdots\!29}a^{21}+\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{13\!\cdots\!23}{22\!\cdots\!29}a^{19}-\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{37\!\cdots\!22}{52\!\cdots\!03}a^{17}+\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{15\!\cdots\!72}{22\!\cdots\!29}a^{15}-\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{11\!\cdots\!35}{22\!\cdots\!29}a^{13}+\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{62\!\cdots\!32}{22\!\cdots\!29}a^{11}-\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{23\!\cdots\!61}{22\!\cdots\!29}a^{9}+\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{63\!\cdots\!66}{22\!\cdots\!29}a^{7}-\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{91\!\cdots\!01}{22\!\cdots\!29}a^{5}+\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{92\!\cdots\!83}{22\!\cdots\!29}a^{3}-\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}-\frac{18\!\cdots\!31}{22\!\cdots\!29}a+\frac{49\!\cdots\!10}{22\!\cdots\!29}$, $\frac{10\!\cdots\!74}{22\!\cdots\!29}a^{39}+\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{20\!\cdots\!65}{22\!\cdots\!29}a^{37}-\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{23\!\cdots\!80}{22\!\cdots\!29}a^{35}+\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{18\!\cdots\!40}{22\!\cdots\!29}a^{33}-\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{10\!\cdots\!89}{22\!\cdots\!29}a^{31}+\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{21\!\cdots\!00}{97\!\cdots\!23}a^{29}-\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{18\!\cdots\!90}{22\!\cdots\!29}a^{27}+\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{53\!\cdots\!21}{22\!\cdots\!29}a^{25}-\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{12\!\cdots\!20}{22\!\cdots\!29}a^{23}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{24\!\cdots\!08}{22\!\cdots\!29}a^{21}-\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{38\!\cdots\!76}{22\!\cdots\!29}a^{19}+\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{46\!\cdots\!48}{22\!\cdots\!29}a^{17}-\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{44\!\cdots\!06}{22\!\cdots\!29}a^{15}+\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{31\!\cdots\!08}{22\!\cdots\!29}a^{13}-\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{16\!\cdots\!46}{22\!\cdots\!29}a^{11}+\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{56\!\cdots\!84}{22\!\cdots\!29}a^{9}-\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{13\!\cdots\!30}{22\!\cdots\!29}a^{7}+\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{10\!\cdots\!16}{22\!\cdots\!29}a^{5}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{21\!\cdots\!22}{22\!\cdots\!29}a^{3}+\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}+\frac{24\!\cdots\!58}{22\!\cdots\!29}a+\frac{17\!\cdots\!77}{22\!\cdots\!29}$, $\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}+a+\frac{17\!\cdots\!77}{22\!\cdots\!29}$, $\frac{74\!\cdots\!28}{22\!\cdots\!29}a^{39}-\frac{10\!\cdots\!52}{22\!\cdots\!29}a^{38}-\frac{14\!\cdots\!08}{22\!\cdots\!29}a^{37}+\frac{20\!\cdots\!06}{22\!\cdots\!29}a^{36}+\frac{17\!\cdots\!56}{22\!\cdots\!29}a^{35}-\frac{23\!\cdots\!88}{22\!\cdots\!29}a^{34}-\frac{13\!\cdots\!45}{22\!\cdots\!29}a^{33}+\frac{18\!\cdots\!76}{22\!\cdots\!29}a^{32}+\frac{79\!\cdots\!78}{22\!\cdots\!29}a^{31}-\frac{10\!\cdots\!48}{22\!\cdots\!29}a^{30}-\frac{15\!\cdots\!04}{97\!\cdots\!23}a^{29}+\frac{21\!\cdots\!18}{97\!\cdots\!23}a^{28}+\frac{13\!\cdots\!66}{22\!\cdots\!29}a^{27}-\frac{18\!\cdots\!72}{22\!\cdots\!29}a^{26}-\frac{40\!\cdots\!52}{22\!\cdots\!29}a^{25}+\frac{54\!\cdots\!94}{22\!\cdots\!29}a^{24}+\frac{96\!\cdots\!02}{22\!\cdots\!29}a^{23}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{22}-\frac{18\!\cdots\!83}{22\!\cdots\!29}a^{21}+\frac{25\!\cdots\!72}{22\!\cdots\!29}a^{20}+\frac{29\!\cdots\!42}{22\!\cdots\!29}a^{19}-\frac{39\!\cdots\!60}{22\!\cdots\!29}a^{18}-\frac{36\!\cdots\!20}{22\!\cdots\!29}a^{17}+\frac{49\!\cdots\!72}{22\!\cdots\!29}a^{16}+\frac{35\!\cdots\!41}{22\!\cdots\!29}a^{15}-\frac{47\!\cdots\!20}{22\!\cdots\!29}a^{14}-\frac{25\!\cdots\!04}{22\!\cdots\!29}a^{13}+\frac{34\!\cdots\!46}{22\!\cdots\!29}a^{12}+\frac{14\!\cdots\!48}{22\!\cdots\!29}a^{11}-\frac{18\!\cdots\!24}{22\!\cdots\!29}a^{10}-\frac{53\!\cdots\!26}{22\!\cdots\!29}a^{9}+\frac{71\!\cdots\!46}{22\!\cdots\!29}a^{8}+\frac{14\!\cdots\!54}{22\!\cdots\!29}a^{7}-\frac{18\!\cdots\!60}{22\!\cdots\!29}a^{6}-\frac{20\!\cdots\!20}{22\!\cdots\!29}a^{5}+\frac{26\!\cdots\!24}{22\!\cdots\!29}a^{4}+\frac{18\!\cdots\!84}{22\!\cdots\!29}a^{3}-\frac{27\!\cdots\!52}{22\!\cdots\!29}a^{2}-\frac{71\!\cdots\!68}{22\!\cdots\!29}a+\frac{31\!\cdots\!33}{22\!\cdots\!29}$, $\frac{13\!\cdots\!79}{22\!\cdots\!29}a^{39}+\frac{50\!\cdots\!76}{22\!\cdots\!29}a^{38}-\frac{27\!\cdots\!31}{22\!\cdots\!29}a^{37}-\frac{10\!\cdots\!03}{22\!\cdots\!29}a^{36}+\frac{31\!\cdots\!93}{22\!\cdots\!29}a^{35}+\frac{11\!\cdots\!44}{22\!\cdots\!29}a^{34}-\frac{24\!\cdots\!00}{22\!\cdots\!29}a^{33}-\frac{91\!\cdots\!88}{22\!\cdots\!29}a^{32}+\frac{14\!\cdots\!81}{22\!\cdots\!29}a^{31}+\frac{54\!\cdots\!24}{22\!\cdots\!29}a^{30}-\frac{28\!\cdots\!76}{97\!\cdots\!23}a^{29}-\frac{10\!\cdots\!09}{97\!\cdots\!23}a^{28}+\frac{24\!\cdots\!82}{22\!\cdots\!29}a^{27}+\frac{91\!\cdots\!36}{22\!\cdots\!29}a^{26}-\frac{71\!\cdots\!87}{22\!\cdots\!29}a^{25}-\frac{27\!\cdots\!97}{22\!\cdots\!29}a^{24}+\frac{17\!\cdots\!53}{22\!\cdots\!29}a^{23}+\frac{65\!\cdots\!56}{22\!\cdots\!29}a^{22}-\frac{76\!\cdots\!40}{52\!\cdots\!03}a^{21}-\frac{12\!\cdots\!86}{22\!\cdots\!29}a^{20}+\frac{51\!\cdots\!87}{22\!\cdots\!29}a^{19}+\frac{19\!\cdots\!80}{22\!\cdots\!29}a^{18}-\frac{62\!\cdots\!28}{22\!\cdots\!29}a^{17}-\frac{24\!\cdots\!86}{22\!\cdots\!29}a^{16}+\frac{13\!\cdots\!01}{52\!\cdots\!03}a^{15}+\frac{23\!\cdots\!60}{22\!\cdots\!29}a^{14}-\frac{41\!\cdots\!64}{22\!\cdots\!29}a^{13}-\frac{17\!\cdots\!73}{22\!\cdots\!29}a^{12}+\frac{21\!\cdots\!10}{22\!\cdots\!29}a^{11}+\frac{94\!\cdots\!12}{22\!\cdots\!29}a^{10}-\frac{75\!\cdots\!40}{22\!\cdots\!29}a^{9}-\frac{35\!\cdots\!23}{22\!\cdots\!29}a^{8}+\frac{17\!\cdots\!72}{22\!\cdots\!29}a^{7}+\frac{94\!\cdots\!80}{22\!\cdots\!29}a^{6}-\frac{14\!\cdots\!40}{22\!\cdots\!29}a^{5}-\frac{13\!\cdots\!12}{22\!\cdots\!29}a^{4}+\frac{28\!\cdots\!29}{22\!\cdots\!29}a^{3}+\frac{13\!\cdots\!76}{22\!\cdots\!29}a^{2}+\frac{27\!\cdots\!98}{22\!\cdots\!29}a+\frac{17\!\cdots\!77}{22\!\cdots\!29}$
| 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: | \( 25665787785126296 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 25665787785126296 \cdot 2420}{6\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.263715329235548 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1)
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^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1, 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^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1);
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^40 - 20*x^38 + 231*x^36 - 1812*x^34 + 10709*x^32 - 49280*x^30 + 181674*x^28 - 540148*x^26 + 1304886*x^24 - 2544812*x^22 + 3994121*x^20 - 4954644*x^18 + 4817692*x^16 - 3548680*x^14 + 1959963*x^12 - 753104*x^10 + 202622*x^8 - 30356*x^6 + 3025*x^4 - 60*x^2 + 1);
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
$C_2^2\times C_{10}$ (as 40T7):
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)
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2^2\times C_{10}$ |
| Character table for $C_2^2\times C_{10}$ is not computed |
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]
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 | ${\href{/padicField/5.10.0.1}{10} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.1.0.1}{1} }^{40}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $4$ | $10$ | $80$ | |||
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |