Properties

Label 36.0.400...241.1
Degree $36$
Signature $[0, 18]$
Discriminant $4.001\times 10^{60}$
Root discriminant \(48.24\)
Ramified primes $7,67,167$
Class number $228$ (GRH)
Class group [228] (GRH)
Galois group $C_2\times A_4\times D_6$ (as 36T334)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512)
 
gp: K = bnfinit(y^36 - 14*y^35 + 129*y^34 - 874*y^33 + 4865*y^32 - 22922*y^31 + 94283*y^30 - 342666*y^29 + 1112273*y^28 - 3234952*y^27 + 8438667*y^26 - 19663764*y^25 + 40577370*y^24 - 72857172*y^23 + 109953255*y^22 - 128328007*y^21 + 83898469*y^20 + 71389621*y^19 - 339203441*y^18 + 610359985*y^17 - 658477439*y^16 + 279191379*y^15 + 463357277*y^14 - 1113046997*y^13 + 1096834803*y^12 - 317516043*y^11 - 517559160*y^10 + 549915913*y^9 + 131226304*y^8 - 546397416*y^7 + 300520880*y^6 + 8345200*y^5 - 14423488*y^4 - 397760*y^3 + 206720*y^2 + 13568*y + 512, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512)
 

\( x^{36} - 14 x^{35} + 129 x^{34} - 874 x^{33} + 4865 x^{32} - 22922 x^{31} + 94283 x^{30} - 342666 x^{29} + 1112273 x^{28} - 3234952 x^{27} + 8438667 x^{26} - 19663764 x^{25} + \cdots + 512 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4000715416325500851269158271470993386638594171162518886820241\) \(\medspace = 7^{30}\cdot 67^{12}\cdot 167^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(48.24\)
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:  $7^{5/6}67^{1/2}167^{1/2}\approx 535.3576385241264$
Ramified primes:   \(7\), \(67\), \(167\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{131072}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{18}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{26}-\frac{1}{4}a^{23}-\frac{1}{4}a^{22}-\frac{1}{4}a^{21}-\frac{1}{4}a^{20}+\frac{1}{4}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{27}-\frac{1}{4}a^{24}-\frac{1}{4}a^{23}-\frac{1}{4}a^{22}-\frac{1}{4}a^{21}-\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{2}a^{16}+\frac{1}{4}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{28}-\frac{1}{4}a^{25}-\frac{1}{4}a^{24}-\frac{1}{4}a^{23}-\frac{1}{4}a^{22}-\frac{1}{4}a^{20}-\frac{1}{4}a^{19}-\frac{1}{2}a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{4}a^{12}+\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{29}-\frac{1}{4}a^{25}-\frac{1}{4}a^{24}-\frac{1}{4}a^{22}+\frac{1}{4}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{30}-\frac{1}{8}a^{28}-\frac{1}{8}a^{26}-\frac{1}{4}a^{25}+\frac{1}{8}a^{24}+\frac{1}{8}a^{22}-\frac{1}{8}a^{20}+\frac{1}{4}a^{17}+\frac{1}{8}a^{16}-\frac{1}{8}a^{15}+\frac{1}{8}a^{14}+\frac{3}{8}a^{13}-\frac{1}{8}a^{12}-\frac{3}{8}a^{11}-\frac{3}{8}a^{10}+\frac{1}{8}a^{9}+\frac{3}{8}a^{8}-\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{31}+\frac{1}{16}a^{29}+\frac{1}{16}a^{27}-\frac{1}{16}a^{25}-\frac{1}{4}a^{24}+\frac{1}{16}a^{23}-\frac{1}{8}a^{22}+\frac{3}{16}a^{21}-\frac{1}{8}a^{20}+\frac{1}{8}a^{19}+\frac{1}{4}a^{18}+\frac{3}{16}a^{17}-\frac{1}{16}a^{16}+\frac{7}{16}a^{15}-\frac{5}{16}a^{14}+\frac{1}{16}a^{13}-\frac{5}{16}a^{12}-\frac{1}{16}a^{11}+\frac{1}{16}a^{10}-\frac{5}{16}a^{9}+\frac{5}{16}a^{8}+\frac{5}{16}a^{7}-\frac{5}{16}a^{6}+\frac{3}{8}a^{5}-\frac{7}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{32}+\frac{1}{32}a^{30}-\frac{1}{8}a^{29}+\frac{1}{32}a^{28}-\frac{1}{8}a^{27}+\frac{3}{32}a^{26}-\frac{7}{32}a^{24}+\frac{3}{16}a^{23}+\frac{7}{32}a^{22}+\frac{3}{16}a^{21}+\frac{3}{16}a^{20}-\frac{9}{32}a^{18}-\frac{5}{32}a^{17}-\frac{9}{32}a^{16}-\frac{1}{32}a^{15}+\frac{9}{32}a^{14}+\frac{7}{32}a^{13}-\frac{13}{32}a^{12}-\frac{3}{32}a^{11}+\frac{11}{32}a^{10}-\frac{11}{32}a^{9}+\frac{1}{32}a^{8}+\frac{11}{32}a^{7}-\frac{3}{16}a^{6}-\frac{11}{32}a^{5}-\frac{5}{16}a^{4}+\frac{3}{8}a^{3}$, $\frac{1}{64}a^{33}+\frac{1}{64}a^{31}-\frac{1}{16}a^{30}-\frac{7}{64}a^{29}-\frac{1}{16}a^{28}+\frac{3}{64}a^{27}-\frac{1}{8}a^{26}+\frac{1}{64}a^{25}-\frac{1}{32}a^{24}+\frac{15}{64}a^{23}+\frac{3}{32}a^{22}+\frac{7}{32}a^{21}+\frac{1}{8}a^{20}+\frac{7}{64}a^{19}+\frac{11}{64}a^{18}+\frac{31}{64}a^{17}-\frac{1}{64}a^{16}-\frac{7}{64}a^{15}-\frac{25}{64}a^{14}+\frac{27}{64}a^{13}+\frac{5}{64}a^{12}-\frac{21}{64}a^{11}+\frac{29}{64}a^{10}-\frac{31}{64}a^{9}+\frac{27}{64}a^{8}+\frac{5}{32}a^{7}-\frac{3}{64}a^{6}-\frac{1}{32}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{24832}a^{34}+\frac{89}{12416}a^{33}+\frac{133}{24832}a^{32}-\frac{249}{12416}a^{31}-\frac{1355}{24832}a^{30}+\frac{1231}{12416}a^{29}+\frac{2751}{24832}a^{28}-\frac{1457}{12416}a^{27}-\frac{2259}{24832}a^{26}+\frac{21}{194}a^{25}+\frac{3999}{24832}a^{24}+\frac{1271}{6208}a^{23}+\frac{11}{12416}a^{22}+\frac{1531}{6208}a^{21}+\frac{5247}{24832}a^{20}-\frac{1463}{24832}a^{19}+\frac{7201}{24832}a^{18}-\frac{2007}{24832}a^{17}-\frac{189}{24832}a^{16}-\frac{1531}{24832}a^{15}+\frac{3581}{24832}a^{14}+\frac{7735}{24832}a^{13}-\frac{5823}{24832}a^{12}-\frac{585}{24832}a^{11}-\frac{1337}{24832}a^{10}+\frac{1665}{24832}a^{9}-\frac{675}{6208}a^{8}+\frac{3421}{24832}a^{7}-\frac{49}{194}a^{6}-\frac{1485}{6208}a^{5}-\frac{5}{776}a^{4}-\frac{367}{776}a^{3}+\frac{55}{194}a^{2}+\frac{61}{388}a-\frac{83}{194}$, $\frac{1}{13\!\cdots\!72}a^{35}-\frac{55\!\cdots\!45}{84\!\cdots\!92}a^{34}-\frac{28\!\cdots\!75}{13\!\cdots\!72}a^{33}+\frac{36\!\cdots\!45}{33\!\cdots\!68}a^{32}-\frac{64\!\cdots\!01}{31\!\cdots\!04}a^{31}-\frac{26\!\cdots\!55}{33\!\cdots\!68}a^{30}+\frac{15\!\cdots\!75}{13\!\cdots\!72}a^{29}-\frac{33\!\cdots\!21}{84\!\cdots\!92}a^{28}-\frac{11\!\cdots\!95}{13\!\cdots\!72}a^{27}+\frac{66\!\cdots\!75}{67\!\cdots\!36}a^{26}-\frac{67\!\cdots\!03}{31\!\cdots\!04}a^{25}-\frac{15\!\cdots\!01}{67\!\cdots\!36}a^{24}-\frac{17\!\cdots\!57}{67\!\cdots\!36}a^{23}+\frac{82\!\cdots\!75}{84\!\cdots\!92}a^{22}-\frac{41\!\cdots\!93}{13\!\cdots\!72}a^{21}-\frac{29\!\cdots\!67}{31\!\cdots\!04}a^{20}-\frac{18\!\cdots\!81}{13\!\cdots\!72}a^{19}+\frac{36\!\cdots\!11}{13\!\cdots\!72}a^{18}+\frac{46\!\cdots\!41}{13\!\cdots\!72}a^{17}-\frac{42\!\cdots\!17}{13\!\cdots\!72}a^{16}+\frac{13\!\cdots\!03}{13\!\cdots\!72}a^{15}+\frac{46\!\cdots\!69}{13\!\cdots\!72}a^{14}-\frac{14\!\cdots\!85}{13\!\cdots\!72}a^{13}+\frac{55\!\cdots\!13}{13\!\cdots\!72}a^{12}-\frac{19\!\cdots\!87}{13\!\cdots\!72}a^{11}-\frac{37\!\cdots\!57}{13\!\cdots\!72}a^{10}+\frac{31\!\cdots\!01}{67\!\cdots\!36}a^{9}+\frac{48\!\cdots\!33}{13\!\cdots\!72}a^{8}+\frac{30\!\cdots\!27}{67\!\cdots\!36}a^{7}-\frac{10\!\cdots\!25}{33\!\cdots\!68}a^{6}+\frac{24\!\cdots\!23}{14\!\cdots\!68}a^{5}+\frac{95\!\cdots\!79}{21\!\cdots\!48}a^{4}-\frac{27\!\cdots\!47}{12\!\cdots\!44}a^{3}+\frac{95\!\cdots\!51}{21\!\cdots\!48}a^{2}-\frac{66\!\cdots\!02}{13\!\cdots\!53}a-\frac{14\!\cdots\!87}{52\!\cdots\!12}$ Copy content Toggle raw display

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

$C_{228}$, which has order $228$ (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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{609030628855225602477307881910046176601557254545316750254007356283546480650712762680928679958730926710244275164067110900204739915}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{35} - \frac{2153182908695010457958566190763198916071742008745147376341887724637126822901102900378557524661393568646410985093189859590122740085}{1672055417177030945869912401329937247252640770202735843002622105254872335409888415540723562437352372345279938889528759105052915724544} a^{34} + \frac{79762930469122760434845495368995858224657127364471491352623242873127458915558592830011638478190871634935424644370620909445231928755}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{33} - \frac{67910434380721177382118105064197751189770535511285050990438773063845017487866652894907163409482108159224906742444658855183390967475}{836027708588515472934956200664968623626320385101367921501311052627436167704944207770361781218676186172639969444764379552526457862272} a^{32} + \frac{3037056875246693120714152272732216086472893961192350177500353795470746180523859012250450571985408731401087603751617532377301180870443}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{31} - \frac{1796407286077463930666516659545813885767331252270446706293524669455037206446667215531188613170659930871110904445548551492145922219817}{836027708588515472934956200664968623626320385101367921501311052627436167704944207770361781218676186172639969444764379552526457862272} a^{30} + \frac{59350643620582557801856160881976225500154602844819704617536097851942357293660199343452718694418947957929532092946853085392624683418233}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{29} - \frac{54150456232424094383491602622531895681212307410674024225778407414093661505483401023551262375053004237756518598768987089321795867970931}{1672055417177030945869912401329937247252640770202735843002622105254872335409888415540723562437352372345279938889528759105052915724544} a^{28} + \frac{706042207892923577155404989877451042220312921170234980918287909027388016649579564643751004862820275647987062541547371003569192346421395}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{27} - \frac{1031388232765900042092037082512609335206214465451518992479754578452777348889494020981279075271727741476147845619509482312168679420728305}{3344110834354061891739824802659874494505281540405471686005244210509744670819776831081447124874704744690559877779057518210105831449088} a^{26} + \frac{5407595187711649485559684178554023210391751739974459716046652449781141912451196896377927500080657512944467146079857601061594578619201253}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{25} - \frac{6336130209320089901349053439282849747424327726301610706250046152600242112317909970555826237633524671204534981250982681284448561102133185}{3344110834354061891739824802659874494505281540405471686005244210509744670819776831081447124874704744690559877779057518210105831449088} a^{24} + \frac{13163583863960928827740309911832901411774739212024025805885284314734576044809492060414528745007736363405647686513354271797722070658112565}{3344110834354061891739824802659874494505281540405471686005244210509744670819776831081447124874704744690559877779057518210105831449088} a^{23} - \frac{5960296937758461770279391219019056571774697273570786677965203177558268623578724941728612588921920750882802072755437402728565819050169669}{836027708588515472934956200664968623626320385101367921501311052627436167704944207770361781218676186172639969444764379552526457862272} a^{22} + \frac{72859217752103146623461359608294551541023824402715242893269638885167851779445966908419394470245309892719372350235467426329745862197367741}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{21} - \frac{86943892873594007146682225027318322888559536654642309532283117857504558861473036502743617587604733481570856782320079332177189147310227395}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{20} + \frac{61125340063978522654198689925043998165722453250317737700664952668617196345422195811932418137035679620141740674769137530199314462515318545}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{19} + \frac{37435093252860707831290865093406287772261780432616317296386081970221708086729598270647543284180108471149446784485080501197999476431541577}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{18} - \frac{213476347305372617453656058523573666455840566359973671363132111618682796474617044727643733882754248327315259189576702998538440980650542569}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{17} + \frac{4125808209910068110846232133818362282076753198520174008135094068131671676035211218514401203780159611194557280646437063910138840385540393}{68950738852661069932779892838347927721758382276401478061963798154840096305562408888277260306694943189496079954207371509486718174208} a^{16} - \frac{450351070528161014161438421836438915911668900076785494344479798946252967895787352172101690776784958706838267143766393068959631302694108003}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{15} + \frac{221023966395200355929041629061957012037611998654462681730944868067234282610442969069540765224345426039453708612541785234811577070849167019}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{14} + \frac{264173013896266098699099619493647740908046577463618706761591494655092906909904606698410705754497309870194313852921273727853235696806946197}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{13} - \frac{720076478861026253055455003234560883741470886982211927172023870590341104107513075847478048553696978358899243240909673353013191171227051901}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{12} + \frac{759171305607600299950320704513354698390407292485136759065986576846125569968363201118576090675163386104589466483579658739565667565968278391}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{11} - \frac{277326106290327780038631075133913200722650301165945599134249169566880419759031113273691033983514336268510547421337230998186909467714056555}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{10} - \frac{148875570865996803888353458274687612328918009833902700071365574808561819813404214070600122256099299287941838297595550745610329127388890719}{3344110834354061891739824802659874494505281540405471686005244210509744670819776831081447124874704744690559877779057518210105831449088} a^{9} + \frac{381779407052265614285595595180222537730213969604502443693639086516353282221079484706009806079984539165476843235723117821948540629818951767}{6688221668708123783479649605319748989010563080810943372010488421019489341639553662162894249749409489381119755558115036420211662898176} a^{8} + \frac{18903029542923282409643593359118544659717841688406644642472280925761845976761554524718180855968709602431928470088818708053373562063731375}{3344110834354061891739824802659874494505281540405471686005244210509744670819776831081447124874704744690559877779057518210105831449088} a^{7} - \frac{87235117205616955861193607876963899799576764976119671518589257611862274728092586032699142603874979682970431944012436704019575464842977107}{1672055417177030945869912401329937247252640770202735843002622105254872335409888415540723562437352372345279938889528759105052915724544} a^{6} + \frac{28536211537379171641276501455782626190179887926088985284652802713998740407553510884519524747304898406297607398331309165450056187154456831}{836027708588515472934956200664968623626320385101367921501311052627436167704944207770361781218676186172639969444764379552526457862272} a^{5} - \frac{30502539945203283947427216119744687091881052175172419813777133185756057275330448912392356771713053593199350944469811258419338511118539}{13062932946695554264608690635390134744161256017208873773457985197303690120389753246411902831541815408947499522574443430508225904098} a^{4} - \frac{1114737349438572057293838495909798900649368780714314130542924159361082413873455902835731087034783862037836171040796515512357182704073}{625769242955475653394428293910904658402934419986053833459065159152272580617473209408953429055895348931616743596380523617160522352} a^{3} + \frac{11323778030093334071323614736955834482234224552541066699634978995580412018117802681635044162922551274296228611121317804722168742748409}{104503463573564434116869525083121077953290048137670990187663881578429520963118025971295222652334523271579996180595547444065807232784} a^{2} + \frac{740516031975566747114159616717379025082250259276401007314274228832973119750265030773152718937470280303630996020145486409248250073429}{26125865893391108529217381270780269488322512034417747546915970394607380240779506492823805663083630817894999045148886861016451808196} a + \frac{12058369745097593715440759455720931329805278729516724857067131350042644998986483759917797105824902049002783526786388028483412991021}{26125865893391108529217381270780269488322512034417747546915970394607380240779506492823805663083630817894999045148886861016451808196} \)  (order $14$) Copy content Toggle raw display
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:   $\frac{25\!\cdots\!37}{66\!\cdots\!76}a^{35}-\frac{87\!\cdots\!45}{16\!\cdots\!44}a^{34}+\frac{31\!\cdots\!97}{66\!\cdots\!76}a^{33}-\frac{33\!\cdots\!87}{10\!\cdots\!84}a^{32}+\frac{11\!\cdots\!25}{66\!\cdots\!76}a^{31}-\frac{34\!\cdots\!77}{41\!\cdots\!36}a^{30}+\frac{22\!\cdots\!47}{66\!\cdots\!76}a^{29}-\frac{19\!\cdots\!43}{16\!\cdots\!44}a^{28}+\frac{25\!\cdots\!41}{66\!\cdots\!76}a^{27}-\frac{36\!\cdots\!83}{33\!\cdots\!88}a^{26}+\frac{18\!\cdots\!59}{66\!\cdots\!76}a^{25}-\frac{21\!\cdots\!19}{33\!\cdots\!88}a^{24}+\frac{43\!\cdots\!99}{33\!\cdots\!88}a^{23}-\frac{19\!\cdots\!01}{83\!\cdots\!72}a^{22}+\frac{22\!\cdots\!55}{66\!\cdots\!76}a^{21}-\frac{23\!\cdots\!21}{66\!\cdots\!76}a^{20}+\frac{10\!\cdots\!03}{66\!\cdots\!76}a^{19}+\frac{24\!\cdots\!31}{66\!\cdots\!76}a^{18}-\frac{79\!\cdots\!35}{66\!\cdots\!76}a^{17}+\frac{13\!\cdots\!63}{68\!\cdots\!08}a^{16}-\frac{11\!\cdots\!29}{66\!\cdots\!76}a^{15}+\frac{17\!\cdots\!69}{66\!\cdots\!76}a^{14}+\frac{13\!\cdots\!83}{66\!\cdots\!76}a^{13}-\frac{24\!\cdots\!15}{66\!\cdots\!76}a^{12}+\frac{18\!\cdots\!57}{66\!\cdots\!76}a^{11}+\frac{64\!\cdots\!51}{66\!\cdots\!76}a^{10}-\frac{75\!\cdots\!33}{33\!\cdots\!88}a^{9}+\frac{91\!\cdots\!09}{66\!\cdots\!76}a^{8}+\frac{38\!\cdots\!41}{33\!\cdots\!88}a^{7}-\frac{30\!\cdots\!81}{16\!\cdots\!44}a^{6}+\frac{37\!\cdots\!05}{83\!\cdots\!72}a^{5}+\frac{22\!\cdots\!70}{65\!\cdots\!49}a^{4}-\frac{16\!\cdots\!63}{62\!\cdots\!52}a^{3}-\frac{73\!\cdots\!65}{10\!\cdots\!84}a^{2}-\frac{34\!\cdots\!99}{26\!\cdots\!96}a-\frac{29\!\cdots\!09}{26\!\cdots\!96}$, $\frac{17\!\cdots\!89}{66\!\cdots\!76}a^{35}-\frac{14\!\cdots\!05}{41\!\cdots\!36}a^{34}+\frac{20\!\cdots\!45}{66\!\cdots\!76}a^{33}-\frac{34\!\cdots\!07}{16\!\cdots\!44}a^{32}+\frac{75\!\cdots\!61}{66\!\cdots\!76}a^{31}-\frac{86\!\cdots\!67}{16\!\cdots\!44}a^{30}+\frac{13\!\cdots\!31}{66\!\cdots\!76}a^{29}-\frac{30\!\cdots\!05}{41\!\cdots\!36}a^{28}+\frac{15\!\cdots\!73}{66\!\cdots\!76}a^{27}-\frac{22\!\cdots\!61}{33\!\cdots\!88}a^{26}+\frac{11\!\cdots\!63}{66\!\cdots\!76}a^{25}-\frac{12\!\cdots\!69}{33\!\cdots\!88}a^{24}+\frac{24\!\cdots\!99}{33\!\cdots\!88}a^{23}-\frac{51\!\cdots\!97}{41\!\cdots\!36}a^{22}+\frac{11\!\cdots\!27}{66\!\cdots\!76}a^{21}-\frac{10\!\cdots\!89}{66\!\cdots\!76}a^{20}+\frac{71\!\cdots\!23}{66\!\cdots\!76}a^{19}+\frac{21\!\cdots\!67}{66\!\cdots\!76}a^{18}-\frac{51\!\cdots\!07}{66\!\cdots\!76}a^{17}+\frac{72\!\cdots\!15}{68\!\cdots\!08}a^{16}-\frac{49\!\cdots\!49}{66\!\cdots\!76}a^{15}-\frac{22\!\cdots\!51}{66\!\cdots\!76}a^{14}+\frac{11\!\cdots\!51}{66\!\cdots\!76}a^{13}-\frac{14\!\cdots\!59}{66\!\cdots\!76}a^{12}+\frac{72\!\cdots\!77}{66\!\cdots\!76}a^{11}+\frac{62\!\cdots\!07}{66\!\cdots\!76}a^{10}-\frac{63\!\cdots\!83}{33\!\cdots\!88}a^{9}+\frac{43\!\cdots\!81}{66\!\cdots\!76}a^{8}+\frac{40\!\cdots\!75}{33\!\cdots\!88}a^{7}-\frac{20\!\cdots\!81}{16\!\cdots\!44}a^{6}-\frac{57\!\cdots\!77}{83\!\cdots\!72}a^{5}+\frac{53\!\cdots\!69}{10\!\cdots\!84}a^{4}-\frac{28\!\cdots\!33}{62\!\cdots\!52}a^{3}-\frac{14\!\cdots\!85}{10\!\cdots\!84}a^{2}+\frac{20\!\cdots\!98}{65\!\cdots\!49}a-\frac{83\!\cdots\!35}{26\!\cdots\!96}$, $\frac{10\!\cdots\!17}{13\!\cdots\!72}a^{35}-\frac{30\!\cdots\!93}{33\!\cdots\!68}a^{34}+\frac{10\!\cdots\!29}{13\!\cdots\!72}a^{33}-\frac{19\!\cdots\!47}{42\!\cdots\!96}a^{32}+\frac{72\!\cdots\!55}{31\!\cdots\!04}a^{31}-\frac{82\!\cdots\!55}{84\!\cdots\!92}a^{30}+\frac{47\!\cdots\!91}{13\!\cdots\!72}a^{29}-\frac{37\!\cdots\!27}{33\!\cdots\!68}a^{28}+\frac{40\!\cdots\!45}{13\!\cdots\!72}a^{27}-\frac{47\!\cdots\!43}{67\!\cdots\!36}a^{26}+\frac{41\!\cdots\!45}{31\!\cdots\!04}a^{25}-\frac{11\!\cdots\!87}{67\!\cdots\!36}a^{24}+\frac{80\!\cdots\!35}{67\!\cdots\!36}a^{23}+\frac{13\!\cdots\!75}{16\!\cdots\!84}a^{22}-\frac{39\!\cdots\!45}{13\!\cdots\!72}a^{21}+\frac{22\!\cdots\!05}{31\!\cdots\!04}a^{20}-\frac{17\!\cdots\!41}{13\!\cdots\!72}a^{19}+\frac{23\!\cdots\!31}{13\!\cdots\!72}a^{18}-\frac{17\!\cdots\!31}{13\!\cdots\!72}a^{17}-\frac{11\!\cdots\!81}{13\!\cdots\!72}a^{16}+\frac{60\!\cdots\!63}{13\!\cdots\!72}a^{15}-\frac{10\!\cdots\!79}{13\!\cdots\!72}a^{14}+\frac{94\!\cdots\!43}{13\!\cdots\!72}a^{13}-\frac{42\!\cdots\!31}{13\!\cdots\!72}a^{12}-\frac{12\!\cdots\!39}{13\!\cdots\!72}a^{11}+\frac{18\!\cdots\!43}{13\!\cdots\!72}a^{10}-\frac{48\!\cdots\!61}{67\!\cdots\!36}a^{9}-\frac{66\!\cdots\!35}{13\!\cdots\!72}a^{8}+\frac{61\!\cdots\!01}{67\!\cdots\!36}a^{7}-\frac{30\!\cdots\!73}{33\!\cdots\!68}a^{6}-\frac{96\!\cdots\!99}{14\!\cdots\!68}a^{5}+\frac{20\!\cdots\!73}{52\!\cdots\!12}a^{4}+\frac{69\!\cdots\!85}{12\!\cdots\!44}a^{3}-\frac{32\!\cdots\!05}{21\!\cdots\!48}a^{2}-\frac{93\!\cdots\!19}{52\!\cdots\!12}a+\frac{36\!\cdots\!91}{52\!\cdots\!12}$, $\frac{61\!\cdots\!41}{67\!\cdots\!36}a^{35}-\frac{10\!\cdots\!63}{84\!\cdots\!92}a^{34}+\frac{78\!\cdots\!53}{67\!\cdots\!36}a^{33}-\frac{13\!\cdots\!25}{16\!\cdots\!84}a^{32}+\frac{68\!\cdots\!47}{15\!\cdots\!52}a^{31}-\frac{34\!\cdots\!81}{16\!\cdots\!84}a^{30}+\frac{56\!\cdots\!35}{67\!\cdots\!36}a^{29}-\frac{25\!\cdots\!75}{84\!\cdots\!92}a^{28}+\frac{66\!\cdots\!89}{67\!\cdots\!36}a^{27}-\frac{96\!\cdots\!13}{33\!\cdots\!68}a^{26}+\frac{11\!\cdots\!85}{15\!\cdots\!52}a^{25}-\frac{58\!\cdots\!81}{33\!\cdots\!68}a^{24}+\frac{12\!\cdots\!63}{33\!\cdots\!68}a^{23}-\frac{26\!\cdots\!67}{42\!\cdots\!96}a^{22}+\frac{64\!\cdots\!07}{67\!\cdots\!36}a^{21}-\frac{17\!\cdots\!71}{15\!\cdots\!52}a^{20}+\frac{47\!\cdots\!79}{67\!\cdots\!36}a^{19}+\frac{45\!\cdots\!71}{67\!\cdots\!36}a^{18}-\frac{20\!\cdots\!03}{67\!\cdots\!36}a^{17}+\frac{37\!\cdots\!43}{69\!\cdots\!88}a^{16}-\frac{38\!\cdots\!69}{67\!\cdots\!36}a^{15}+\frac{15\!\cdots\!17}{67\!\cdots\!36}a^{14}+\frac{28\!\cdots\!71}{67\!\cdots\!36}a^{13}-\frac{65\!\cdots\!95}{67\!\cdots\!36}a^{12}+\frac{62\!\cdots\!33}{67\!\cdots\!36}a^{11}-\frac{16\!\cdots\!53}{67\!\cdots\!36}a^{10}-\frac{15\!\cdots\!71}{33\!\cdots\!68}a^{9}+\frac{30\!\cdots\!85}{67\!\cdots\!36}a^{8}+\frac{47\!\cdots\!83}{33\!\cdots\!68}a^{7}-\frac{79\!\cdots\!53}{16\!\cdots\!84}a^{6}+\frac{17\!\cdots\!25}{74\!\cdots\!84}a^{5}+\frac{27\!\cdots\!85}{21\!\cdots\!48}a^{4}-\frac{17\!\cdots\!21}{31\!\cdots\!36}a^{3}-\frac{77\!\cdots\!05}{10\!\cdots\!24}a^{2}-\frac{99\!\cdots\!48}{13\!\cdots\!53}a-\frac{44\!\cdots\!11}{26\!\cdots\!06}$, $\frac{56\!\cdots\!69}{13\!\cdots\!72}a^{35}-\frac{19\!\cdots\!59}{33\!\cdots\!68}a^{34}+\frac{74\!\cdots\!01}{13\!\cdots\!72}a^{33}-\frac{63\!\cdots\!91}{16\!\cdots\!84}a^{32}+\frac{65\!\cdots\!67}{31\!\cdots\!04}a^{31}-\frac{16\!\cdots\!05}{16\!\cdots\!84}a^{30}+\frac{55\!\cdots\!91}{13\!\cdots\!72}a^{29}-\frac{50\!\cdots\!93}{33\!\cdots\!68}a^{28}+\frac{65\!\cdots\!81}{13\!\cdots\!72}a^{27}-\frac{95\!\cdots\!55}{67\!\cdots\!36}a^{26}+\frac{11\!\cdots\!17}{31\!\cdots\!04}a^{25}-\frac{58\!\cdots\!79}{67\!\cdots\!36}a^{24}+\frac{12\!\cdots\!75}{67\!\cdots\!36}a^{23}-\frac{54\!\cdots\!95}{16\!\cdots\!84}a^{22}+\frac{66\!\cdots\!23}{13\!\cdots\!72}a^{21}-\frac{18\!\cdots\!79}{31\!\cdots\!04}a^{20}+\frac{52\!\cdots\!75}{13\!\cdots\!72}a^{19}+\frac{40\!\cdots\!63}{13\!\cdots\!72}a^{18}-\frac{20\!\cdots\!67}{13\!\cdots\!72}a^{17}+\frac{38\!\cdots\!23}{13\!\cdots\!76}a^{16}-\frac{41\!\cdots\!57}{13\!\cdots\!72}a^{15}+\frac{18\!\cdots\!77}{13\!\cdots\!72}a^{14}+\frac{27\!\cdots\!15}{13\!\cdots\!72}a^{13}-\frac{69\!\cdots\!35}{13\!\cdots\!72}a^{12}+\frac{69\!\cdots\!01}{13\!\cdots\!72}a^{11}-\frac{21\!\cdots\!33}{13\!\cdots\!72}a^{10}-\frac{16\!\cdots\!21}{67\!\cdots\!36}a^{9}+\frac{38\!\cdots\!81}{13\!\cdots\!72}a^{8}+\frac{25\!\cdots\!93}{67\!\cdots\!36}a^{7}-\frac{88\!\cdots\!97}{33\!\cdots\!68}a^{6}+\frac{23\!\cdots\!21}{14\!\cdots\!68}a^{5}+\frac{72\!\cdots\!45}{10\!\cdots\!24}a^{4}-\frac{26\!\cdots\!87}{12\!\cdots\!44}a^{3}+\frac{20\!\cdots\!87}{21\!\cdots\!48}a^{2}+\frac{20\!\cdots\!41}{52\!\cdots\!12}a-\frac{57\!\cdots\!57}{52\!\cdots\!12}$, $\frac{10\!\cdots\!71}{13\!\cdots\!76}a^{35}-\frac{33\!\cdots\!15}{33\!\cdots\!68}a^{34}+\frac{12\!\cdots\!91}{13\!\cdots\!72}a^{33}-\frac{51\!\cdots\!09}{84\!\cdots\!92}a^{32}+\frac{10\!\cdots\!89}{31\!\cdots\!04}a^{31}-\frac{83\!\cdots\!87}{52\!\cdots\!12}a^{30}+\frac{86\!\cdots\!01}{13\!\cdots\!72}a^{29}-\frac{78\!\cdots\!25}{33\!\cdots\!68}a^{28}+\frac{10\!\cdots\!07}{13\!\cdots\!72}a^{27}-\frac{14\!\cdots\!85}{67\!\cdots\!36}a^{26}+\frac{17\!\cdots\!51}{31\!\cdots\!04}a^{25}-\frac{86\!\cdots\!49}{67\!\cdots\!36}a^{24}+\frac{17\!\cdots\!61}{67\!\cdots\!36}a^{23}-\frac{77\!\cdots\!19}{16\!\cdots\!84}a^{22}+\frac{90\!\cdots\!33}{13\!\cdots\!72}a^{21}-\frac{23\!\cdots\!29}{31\!\cdots\!04}a^{20}+\frac{52\!\cdots\!81}{13\!\cdots\!72}a^{19}+\frac{87\!\cdots\!93}{13\!\cdots\!72}a^{18}-\frac{31\!\cdots\!21}{13\!\cdots\!72}a^{17}+\frac{51\!\cdots\!41}{13\!\cdots\!72}a^{16}-\frac{50\!\cdots\!75}{13\!\cdots\!72}a^{15}+\frac{12\!\cdots\!07}{13\!\cdots\!72}a^{14}+\frac{50\!\cdots\!57}{13\!\cdots\!72}a^{13}-\frac{96\!\cdots\!57}{13\!\cdots\!72}a^{12}+\frac{81\!\cdots\!87}{13\!\cdots\!72}a^{11}-\frac{66\!\cdots\!67}{13\!\cdots\!72}a^{10}-\frac{27\!\cdots\!47}{67\!\cdots\!36}a^{9}+\frac{40\!\cdots\!15}{13\!\cdots\!72}a^{8}+\frac{12\!\cdots\!99}{67\!\cdots\!36}a^{7}-\frac{12\!\cdots\!11}{33\!\cdots\!68}a^{6}+\frac{18\!\cdots\!47}{14\!\cdots\!68}a^{5}+\frac{52\!\cdots\!03}{10\!\cdots\!24}a^{4}-\frac{62\!\cdots\!65}{12\!\cdots\!44}a^{3}-\frac{21\!\cdots\!03}{21\!\cdots\!48}a^{2}-\frac{13\!\cdots\!33}{52\!\cdots\!12}a+\frac{24\!\cdots\!69}{52\!\cdots\!12}$, $\frac{18\!\cdots\!09}{67\!\cdots\!36}a^{35}-\frac{78\!\cdots\!13}{21\!\cdots\!48}a^{34}+\frac{22\!\cdots\!81}{67\!\cdots\!36}a^{33}-\frac{37\!\cdots\!75}{16\!\cdots\!84}a^{32}+\frac{19\!\cdots\!03}{15\!\cdots\!52}a^{31}-\frac{96\!\cdots\!91}{16\!\cdots\!84}a^{30}+\frac{15\!\cdots\!19}{67\!\cdots\!36}a^{29}-\frac{34\!\cdots\!55}{42\!\cdots\!96}a^{28}+\frac{17\!\cdots\!53}{67\!\cdots\!36}a^{27}-\frac{25\!\cdots\!41}{33\!\cdots\!68}a^{26}+\frac{30\!\cdots\!69}{15\!\cdots\!52}a^{25}-\frac{14\!\cdots\!05}{33\!\cdots\!68}a^{24}+\frac{29\!\cdots\!19}{33\!\cdots\!68}a^{23}-\frac{64\!\cdots\!87}{42\!\cdots\!96}a^{22}+\frac{14\!\cdots\!15}{67\!\cdots\!36}a^{21}-\frac{34\!\cdots\!99}{15\!\cdots\!52}a^{20}+\frac{49\!\cdots\!79}{67\!\cdots\!36}a^{19}+\frac{20\!\cdots\!63}{67\!\cdots\!36}a^{18}-\frac{58\!\cdots\!39}{67\!\cdots\!36}a^{17}+\frac{89\!\cdots\!07}{69\!\cdots\!88}a^{16}-\frac{73\!\cdots\!73}{67\!\cdots\!36}a^{15}-\frac{27\!\cdots\!11}{67\!\cdots\!36}a^{14}+\frac{11\!\cdots\!27}{67\!\cdots\!36}a^{13}-\frac{17\!\cdots\!19}{67\!\cdots\!36}a^{12}+\frac{11\!\cdots\!25}{67\!\cdots\!36}a^{11}+\frac{32\!\cdots\!63}{67\!\cdots\!36}a^{10}-\frac{65\!\cdots\!87}{33\!\cdots\!68}a^{9}+\frac{68\!\cdots\!33}{67\!\cdots\!36}a^{8}+\frac{34\!\cdots\!95}{33\!\cdots\!68}a^{7}-\frac{24\!\cdots\!47}{16\!\cdots\!84}a^{6}+\frac{17\!\cdots\!63}{74\!\cdots\!84}a^{5}+\frac{90\!\cdots\!27}{21\!\cdots\!48}a^{4}-\frac{28\!\cdots\!53}{31\!\cdots\!36}a^{3}-\frac{98\!\cdots\!11}{10\!\cdots\!24}a^{2}+\frac{47\!\cdots\!85}{26\!\cdots\!06}a+\frac{97\!\cdots\!33}{26\!\cdots\!06}$, $\frac{36\!\cdots\!31}{67\!\cdots\!36}a^{35}-\frac{12\!\cdots\!41}{16\!\cdots\!84}a^{34}+\frac{47\!\cdots\!99}{67\!\cdots\!36}a^{33}-\frac{39\!\cdots\!85}{84\!\cdots\!92}a^{32}+\frac{41\!\cdots\!77}{15\!\cdots\!52}a^{31}-\frac{10\!\cdots\!91}{84\!\cdots\!92}a^{30}+\frac{34\!\cdots\!85}{67\!\cdots\!36}a^{29}-\frac{31\!\cdots\!99}{16\!\cdots\!84}a^{28}+\frac{40\!\cdots\!95}{67\!\cdots\!36}a^{27}-\frac{58\!\cdots\!37}{33\!\cdots\!68}a^{26}+\frac{71\!\cdots\!71}{15\!\cdots\!52}a^{25}-\frac{35\!\cdots\!65}{33\!\cdots\!68}a^{24}+\frac{73\!\cdots\!89}{33\!\cdots\!68}a^{23}-\frac{33\!\cdots\!21}{84\!\cdots\!92}a^{22}+\frac{40\!\cdots\!81}{67\!\cdots\!36}a^{21}-\frac{10\!\cdots\!01}{15\!\cdots\!52}a^{20}+\frac{30\!\cdots\!49}{67\!\cdots\!36}a^{19}+\frac{25\!\cdots\!13}{67\!\cdots\!36}a^{18}-\frac{12\!\cdots\!57}{67\!\cdots\!36}a^{17}+\frac{22\!\cdots\!41}{69\!\cdots\!88}a^{16}-\frac{24\!\cdots\!71}{67\!\cdots\!36}a^{15}+\frac{10\!\cdots\!35}{67\!\cdots\!36}a^{14}+\frac{16\!\cdots\!17}{67\!\cdots\!36}a^{13}-\frac{40\!\cdots\!01}{67\!\cdots\!36}a^{12}+\frac{39\!\cdots\!27}{67\!\cdots\!36}a^{11}-\frac{12\!\cdots\!91}{67\!\cdots\!36}a^{10}-\frac{89\!\cdots\!91}{33\!\cdots\!68}a^{9}+\frac{19\!\cdots\!83}{67\!\cdots\!36}a^{8}+\frac{22\!\cdots\!39}{33\!\cdots\!68}a^{7}-\frac{48\!\cdots\!07}{16\!\cdots\!84}a^{6}+\frac{11\!\cdots\!19}{74\!\cdots\!84}a^{5}-\frac{17\!\cdots\!89}{10\!\cdots\!24}a^{4}-\frac{10\!\cdots\!91}{31\!\cdots\!36}a^{3}+\frac{11\!\cdots\!15}{10\!\cdots\!24}a^{2}-\frac{28\!\cdots\!82}{13\!\cdots\!53}a-\frac{31\!\cdots\!59}{26\!\cdots\!06}$, $\frac{41\!\cdots\!01}{33\!\cdots\!68}a^{35}-\frac{60\!\cdots\!65}{33\!\cdots\!68}a^{34}+\frac{56\!\cdots\!71}{33\!\cdots\!68}a^{33}-\frac{38\!\cdots\!01}{33\!\cdots\!68}a^{32}+\frac{51\!\cdots\!33}{78\!\cdots\!76}a^{31}-\frac{10\!\cdots\!01}{33\!\cdots\!68}a^{30}+\frac{43\!\cdots\!65}{33\!\cdots\!68}a^{29}-\frac{16\!\cdots\!99}{33\!\cdots\!68}a^{28}+\frac{52\!\cdots\!31}{33\!\cdots\!68}a^{27}-\frac{15\!\cdots\!63}{33\!\cdots\!68}a^{26}+\frac{96\!\cdots\!61}{78\!\cdots\!76}a^{25}-\frac{97\!\cdots\!89}{33\!\cdots\!68}a^{24}+\frac{10\!\cdots\!29}{16\!\cdots\!84}a^{23}-\frac{18\!\cdots\!51}{16\!\cdots\!84}a^{22}+\frac{59\!\cdots\!99}{33\!\cdots\!68}a^{21}-\frac{10\!\cdots\!37}{49\!\cdots\!36}a^{20}+\frac{29\!\cdots\!73}{16\!\cdots\!84}a^{19}+\frac{79\!\cdots\!37}{16\!\cdots\!84}a^{18}-\frac{39\!\cdots\!03}{84\!\cdots\!92}a^{17}+\frac{20\!\cdots\!15}{21\!\cdots\!84}a^{16}-\frac{19\!\cdots\!19}{16\!\cdots\!84}a^{15}+\frac{58\!\cdots\!95}{84\!\cdots\!92}a^{14}+\frac{59\!\cdots\!52}{13\!\cdots\!53}a^{13}-\frac{17\!\cdots\!55}{10\!\cdots\!24}a^{12}+\frac{33\!\cdots\!19}{16\!\cdots\!84}a^{11}-\frac{25\!\cdots\!41}{26\!\cdots\!06}a^{10}-\frac{17\!\cdots\!27}{33\!\cdots\!68}a^{9}+\frac{33\!\cdots\!37}{33\!\cdots\!68}a^{8}-\frac{41\!\cdots\!47}{33\!\cdots\!68}a^{7}-\frac{66\!\cdots\!31}{84\!\cdots\!92}a^{6}+\frac{50\!\cdots\!49}{74\!\cdots\!84}a^{5}-\frac{67\!\cdots\!45}{52\!\cdots\!12}a^{4}-\frac{25\!\cdots\!49}{63\!\cdots\!72}a^{3}+\frac{21\!\cdots\!57}{52\!\cdots\!12}a^{2}+\frac{49\!\cdots\!43}{52\!\cdots\!12}a+\frac{35\!\cdots\!73}{26\!\cdots\!06}$, $\frac{56\!\cdots\!51}{67\!\cdots\!36}a^{35}-\frac{39\!\cdots\!67}{33\!\cdots\!68}a^{34}+\frac{73\!\cdots\!51}{67\!\cdots\!36}a^{33}-\frac{25\!\cdots\!57}{33\!\cdots\!68}a^{32}+\frac{64\!\cdots\!21}{15\!\cdots\!52}a^{31}-\frac{65\!\cdots\!81}{33\!\cdots\!68}a^{30}+\frac{54\!\cdots\!17}{67\!\cdots\!36}a^{29}-\frac{98\!\cdots\!89}{33\!\cdots\!68}a^{28}+\frac{64\!\cdots\!51}{67\!\cdots\!36}a^{27}-\frac{46\!\cdots\!03}{16\!\cdots\!84}a^{26}+\frac{11\!\cdots\!39}{15\!\cdots\!52}a^{25}-\frac{89\!\cdots\!31}{52\!\cdots\!12}a^{24}+\frac{11\!\cdots\!41}{33\!\cdots\!68}a^{23}-\frac{10\!\cdots\!01}{16\!\cdots\!84}a^{22}+\frac{64\!\cdots\!85}{67\!\cdots\!36}a^{21}-\frac{17\!\cdots\!63}{15\!\cdots\!52}a^{20}+\frac{51\!\cdots\!15}{67\!\cdots\!36}a^{19}+\frac{37\!\cdots\!55}{67\!\cdots\!36}a^{18}-\frac{19\!\cdots\!99}{67\!\cdots\!36}a^{17}+\frac{36\!\cdots\!67}{69\!\cdots\!88}a^{16}-\frac{39\!\cdots\!85}{67\!\cdots\!36}a^{15}+\frac{17\!\cdots\!77}{67\!\cdots\!36}a^{14}+\frac{25\!\cdots\!03}{67\!\cdots\!36}a^{13}-\frac{64\!\cdots\!55}{67\!\cdots\!36}a^{12}+\frac{65\!\cdots\!09}{67\!\cdots\!36}a^{11}-\frac{21\!\cdots\!97}{67\!\cdots\!36}a^{10}-\frac{34\!\cdots\!41}{84\!\cdots\!92}a^{9}+\frac{32\!\cdots\!87}{67\!\cdots\!36}a^{8}+\frac{14\!\cdots\!01}{16\!\cdots\!84}a^{7}-\frac{77\!\cdots\!01}{16\!\cdots\!84}a^{6}+\frac{10\!\cdots\!61}{37\!\cdots\!92}a^{5}-\frac{93\!\cdots\!17}{10\!\cdots\!24}a^{4}-\frac{58\!\cdots\!11}{63\!\cdots\!72}a^{3}+\frac{40\!\cdots\!89}{10\!\cdots\!24}a^{2}+\frac{59\!\cdots\!23}{52\!\cdots\!12}a-\frac{25\!\cdots\!99}{13\!\cdots\!53}$, $\frac{87\!\cdots\!39}{33\!\cdots\!68}a^{35}-\frac{57\!\cdots\!81}{16\!\cdots\!84}a^{34}+\frac{10\!\cdots\!99}{33\!\cdots\!68}a^{33}-\frac{33\!\cdots\!63}{16\!\cdots\!84}a^{32}+\frac{83\!\cdots\!33}{78\!\cdots\!76}a^{31}-\frac{82\!\cdots\!27}{16\!\cdots\!84}a^{30}+\frac{65\!\cdots\!01}{33\!\cdots\!68}a^{29}-\frac{11\!\cdots\!27}{16\!\cdots\!84}a^{28}+\frac{72\!\cdots\!83}{33\!\cdots\!68}a^{27}-\frac{25\!\cdots\!57}{42\!\cdots\!96}a^{26}+\frac{11\!\cdots\!51}{78\!\cdots\!76}a^{25}-\frac{27\!\cdots\!73}{84\!\cdots\!92}a^{24}+\frac{10\!\cdots\!69}{16\!\cdots\!84}a^{23}-\frac{87\!\cdots\!71}{84\!\cdots\!92}a^{22}+\frac{45\!\cdots\!17}{33\!\cdots\!68}a^{21}-\frac{86\!\cdots\!23}{78\!\cdots\!76}a^{20}-\frac{10\!\cdots\!13}{33\!\cdots\!68}a^{19}+\frac{10\!\cdots\!63}{33\!\cdots\!68}a^{18}-\frac{22\!\cdots\!15}{33\!\cdots\!68}a^{17}+\frac{27\!\cdots\!59}{33\!\cdots\!68}a^{16}-\frac{14\!\cdots\!45}{33\!\cdots\!68}a^{15}-\frac{18\!\cdots\!91}{33\!\cdots\!68}a^{14}+\frac{52\!\cdots\!43}{33\!\cdots\!68}a^{13}-\frac{56\!\cdots\!79}{33\!\cdots\!68}a^{12}+\frac{15\!\cdots\!73}{33\!\cdots\!68}a^{11}+\frac{41\!\cdots\!71}{33\!\cdots\!68}a^{10}-\frac{13\!\cdots\!55}{84\!\cdots\!92}a^{9}+\frac{22\!\cdots\!03}{33\!\cdots\!68}a^{8}+\frac{59\!\cdots\!51}{42\!\cdots\!96}a^{7}-\frac{74\!\cdots\!59}{84\!\cdots\!92}a^{6}-\frac{81\!\cdots\!87}{18\!\cdots\!96}a^{5}+\frac{50\!\cdots\!33}{10\!\cdots\!24}a^{4}+\frac{37\!\cdots\!51}{31\!\cdots\!36}a^{3}-\frac{19\!\cdots\!35}{52\!\cdots\!12}a^{2}-\frac{16\!\cdots\!17}{26\!\cdots\!06}a+\frac{30\!\cdots\!21}{13\!\cdots\!53}$, $\frac{42\!\cdots\!59}{20\!\cdots\!04}a^{35}-\frac{59\!\cdots\!21}{20\!\cdots\!04}a^{34}+\frac{54\!\cdots\!13}{20\!\cdots\!04}a^{33}-\frac{37\!\cdots\!69}{20\!\cdots\!04}a^{32}+\frac{48\!\cdots\!87}{47\!\cdots\!28}a^{31}-\frac{97\!\cdots\!01}{20\!\cdots\!04}a^{30}+\frac{40\!\cdots\!15}{20\!\cdots\!04}a^{29}-\frac{14\!\cdots\!35}{20\!\cdots\!04}a^{28}+\frac{47\!\cdots\!05}{20\!\cdots\!04}a^{27}-\frac{13\!\cdots\!07}{20\!\cdots\!04}a^{26}+\frac{84\!\cdots\!35}{47\!\cdots\!28}a^{25}-\frac{84\!\cdots\!33}{20\!\cdots\!04}a^{24}+\frac{87\!\cdots\!53}{10\!\cdots\!52}a^{23}-\frac{16\!\cdots\!63}{10\!\cdots\!16}a^{22}+\frac{47\!\cdots\!09}{20\!\cdots\!04}a^{21}-\frac{64\!\cdots\!15}{23\!\cdots\!64}a^{20}+\frac{93\!\cdots\!39}{50\!\cdots\!76}a^{19}+\frac{71\!\cdots\!71}{50\!\cdots\!76}a^{18}-\frac{72\!\cdots\!55}{10\!\cdots\!52}a^{17}+\frac{13\!\cdots\!41}{10\!\cdots\!52}a^{16}-\frac{71\!\cdots\!03}{50\!\cdots\!76}a^{15}+\frac{63\!\cdots\!77}{10\!\cdots\!52}a^{14}+\frac{95\!\cdots\!25}{10\!\cdots\!52}a^{13}-\frac{23\!\cdots\!85}{10\!\cdots\!52}a^{12}+\frac{11\!\cdots\!13}{50\!\cdots\!76}a^{11}-\frac{75\!\cdots\!07}{10\!\cdots\!52}a^{10}-\frac{21\!\cdots\!87}{20\!\cdots\!04}a^{9}+\frac{23\!\cdots\!43}{20\!\cdots\!04}a^{8}+\frac{48\!\cdots\!33}{20\!\cdots\!32}a^{7}-\frac{72\!\cdots\!21}{63\!\cdots\!72}a^{6}+\frac{29\!\cdots\!21}{44\!\cdots\!52}a^{5}-\frac{14\!\cdots\!35}{15\!\cdots\!18}a^{4}-\frac{15\!\cdots\!33}{63\!\cdots\!72}a^{3}+\frac{11\!\cdots\!31}{31\!\cdots\!36}a^{2}+\frac{10\!\cdots\!51}{31\!\cdots\!36}a-\frac{53\!\cdots\!89}{15\!\cdots\!18}$, $\frac{18\!\cdots\!07}{67\!\cdots\!36}a^{35}-\frac{12\!\cdots\!39}{33\!\cdots\!68}a^{34}+\frac{23\!\cdots\!59}{67\!\cdots\!36}a^{33}-\frac{79\!\cdots\!17}{33\!\cdots\!68}a^{32}+\frac{20\!\cdots\!97}{15\!\cdots\!52}a^{31}-\frac{20\!\cdots\!01}{33\!\cdots\!68}a^{30}+\frac{17\!\cdots\!85}{67\!\cdots\!36}a^{29}-\frac{30\!\cdots\!29}{33\!\cdots\!68}a^{28}+\frac{19\!\cdots\!99}{67\!\cdots\!36}a^{27}-\frac{14\!\cdots\!05}{16\!\cdots\!84}a^{26}+\frac{34\!\cdots\!91}{15\!\cdots\!52}a^{25}-\frac{43\!\cdots\!45}{84\!\cdots\!92}a^{24}+\frac{35\!\cdots\!29}{33\!\cdots\!68}a^{23}-\frac{31\!\cdots\!97}{16\!\cdots\!84}a^{22}+\frac{18\!\cdots\!97}{67\!\cdots\!36}a^{21}-\frac{49\!\cdots\!07}{15\!\cdots\!52}a^{20}+\frac{12\!\cdots\!79}{67\!\cdots\!36}a^{19}+\frac{15\!\cdots\!83}{67\!\cdots\!36}a^{18}-\frac{61\!\cdots\!71}{67\!\cdots\!36}a^{17}+\frac{10\!\cdots\!43}{67\!\cdots\!36}a^{16}-\frac{10\!\cdots\!77}{67\!\cdots\!36}a^{15}+\frac{37\!\cdots\!29}{67\!\cdots\!36}a^{14}+\frac{91\!\cdots\!87}{67\!\cdots\!36}a^{13}-\frac{19\!\cdots\!71}{67\!\cdots\!36}a^{12}+\frac{17\!\cdots\!33}{67\!\cdots\!36}a^{11}-\frac{34\!\cdots\!13}{67\!\cdots\!36}a^{10}-\frac{12\!\cdots\!57}{84\!\cdots\!92}a^{9}+\frac{88\!\cdots\!31}{67\!\cdots\!36}a^{8}+\frac{92\!\cdots\!79}{16\!\cdots\!84}a^{7}-\frac{24\!\cdots\!23}{16\!\cdots\!84}a^{6}+\frac{23\!\cdots\!31}{37\!\cdots\!92}a^{5}+\frac{58\!\cdots\!13}{52\!\cdots\!12}a^{4}-\frac{15\!\cdots\!05}{63\!\cdots\!72}a^{3}-\frac{42\!\cdots\!85}{10\!\cdots\!24}a^{2}+\frac{66\!\cdots\!89}{52\!\cdots\!12}a+\frac{13\!\cdots\!18}{13\!\cdots\!53}$, $\frac{13\!\cdots\!85}{40\!\cdots\!08}a^{35}-\frac{95\!\cdots\!23}{20\!\cdots\!04}a^{34}+\frac{17\!\cdots\!33}{40\!\cdots\!08}a^{33}-\frac{60\!\cdots\!69}{20\!\cdots\!04}a^{32}+\frac{15\!\cdots\!47}{94\!\cdots\!56}a^{31}-\frac{15\!\cdots\!37}{20\!\cdots\!04}a^{30}+\frac{13\!\cdots\!91}{40\!\cdots\!08}a^{29}-\frac{24\!\cdots\!53}{20\!\cdots\!04}a^{28}+\frac{15\!\cdots\!65}{40\!\cdots\!08}a^{27}-\frac{71\!\cdots\!25}{63\!\cdots\!72}a^{26}+\frac{27\!\cdots\!81}{94\!\cdots\!56}a^{25}-\frac{70\!\cdots\!13}{10\!\cdots\!52}a^{24}+\frac{29\!\cdots\!83}{20\!\cdots\!04}a^{23}-\frac{27\!\cdots\!05}{10\!\cdots\!16}a^{22}+\frac{16\!\cdots\!95}{40\!\cdots\!08}a^{21}-\frac{44\!\cdots\!65}{94\!\cdots\!56}a^{20}+\frac{13\!\cdots\!13}{40\!\cdots\!08}a^{19}+\frac{82\!\cdots\!37}{40\!\cdots\!08}a^{18}-\frac{47\!\cdots\!81}{40\!\cdots\!08}a^{17}+\frac{88\!\cdots\!33}{40\!\cdots\!08}a^{16}-\frac{99\!\cdots\!55}{40\!\cdots\!08}a^{15}+\frac{48\!\cdots\!79}{40\!\cdots\!08}a^{14}+\frac{57\!\cdots\!29}{40\!\cdots\!08}a^{13}-\frac{15\!\cdots\!37}{40\!\cdots\!08}a^{12}+\frac{16\!\cdots\!19}{40\!\cdots\!08}a^{11}-\frac{61\!\cdots\!31}{40\!\cdots\!08}a^{10}-\frac{16\!\cdots\!61}{10\!\cdots\!52}a^{9}+\frac{82\!\cdots\!45}{40\!\cdots\!08}a^{8}+\frac{55\!\cdots\!57}{26\!\cdots\!04}a^{7}-\frac{18\!\cdots\!27}{10\!\cdots\!52}a^{6}+\frac{27\!\cdots\!39}{22\!\cdots\!76}a^{5}-\frac{16\!\cdots\!45}{15\!\cdots\!18}a^{4}-\frac{27\!\cdots\!47}{63\!\cdots\!72}a^{3}+\frac{28\!\cdots\!69}{63\!\cdots\!72}a^{2}+\frac{10\!\cdots\!67}{31\!\cdots\!36}a-\frac{49\!\cdots\!92}{79\!\cdots\!59}$, $\frac{41\!\cdots\!13}{20\!\cdots\!04}a^{35}-\frac{58\!\cdots\!41}{20\!\cdots\!04}a^{34}+\frac{53\!\cdots\!71}{20\!\cdots\!04}a^{33}-\frac{36\!\cdots\!73}{20\!\cdots\!04}a^{32}+\frac{46\!\cdots\!05}{47\!\cdots\!28}a^{31}-\frac{95\!\cdots\!81}{20\!\cdots\!04}a^{30}+\frac{39\!\cdots\!01}{20\!\cdots\!04}a^{29}-\frac{14\!\cdots\!27}{20\!\cdots\!04}a^{28}+\frac{46\!\cdots\!63}{20\!\cdots\!04}a^{27}-\frac{13\!\cdots\!07}{20\!\cdots\!04}a^{26}+\frac{81\!\cdots\!81}{47\!\cdots\!28}a^{25}-\frac{81\!\cdots\!61}{20\!\cdots\!04}a^{24}+\frac{84\!\cdots\!07}{10\!\cdots\!52}a^{23}-\frac{15\!\cdots\!87}{10\!\cdots\!16}a^{22}+\frac{45\!\cdots\!03}{20\!\cdots\!04}a^{21}-\frac{15\!\cdots\!15}{58\!\cdots\!16}a^{20}+\frac{17\!\cdots\!51}{10\!\cdots\!52}a^{19}+\frac{14\!\cdots\!71}{10\!\cdots\!52}a^{18}-\frac{54\!\cdots\!74}{79\!\cdots\!59}a^{17}+\frac{63\!\cdots\!27}{50\!\cdots\!76}a^{16}-\frac{13\!\cdots\!93}{10\!\cdots\!52}a^{15}+\frac{15\!\cdots\!93}{25\!\cdots\!88}a^{14}+\frac{47\!\cdots\!79}{50\!\cdots\!76}a^{13}-\frac{11\!\cdots\!67}{50\!\cdots\!76}a^{12}+\frac{23\!\cdots\!25}{10\!\cdots\!52}a^{11}-\frac{34\!\cdots\!35}{50\!\cdots\!76}a^{10}-\frac{20\!\cdots\!35}{20\!\cdots\!04}a^{9}+\frac{22\!\cdots\!77}{20\!\cdots\!04}a^{8}+\frac{51\!\cdots\!05}{20\!\cdots\!32}a^{7}-\frac{28\!\cdots\!07}{25\!\cdots\!88}a^{6}+\frac{28\!\cdots\!53}{44\!\cdots\!52}a^{5}+\frac{19\!\cdots\!99}{12\!\cdots\!44}a^{4}-\frac{15\!\cdots\!49}{63\!\cdots\!72}a^{3}-\frac{15\!\cdots\!39}{31\!\cdots\!36}a^{2}-\frac{14\!\cdots\!17}{31\!\cdots\!36}a+\frac{12\!\cdots\!99}{15\!\cdots\!18}$, $\frac{15\!\cdots\!49}{13\!\cdots\!72}a^{35}-\frac{52\!\cdots\!43}{33\!\cdots\!68}a^{34}+\frac{19\!\cdots\!53}{13\!\cdots\!72}a^{33}-\frac{16\!\cdots\!67}{16\!\cdots\!84}a^{32}+\frac{17\!\cdots\!07}{31\!\cdots\!04}a^{31}-\frac{43\!\cdots\!45}{16\!\cdots\!84}a^{30}+\frac{14\!\cdots\!15}{13\!\cdots\!72}a^{29}-\frac{13\!\cdots\!41}{33\!\cdots\!68}a^{28}+\frac{16\!\cdots\!85}{13\!\cdots\!72}a^{27}-\frac{24\!\cdots\!31}{67\!\cdots\!36}a^{26}+\frac{30\!\cdots\!89}{31\!\cdots\!04}a^{25}-\frac{15\!\cdots\!55}{67\!\cdots\!36}a^{24}+\frac{31\!\cdots\!95}{67\!\cdots\!36}a^{23}-\frac{14\!\cdots\!91}{16\!\cdots\!84}a^{22}+\frac{17\!\cdots\!83}{13\!\cdots\!72}a^{21}-\frac{46\!\cdots\!47}{31\!\cdots\!04}a^{20}+\frac{13\!\cdots\!67}{13\!\cdots\!72}a^{19}+\frac{10\!\cdots\!71}{13\!\cdots\!72}a^{18}-\frac{51\!\cdots\!63}{13\!\cdots\!72}a^{17}+\frac{93\!\cdots\!11}{13\!\cdots\!72}a^{16}-\frac{10\!\cdots\!49}{13\!\cdots\!72}a^{15}+\frac{46\!\cdots\!09}{13\!\cdots\!72}a^{14}+\frac{67\!\cdots\!31}{13\!\cdots\!72}a^{13}-\frac{17\!\cdots\!39}{13\!\cdots\!72}a^{12}+\frac{17\!\cdots\!33}{13\!\cdots\!72}a^{11}-\frac{54\!\cdots\!29}{13\!\cdots\!72}a^{10}-\frac{37\!\cdots\!69}{67\!\cdots\!36}a^{9}+\frac{86\!\cdots\!17}{13\!\cdots\!72}a^{8}+\frac{80\!\cdots\!61}{67\!\cdots\!36}a^{7}-\frac{20\!\cdots\!61}{33\!\cdots\!68}a^{6}+\frac{53\!\cdots\!25}{14\!\cdots\!68}a^{5}-\frac{59\!\cdots\!79}{10\!\cdots\!24}a^{4}-\frac{20\!\cdots\!21}{12\!\cdots\!44}a^{3}+\frac{67\!\cdots\!19}{21\!\cdots\!48}a^{2}+\frac{12\!\cdots\!49}{52\!\cdots\!12}a+\frac{13\!\cdots\!19}{52\!\cdots\!12}$, $\frac{21\!\cdots\!03}{13\!\cdots\!72}a^{35}-\frac{36\!\cdots\!89}{16\!\cdots\!84}a^{34}+\frac{27\!\cdots\!71}{13\!\cdots\!72}a^{33}-\frac{46\!\cdots\!27}{33\!\cdots\!68}a^{32}+\frac{23\!\cdots\!49}{31\!\cdots\!04}a^{31}-\frac{12\!\cdots\!71}{33\!\cdots\!68}a^{30}+\frac{19\!\cdots\!17}{13\!\cdots\!72}a^{29}-\frac{90\!\cdots\!51}{16\!\cdots\!84}a^{28}+\frac{23\!\cdots\!03}{13\!\cdots\!72}a^{27}-\frac{34\!\cdots\!27}{67\!\cdots\!36}a^{26}+\frac{41\!\cdots\!27}{31\!\cdots\!04}a^{25}-\frac{20\!\cdots\!87}{67\!\cdots\!36}a^{24}+\frac{42\!\cdots\!81}{67\!\cdots\!36}a^{23}-\frac{11\!\cdots\!31}{10\!\cdots\!24}a^{22}+\frac{23\!\cdots\!13}{13\!\cdots\!72}a^{21}-\frac{63\!\cdots\!65}{31\!\cdots\!04}a^{20}+\frac{17\!\cdots\!61}{13\!\cdots\!72}a^{19}+\frac{14\!\cdots\!45}{13\!\cdots\!72}a^{18}-\frac{70\!\cdots\!25}{13\!\cdots\!72}a^{17}+\frac{12\!\cdots\!17}{13\!\cdots\!72}a^{16}-\frac{13\!\cdots\!35}{13\!\cdots\!72}a^{15}+\frac{60\!\cdots\!07}{13\!\cdots\!72}a^{14}+\frac{93\!\cdots\!69}{13\!\cdots\!72}a^{13}-\frac{23\!\cdots\!21}{13\!\cdots\!72}a^{12}+\frac{23\!\cdots\!55}{13\!\cdots\!72}a^{11}-\frac{70\!\cdots\!59}{13\!\cdots\!72}a^{10}-\frac{51\!\cdots\!13}{67\!\cdots\!36}a^{9}+\frac{11\!\cdots\!67}{13\!\cdots\!72}a^{8}+\frac{13\!\cdots\!13}{67\!\cdots\!36}a^{7}-\frac{27\!\cdots\!07}{33\!\cdots\!68}a^{6}+\frac{68\!\cdots\!61}{14\!\cdots\!68}a^{5}-\frac{10\!\cdots\!36}{13\!\cdots\!53}a^{4}-\frac{66\!\cdots\!31}{12\!\cdots\!44}a^{3}-\frac{15\!\cdots\!51}{21\!\cdots\!48}a^{2}-\frac{44\!\cdots\!39}{26\!\cdots\!06}a+\frac{98\!\cdots\!11}{52\!\cdots\!12}$ Copy content Toggle raw display (assuming GRH)
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:  \( 154416550498474.53 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 154416550498474.53 \cdot 228}{14\cdot\sqrt{4000715416325500851269158271470993386638594171162518886820241}}\cr\approx \mathstrut & 0.292866339099107 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512)
 
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^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512, 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^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512);
 
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^36 - 14*x^35 + 129*x^34 - 874*x^33 + 4865*x^32 - 22922*x^31 + 94283*x^30 - 342666*x^29 + 1112273*x^28 - 3234952*x^27 + 8438667*x^26 - 19663764*x^25 + 40577370*x^24 - 72857172*x^23 + 109953255*x^22 - 128328007*x^21 + 83898469*x^20 + 71389621*x^19 - 339203441*x^18 + 610359985*x^17 - 658477439*x^16 + 279191379*x^15 + 463357277*x^14 - 1113046997*x^13 + 1096834803*x^12 - 317516043*x^11 - 517559160*x^10 + 549915913*x^9 + 131226304*x^8 - 546397416*x^7 + 300520880*x^6 + 8345200*x^5 - 14423488*x^4 - 397760*x^3 + 206720*x^2 + 13568*x + 512);
 
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

$C_2\times A_4\times D_6$ (as 36T334):

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 288
The 48 conjugacy class representatives for $C_2\times A_4\times D_6$
Character table for $C_2\times A_4\times D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.3.469.1, 6.0.400967.1, 6.6.2806769.1, 6.0.1539727.2, \(\Q(\zeta_{7})\), 9.9.247691263309.1, 12.0.7877952219361.1, 18.0.429456733437258587406367.1, 18.0.285739835154984956039631466703.1, 18.18.2000178846084894692277420266921.1

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 36 siblings: deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, some data not computed
Minimal sibling: not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{4}{,}\,{\href{/padicField/2.3.0.1}{3} }^{4}$ ${\href{/padicField/3.6.0.1}{6} }^{6}$ ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{18}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
\(67\) Copy content Toggle raw display 67.6.0.1$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
67.6.0.1$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
67.12.6.1$x^{12} + 402 x^{10} + 126 x^{9} + 67433 x^{8} + 110 x^{7} + 5992969 x^{6} - 3453840 x^{5} + 298670675 x^{4} - 304805096 x^{3} + 8036896364 x^{2} - 7423678304 x + 93605576188$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
67.12.6.1$x^{12} + 402 x^{10} + 126 x^{9} + 67433 x^{8} + 110 x^{7} + 5992969 x^{6} - 3453840 x^{5} + 298670675 x^{4} - 304805096 x^{3} + 8036896364 x^{2} - 7423678304 x + 93605576188$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(167\) Copy content Toggle raw display 167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$