Normalized defining polynomial

\( x^{27} - 2 x^{26} + x^{25} + 18 x^{24} + 107 x^{23} + 180 x^{22} + 332 x^{21} + 290 x^{20} + 295 x^{19} + \cdots - 1 \) x^27 - 2*x^26 + x^25 + 18*x^24 + 107*x^23 + 180*x^22 + 332*x^21 + 290*x^20 + 295*x^19 - 179*x^18 - 772*x^17 - 1774*x^16 - 2092*x^15 - 2320*x^14 - 811*x^13 + 489*x^12 + 3551*x^11 + 5424*x^10 + 7617*x^9 + 6962*x^8 + 6380*x^7 + 2824*x^6 + 2291*x^5 - 521*x^4 + 623*x^3 - 221*x^2 + 207*x - 1

Invariants

Degree:		$27$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[1, 13]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(-3639553781467035763182087002112051895733239\) -3639553781467035763182087002112051895733239 \(\medspace = -\,1879^{13}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(37.70\)	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:		$1879^{1/2}\approx 43.347433603386484$
Ramified primes:		\(1879\) 1879	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{-1879}) \)
$\card{ \Aut(K/\Q) }$:		$1$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{9}a^{5}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{4}{9}a^{2}-\frac{1}{9}a+\frac{4}{9}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{1}{3}a^{7}-\frac{1}{9}a^{6}-\frac{1}{3}a^{5}+\frac{4}{9}a^{4}+\frac{4}{9}a^{3}-\frac{1}{9}a^{2}+\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{12}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{4}{9}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{9}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}-\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{27}a^{18}+\frac{1}{27}a^{16}+\frac{1}{27}a^{15}-\frac{4}{27}a^{12}-\frac{4}{27}a^{10}-\frac{2}{27}a^{9}-\frac{1}{9}a^{8}+\frac{5}{27}a^{7}+\frac{4}{9}a^{6}+\frac{1}{9}a^{5}-\frac{5}{27}a^{4}+\frac{1}{9}a^{3}+\frac{8}{27}a-\frac{7}{27}$, $\frac{1}{27}a^{19}+\frac{1}{27}a^{17}+\frac{1}{27}a^{16}-\frac{1}{27}a^{13}+\frac{2}{27}a^{11}+\frac{4}{27}a^{10}+\frac{2}{27}a^{8}+\frac{1}{9}a^{7}-\frac{2}{9}a^{6}-\frac{8}{27}a^{5}-\frac{2}{9}a^{4}+\frac{1}{9}a^{3}+\frac{11}{27}a^{2}-\frac{10}{27}a+\frac{4}{9}$, $\frac{1}{27}a^{20}+\frac{1}{27}a^{17}-\frac{1}{27}a^{16}-\frac{1}{27}a^{15}-\frac{1}{27}a^{14}-\frac{1}{9}a^{12}+\frac{4}{27}a^{11}+\frac{4}{27}a^{10}+\frac{4}{27}a^{9}-\frac{1}{9}a^{8}-\frac{11}{27}a^{7}+\frac{7}{27}a^{6}-\frac{1}{3}a^{5}-\frac{10}{27}a^{4}+\frac{8}{27}a^{3}-\frac{10}{27}a^{2}+\frac{4}{27}a-\frac{11}{27}$, $\frac{1}{81}a^{21}-\frac{1}{81}a^{19}+\frac{4}{81}a^{17}+\frac{1}{27}a^{16}+\frac{4}{81}a^{15}-\frac{1}{27}a^{14}+\frac{1}{81}a^{13}-\frac{4}{81}a^{12}+\frac{2}{81}a^{11}+\frac{4}{81}a^{10}-\frac{10}{81}a^{9}-\frac{4}{81}a^{8}+\frac{29}{81}a^{7}-\frac{1}{27}a^{6}-\frac{35}{81}a^{5}+\frac{40}{81}a^{4}+\frac{29}{81}a^{3}-\frac{16}{81}a^{2}+\frac{7}{27}a+\frac{28}{81}$, $\frac{1}{1053}a^{22}-\frac{1}{351}a^{21}+\frac{14}{1053}a^{20}+\frac{2}{351}a^{19}+\frac{7}{1053}a^{18}+\frac{4}{117}a^{17}+\frac{22}{1053}a^{16}+\frac{2}{117}a^{15}+\frac{49}{1053}a^{14}+\frac{53}{1053}a^{13}-\frac{34}{1053}a^{12}+\frac{82}{1053}a^{11}+\frac{47}{1053}a^{10}+\frac{53}{1053}a^{9}-\frac{4}{81}a^{8}-\frac{146}{351}a^{7}+\frac{475}{1053}a^{6}-\frac{176}{1053}a^{5}+\frac{419}{1053}a^{4}+\frac{503}{1053}a^{3}+\frac{2}{27}a^{2}-\frac{278}{1053}a+\frac{32}{117}$, $\frac{1}{660231}a^{23}-\frac{14}{220077}a^{22}+\frac{1036}{220077}a^{21}-\frac{46}{24453}a^{20}-\frac{2446}{220077}a^{19}+\frac{727}{220077}a^{18}-\frac{31633}{660231}a^{17}-\frac{137}{220077}a^{16}+\frac{12971}{660231}a^{15}-\frac{9268}{660231}a^{14}+\frac{4907}{220077}a^{13}-\frac{35428}{220077}a^{12}+\frac{3077}{60021}a^{11}-\frac{22949}{220077}a^{10}-\frac{4742}{50787}a^{9}+\frac{41201}{660231}a^{8}+\frac{97589}{220077}a^{7}+\frac{160855}{660231}a^{6}-\frac{92767}{220077}a^{5}+\frac{1106}{2717}a^{4}-\frac{13834}{50787}a^{3}+\frac{9134}{24453}a^{2}-\frac{73471}{220077}a-\frac{16976}{50787}$, $\frac{1}{660231}a^{24}+\frac{10}{73359}a^{22}+\frac{80}{20007}a^{21}-\frac{137}{24453}a^{20}-\frac{1328}{73359}a^{19}+\frac{2285}{660231}a^{18}+\frac{3998}{73359}a^{17}-\frac{7426}{660231}a^{16}+\frac{15731}{660231}a^{15}+\frac{734}{16929}a^{14}-\frac{71}{16929}a^{13}-\frac{2207}{660231}a^{12}+\frac{28102}{220077}a^{11}+\frac{60769}{660231}a^{10}-\frac{38677}{660231}a^{9}-\frac{2048}{24453}a^{8}+\frac{176647}{660231}a^{7}-\frac{1321}{24453}a^{6}-\frac{3008}{24453}a^{5}+\frac{181892}{660231}a^{4}-\frac{67403}{220077}a^{3}+\frac{1822}{16929}a^{2}+\frac{260530}{660231}a-\frac{11281}{73359}$, $\frac{1}{2748541653}a^{25}-\frac{379}{2748541653}a^{24}+\frac{37}{249867423}a^{23}-\frac{70184}{305393517}a^{22}-\frac{4644260}{916180551}a^{21}-\frac{53558}{23491809}a^{20}-\frac{2391217}{2748541653}a^{19}-\frac{37820729}{2748541653}a^{18}+\frac{14361475}{916180551}a^{17}+\frac{40660544}{916180551}a^{16}+\frac{61609466}{2748541653}a^{15}+\frac{95881726}{2748541653}a^{14}+\frac{15152659}{2748541653}a^{13}-\frac{294418369}{2748541653}a^{12}-\frac{4819130}{48220029}a^{11}+\frac{161806582}{2748541653}a^{10}+\frac{39508234}{916180551}a^{9}+\frac{25391468}{2748541653}a^{8}-\frac{281013985}{2748541653}a^{7}+\frac{1281820217}{2748541653}a^{6}+\frac{732597692}{2748541653}a^{5}-\frac{792823778}{2748541653}a^{4}-\frac{903329036}{2748541653}a^{3}+\frac{516456796}{2748541653}a^{2}-\frac{355118539}{2748541653}a+\frac{1359261317}{2748541653}$, $\frac{1}{20\!\cdots\!01}a^{26}-\frac{77512}{618656497435197}a^{25}-\frac{372163378}{887637583276587}a^{24}+\frac{11356514027}{20\!\cdots\!01}a^{23}+\frac{2064472311619}{68\!\cdots\!67}a^{22}-\frac{1439099449343}{618656497435197}a^{21}-\frac{7475350304912}{887637583276587}a^{20}-\frac{50445961554746}{68\!\cdots\!67}a^{19}-\frac{161504644084115}{20\!\cdots\!01}a^{18}+\frac{53867607631555}{68\!\cdots\!67}a^{17}-\frac{207808209734284}{20\!\cdots\!01}a^{16}-\frac{170618308883543}{68\!\cdots\!67}a^{15}+\frac{42868602822712}{887637583276587}a^{14}-\frac{6383639394283}{295879194425529}a^{13}-\frac{105453720071077}{10\!\cdots\!79}a^{12}-\frac{26\!\cdots\!36}{20\!\cdots\!01}a^{11}-\frac{463213092146975}{20\!\cdots\!01}a^{10}-\frac{15\!\cdots\!83}{20\!\cdots\!01}a^{9}-\frac{99420778868012}{15\!\cdots\!77}a^{8}+\frac{305705949961427}{18\!\cdots\!91}a^{7}-\frac{10\!\cdots\!50}{20\!\cdots\!01}a^{6}-\frac{33\!\cdots\!46}{68\!\cdots\!67}a^{5}-\frac{25\!\cdots\!26}{20\!\cdots\!01}a^{4}-\frac{60\!\cdots\!13}{20\!\cdots\!01}a^{3}+\frac{92268633458152}{206218832478399}a^{2}+\frac{412509601318531}{20\!\cdots\!01}a+\frac{21\!\cdots\!21}{20\!\cdots\!01}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 - 1/3, 1/3*a^9 - 1/3*a, 1/3*a^10 - 1/3*a^2, 1/3*a^11 - 1/3*a^3, 1/3*a^12 - 1/3*a^4, 1/9*a^13 - 1/9*a^11 - 1/9*a^10 + 1/9*a^9 - 1/9*a^8 - 1/3*a^7 - 1/3*a^6 - 1/9*a^5 - 1/3*a^4 + 4/9*a^3 + 4/9*a^2 - 1/9*a + 4/9, 1/9*a^14 - 1/9*a^12 - 1/9*a^11 + 1/9*a^10 - 1/9*a^9 - 1/3*a^7 - 1/9*a^6 - 1/3*a^5 + 4/9*a^4 + 4/9*a^3 - 1/9*a^2 + 4/9*a - 1/3, 1/9*a^15 - 1/9*a^12 + 1/9*a^10 + 1/9*a^9 - 1/9*a^8 - 4/9*a^7 + 1/3*a^6 + 1/3*a^5 + 1/9*a^4 + 1/3*a^3 - 4/9*a^2 - 4/9*a + 1/9, 1/9*a^16 + 1/9*a^8 - 2/9, 1/9*a^17 + 1/9*a^9 - 2/9*a, 1/27*a^18 + 1/27*a^16 + 1/27*a^15 - 4/27*a^12 - 4/27*a^10 - 2/27*a^9 - 1/9*a^8 + 5/27*a^7 + 4/9*a^6 + 1/9*a^5 - 5/27*a^4 + 1/9*a^3 + 8/27*a - 7/27, 1/27*a^19 + 1/27*a^17 + 1/27*a^16 - 1/27*a^13 + 2/27*a^11 + 4/27*a^10 + 2/27*a^8 + 1/9*a^7 - 2/9*a^6 - 8/27*a^5 - 2/9*a^4 + 1/9*a^3 + 11/27*a^2 - 10/27*a + 4/9, 1/27*a^20 + 1/27*a^17 - 1/27*a^16 - 1/27*a^15 - 1/27*a^14 - 1/9*a^12 + 4/27*a^11 + 4/27*a^10 + 4/27*a^9 - 1/9*a^8 - 11/27*a^7 + 7/27*a^6 - 1/3*a^5 - 10/27*a^4 + 8/27*a^3 - 10/27*a^2 + 4/27*a - 11/27, 1/81*a^21 - 1/81*a^19 + 4/81*a^17 + 1/27*a^16 + 4/81*a^15 - 1/27*a^14 + 1/81*a^13 - 4/81*a^12 + 2/81*a^11 + 4/81*a^10 - 10/81*a^9 - 4/81*a^8 + 29/81*a^7 - 1/27*a^6 - 35/81*a^5 + 40/81*a^4 + 29/81*a^3 - 16/81*a^2 + 7/27*a + 28/81, 1/1053*a^22 - 1/351*a^21 + 14/1053*a^20 + 2/351*a^19 + 7/1053*a^18 + 4/117*a^17 + 22/1053*a^16 + 2/117*a^15 + 49/1053*a^14 + 53/1053*a^13 - 34/1053*a^12 + 82/1053*a^11 + 47/1053*a^10 + 53/1053*a^9 - 4/81*a^8 - 146/351*a^7 + 475/1053*a^6 - 176/1053*a^5 + 419/1053*a^4 + 503/1053*a^3 + 2/27*a^2 - 278/1053*a + 32/117, 1/660231*a^23 - 14/220077*a^22 + 1036/220077*a^21 - 46/24453*a^20 - 2446/220077*a^19 + 727/220077*a^18 - 31633/660231*a^17 - 137/220077*a^16 + 12971/660231*a^15 - 9268/660231*a^14 + 4907/220077*a^13 - 35428/220077*a^12 + 3077/60021*a^11 - 22949/220077*a^10 - 4742/50787*a^9 + 41201/660231*a^8 + 97589/220077*a^7 + 160855/660231*a^6 - 92767/220077*a^5 + 1106/2717*a^4 - 13834/50787*a^3 + 9134/24453*a^2 - 73471/220077*a - 16976/50787, 1/660231*a^24 + 10/73359*a^22 + 80/20007*a^21 - 137/24453*a^20 - 1328/73359*a^19 + 2285/660231*a^18 + 3998/73359*a^17 - 7426/660231*a^16 + 15731/660231*a^15 + 734/16929*a^14 - 71/16929*a^13 - 2207/660231*a^12 + 28102/220077*a^11 + 60769/660231*a^10 - 38677/660231*a^9 - 2048/24453*a^8 + 176647/660231*a^7 - 1321/24453*a^6 - 3008/24453*a^5 + 181892/660231*a^4 - 67403/220077*a^3 + 1822/16929*a^2 + 260530/660231*a - 11281/73359, 1/2748541653*a^25 - 379/2748541653*a^24 + 37/249867423*a^23 - 70184/305393517*a^22 - 4644260/916180551*a^21 - 53558/23491809*a^20 - 2391217/2748541653*a^19 - 37820729/2748541653*a^18 + 14361475/916180551*a^17 + 40660544/916180551*a^16 + 61609466/2748541653*a^15 + 95881726/2748541653*a^14 + 15152659/2748541653*a^13 - 294418369/2748541653*a^12 - 4819130/48220029*a^11 + 161806582/2748541653*a^10 + 39508234/916180551*a^9 + 25391468/2748541653*a^8 - 281013985/2748541653*a^7 + 1281820217/2748541653*a^6 + 732597692/2748541653*a^5 - 792823778/2748541653*a^4 - 903329036/2748541653*a^3 + 516456796/2748541653*a^2 - 355118539/2748541653*a + 1359261317/2748541653, 1/20415664415361501*a^26 - 77512/618656497435197*a^25 - 372163378/887637583276587*a^24 + 11356514027/20415664415361501*a^23 + 2064472311619/6805221471787167*a^22 - 1439099449343/618656497435197*a^21 - 7475350304912/887637583276587*a^20 - 50445961554746/6805221471787167*a^19 - 161504644084115/20415664415361501*a^18 + 53867607631555/6805221471787167*a^17 - 207808209734284/20415664415361501*a^16 - 170618308883543/6805221471787167*a^15 + 42868602822712/887637583276587*a^14 - 6383639394283/295879194425529*a^13 - 105453720071077/1074508653440079*a^12 - 2684484427036436/20415664415361501*a^11 - 463213092146975/20415664415361501*a^10 - 1560623540623783/20415664415361501*a^9 - 99420778868012/1570435724258577*a^8 + 305705949961427/1855969492305591*a^7 - 10172989146730250/20415664415361501*a^6 - 3392601989309746/6805221471787167*a^5 - 2572942807864126/20415664415361501*a^4 - 6022331046167413/20415664415361501*a^3 + 92268633458152/206218832478399*a^2 + 412509601318531/20415664415361501*a + 2159165400573521/20415664415361501

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

Unit group

Rank:		$13$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	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:		$\frac{74313215719169}{68\!\cdots\!67}a^{26}-\frac{17827932524626}{756135719087463}a^{25}+\frac{87533871614216}{68\!\cdots\!67}a^{24}+\frac{103499837832202}{523478574752859}a^{23}+\frac{25\!\cdots\!87}{22\!\cdots\!89}a^{22}+\frac{19013803111712}{10853622762021}a^{21}+\frac{21\!\cdots\!20}{68\!\cdots\!67}a^{20}+\frac{52\!\cdots\!00}{22\!\cdots\!89}a^{19}+\frac{13\!\cdots\!38}{618656497435197}a^{18}-\frac{531609339815269}{174492858250953}a^{17}-\frac{59\!\cdots\!30}{68\!\cdots\!67}a^{16}-\frac{13\!\cdots\!13}{756135719087463}a^{15}-\frac{12\!\cdots\!18}{68\!\cdots\!67}a^{14}-\frac{14\!\cdots\!09}{756135719087463}a^{13}-\frac{66\!\cdots\!97}{68\!\cdots\!67}a^{12}+\frac{77\!\cdots\!85}{68\!\cdots\!67}a^{11}+\frac{27\!\cdots\!12}{68\!\cdots\!67}a^{10}+\frac{36\!\cdots\!68}{68\!\cdots\!67}a^{9}+\frac{47\!\cdots\!77}{68\!\cdots\!67}a^{8}+\frac{36\!\cdots\!82}{68\!\cdots\!67}a^{7}+\frac{29\!\cdots\!39}{68\!\cdots\!67}a^{6}+\frac{39\!\cdots\!61}{756135719087463}a^{5}+\frac{36\!\cdots\!72}{68\!\cdots\!67}a^{4}-\frac{13\!\cdots\!23}{68\!\cdots\!67}a^{3}+\frac{951872595849139}{22\!\cdots\!89}a^{2}-\frac{32\!\cdots\!47}{68\!\cdots\!67}a+\frac{86\!\cdots\!73}{68\!\cdots\!67}$, $\frac{270904564493110}{20\!\cdots\!01}a^{26}-\frac{215643172835603}{68\!\cdots\!67}a^{25}+\frac{340132787248849}{20\!\cdots\!01}a^{24}+\frac{461064442144315}{18\!\cdots\!91}a^{23}+\frac{90\!\cdots\!79}{68\!\cdots\!67}a^{22}+\frac{502678572436633}{295879194425529}a^{21}+\frac{56\!\cdots\!13}{20\!\cdots\!01}a^{20}+\frac{713527010185163}{618656497435197}a^{19}+\frac{24\!\cdots\!48}{20\!\cdots\!01}a^{18}-\frac{28\!\cdots\!30}{68\!\cdots\!67}a^{17}-\frac{98\!\cdots\!44}{10\!\cdots\!79}a^{16}-\frac{11\!\cdots\!12}{68\!\cdots\!67}a^{15}-\frac{26\!\cdots\!27}{20\!\cdots\!01}a^{14}-\frac{75\!\cdots\!52}{68\!\cdots\!67}a^{13}+\frac{14\!\cdots\!55}{20\!\cdots\!01}a^{12}+\frac{24\!\cdots\!91}{20\!\cdots\!01}a^{11}+\frac{32\!\cdots\!22}{887637583276587}a^{10}+\frac{88\!\cdots\!61}{20\!\cdots\!01}a^{9}+\frac{11\!\cdots\!55}{20\!\cdots\!01}a^{8}+\frac{64\!\cdots\!01}{20\!\cdots\!01}a^{7}+\frac{55\!\cdots\!70}{20\!\cdots\!01}a^{6}-\frac{13\!\cdots\!29}{523478574752859}a^{5}+\frac{33\!\cdots\!97}{20\!\cdots\!01}a^{4}-\frac{18\!\cdots\!13}{20\!\cdots\!01}a^{3}+\frac{894439006827653}{98626398141843}a^{2}-\frac{46\!\cdots\!15}{10\!\cdots\!79}a+\frac{21\!\cdots\!92}{20\!\cdots\!01}$, $\frac{60292762415432}{20\!\cdots\!01}a^{26}-\frac{568365644311}{523478574752859}a^{25}-\frac{155091094045684}{20\!\cdots\!01}a^{24}+\frac{2705692856591}{46717767540873}a^{23}+\frac{27\!\cdots\!77}{68\!\cdots\!67}a^{22}+\frac{537095260805341}{523478574752859}a^{21}+\frac{26\!\cdots\!47}{15\!\cdots\!77}a^{20}+\frac{14\!\cdots\!89}{68\!\cdots\!67}a^{19}+\frac{33\!\cdots\!23}{18\!\cdots\!91}a^{18}+\frac{17\!\cdots\!73}{68\!\cdots\!67}a^{17}-\frac{65\!\cdots\!79}{18\!\cdots\!91}a^{16}-\frac{61\!\cdots\!17}{68\!\cdots\!67}a^{15}-\frac{27\!\cdots\!98}{20\!\cdots\!01}a^{14}-\frac{50\!\cdots\!63}{358169551146693}a^{13}-\frac{19\!\cdots\!66}{20\!\cdots\!01}a^{12}+\frac{24\!\cdots\!88}{20\!\cdots\!01}a^{11}+\frac{31\!\cdots\!51}{20\!\cdots\!01}a^{10}+\frac{65\!\cdots\!52}{20\!\cdots\!01}a^{9}+\frac{82\!\cdots\!19}{18\!\cdots\!91}a^{8}+\frac{96\!\cdots\!77}{20\!\cdots\!01}a^{7}+\frac{81\!\cdots\!78}{20\!\cdots\!01}a^{6}+\frac{17\!\cdots\!58}{68\!\cdots\!67}a^{5}+\frac{21\!\cdots\!93}{20\!\cdots\!01}a^{4}+\frac{25\!\cdots\!06}{10\!\cdots\!79}a^{3}-\frac{46\!\cdots\!78}{22\!\cdots\!89}a^{2}-\frac{156658438335856}{10\!\cdots\!79}a+\frac{16\!\cdots\!43}{20\!\cdots\!01}$, $\frac{79432750765169}{68\!\cdots\!67}a^{26}-\frac{55396093475908}{22\!\cdots\!89}a^{25}+\frac{99604340046698}{68\!\cdots\!67}a^{24}+\frac{128075923733177}{618656497435197}a^{23}+\frac{253746663834130}{206218832478399}a^{22}+\frac{45\!\cdots\!85}{22\!\cdots\!89}a^{21}+\frac{25\!\cdots\!58}{68\!\cdots\!67}a^{20}+\frac{70\!\cdots\!18}{22\!\cdots\!89}a^{19}+\frac{20\!\cdots\!07}{618656497435197}a^{18}-\frac{399823497909461}{174492858250953}a^{17}-\frac{59\!\cdots\!54}{68\!\cdots\!67}a^{16}-\frac{15\!\cdots\!75}{756135719087463}a^{15}-\frac{15\!\cdots\!70}{68\!\cdots\!67}a^{14}-\frac{58\!\cdots\!79}{22\!\cdots\!89}a^{13}-\frac{53\!\cdots\!97}{68\!\cdots\!67}a^{12}+\frac{39\!\cdots\!48}{68\!\cdots\!67}a^{11}+\frac{27\!\cdots\!53}{68\!\cdots\!67}a^{10}+\frac{40\!\cdots\!08}{68\!\cdots\!67}a^{9}+\frac{25\!\cdots\!67}{295879194425529}a^{8}+\frac{51\!\cdots\!03}{68\!\cdots\!67}a^{7}+\frac{47\!\cdots\!06}{68\!\cdots\!67}a^{6}+\frac{65\!\cdots\!44}{22\!\cdots\!89}a^{5}+\frac{15\!\cdots\!86}{618656497435197}a^{4}-\frac{48\!\cdots\!65}{68\!\cdots\!67}a^{3}+\frac{20\!\cdots\!31}{252045239695821}a^{2}-\frac{21\!\cdots\!85}{68\!\cdots\!67}a+\frac{89997611574578}{32560868286063}$, $\frac{32916582370019}{20\!\cdots\!01}a^{26}-\frac{35145815796745}{68\!\cdots\!67}a^{25}+\frac{78558887334515}{20\!\cdots\!01}a^{24}+\frac{641765475775978}{20\!\cdots\!01}a^{23}+\frac{922935601287002}{68\!\cdots\!67}a^{22}+\frac{341538492345295}{68\!\cdots\!67}a^{21}+\frac{578918271784279}{20\!\cdots\!01}a^{20}-\frac{21\!\cdots\!64}{68\!\cdots\!67}a^{19}-\frac{228755534000447}{887637583276587}a^{18}-\frac{226492276457636}{295879194425529}a^{17}-\frac{18\!\cdots\!80}{20\!\cdots\!01}a^{16}-\frac{55\!\cdots\!47}{68\!\cdots\!67}a^{15}+\frac{18\!\cdots\!29}{20\!\cdots\!01}a^{14}+\frac{11\!\cdots\!71}{68\!\cdots\!67}a^{13}+\frac{62\!\cdots\!55}{18\!\cdots\!91}a^{12}+\frac{41\!\cdots\!31}{20\!\cdots\!01}a^{11}+\frac{33\!\cdots\!66}{10\!\cdots\!79}a^{10}+\frac{16\!\cdots\!47}{18\!\cdots\!91}a^{9}-\frac{16\!\cdots\!09}{20\!\cdots\!01}a^{8}-\frac{92\!\cdots\!86}{15\!\cdots\!77}a^{7}-\frac{12\!\cdots\!41}{20\!\cdots\!01}a^{6}-\frac{46\!\cdots\!55}{68\!\cdots\!67}a^{5}-\frac{34\!\cdots\!96}{20\!\cdots\!01}a^{4}-\frac{83\!\cdots\!58}{18\!\cdots\!91}a^{3}-\frac{28\!\cdots\!83}{22\!\cdots\!89}a^{2}-\frac{13\!\cdots\!49}{887637583276587}a+\frac{832933199126521}{15\!\cdots\!77}$, $\frac{93293463115393}{68\!\cdots\!67}a^{26}-\frac{88834027363043}{22\!\cdots\!89}a^{25}+\frac{232558387878385}{68\!\cdots\!67}a^{24}+\frac{16\!\cdots\!95}{68\!\cdots\!67}a^{23}+\frac{28\!\cdots\!01}{22\!\cdots\!89}a^{22}+\frac{26\!\cdots\!53}{22\!\cdots\!89}a^{21}+\frac{14\!\cdots\!83}{68\!\cdots\!67}a^{20}-\frac{116434588301021}{22\!\cdots\!89}a^{19}+\frac{23\!\cdots\!65}{68\!\cdots\!67}a^{18}-\frac{12\!\cdots\!01}{22\!\cdots\!89}a^{17}-\frac{52\!\cdots\!62}{618656497435197}a^{16}-\frac{10\!\cdots\!65}{756135719087463}a^{15}-\frac{44\!\cdots\!34}{68\!\cdots\!67}a^{14}-\frac{11\!\cdots\!56}{22\!\cdots\!89}a^{13}+\frac{10\!\cdots\!76}{68\!\cdots\!67}a^{12}+\frac{10\!\cdots\!51}{68\!\cdots\!67}a^{11}+\frac{23\!\cdots\!78}{618656497435197}a^{10}+\frac{20\!\cdots\!96}{68\!\cdots\!67}a^{9}+\frac{23\!\cdots\!86}{68\!\cdots\!67}a^{8}+\frac{25\!\cdots\!57}{68\!\cdots\!67}a^{7}+\frac{20\!\cdots\!83}{68\!\cdots\!67}a^{6}-\frac{66\!\cdots\!60}{206218832478399}a^{5}+\frac{991140679826888}{68\!\cdots\!67}a^{4}-\frac{77\!\cdots\!04}{295879194425529}a^{3}+\frac{87\!\cdots\!85}{756135719087463}a^{2}-\frac{37\!\cdots\!91}{68\!\cdots\!67}a+\frac{49\!\cdots\!04}{68\!\cdots\!67}$, $\frac{26741842526257}{20\!\cdots\!01}a^{26}-\frac{48805747136444}{68\!\cdots\!67}a^{25}+\frac{194956643213788}{20\!\cdots\!01}a^{24}+\frac{23928059179514}{10\!\cdots\!79}a^{23}+\frac{355877398005199}{68\!\cdots\!67}a^{22}-\frac{17\!\cdots\!10}{68\!\cdots\!67}a^{21}-\frac{91\!\cdots\!90}{20\!\cdots\!01}a^{20}-\frac{79\!\cdots\!66}{68\!\cdots\!67}a^{19}-\frac{23\!\cdots\!66}{20\!\cdots\!01}a^{18}-\frac{11\!\cdots\!46}{68\!\cdots\!67}a^{17}-\frac{10\!\cdots\!12}{20\!\cdots\!01}a^{16}+\frac{92\!\cdots\!18}{68\!\cdots\!67}a^{15}+\frac{11\!\cdots\!35}{20\!\cdots\!01}a^{14}+\frac{50\!\cdots\!40}{68\!\cdots\!67}a^{13}+\frac{21\!\cdots\!18}{20\!\cdots\!01}a^{12}+\frac{12\!\cdots\!27}{20\!\cdots\!01}a^{11}+\frac{52\!\cdots\!68}{20\!\cdots\!01}a^{10}-\frac{16\!\cdots\!32}{20\!\cdots\!01}a^{9}-\frac{18\!\cdots\!52}{10\!\cdots\!79}a^{8}-\frac{57\!\cdots\!42}{20\!\cdots\!01}a^{7}-\frac{58\!\cdots\!31}{20\!\cdots\!01}a^{6}-\frac{19\!\cdots\!28}{68\!\cdots\!67}a^{5}-\frac{16\!\cdots\!33}{10\!\cdots\!79}a^{4}-\frac{25\!\cdots\!54}{20\!\cdots\!01}a^{3}-\frac{34\!\cdots\!32}{22\!\cdots\!89}a^{2}-\frac{85\!\cdots\!71}{20\!\cdots\!01}a-\frac{10\!\cdots\!50}{20\!\cdots\!01}$, $\frac{10576751116519}{22\!\cdots\!89}a^{26}-\frac{21272363381998}{22\!\cdots\!89}a^{25}+\frac{680980178168}{119389850382231}a^{24}+\frac{63085086253796}{756135719087463}a^{23}+\frac{6571821532217}{13265538931359}a^{22}+\frac{644314561462391}{756135719087463}a^{21}+\frac{37\!\cdots\!96}{22\!\cdots\!89}a^{20}+\frac{37\!\cdots\!50}{22\!\cdots\!89}a^{19}+\frac{13\!\cdots\!85}{756135719087463}a^{18}-\frac{20400664882436}{68739610826133}a^{17}-\frac{71\!\cdots\!58}{22\!\cdots\!89}a^{16}-\frac{18\!\cdots\!34}{22\!\cdots\!89}a^{15}-\frac{24\!\cdots\!42}{22\!\cdots\!89}a^{14}-\frac{30\!\cdots\!86}{22\!\cdots\!89}a^{13}-\frac{55\!\cdots\!61}{756135719087463}a^{12}-\frac{27\!\cdots\!65}{22\!\cdots\!89}a^{11}+\frac{11\!\cdots\!32}{84015079898607}a^{10}+\frac{59\!\cdots\!90}{22\!\cdots\!89}a^{9}+\frac{89\!\cdots\!44}{22\!\cdots\!89}a^{8}+\frac{93\!\cdots\!76}{22\!\cdots\!89}a^{7}+\frac{49\!\cdots\!06}{119389850382231}a^{6}+\frac{58\!\cdots\!93}{22\!\cdots\!89}a^{5}+\frac{44\!\cdots\!07}{22\!\cdots\!89}a^{4}+\frac{10\!\cdots\!44}{206218832478399}a^{3}+\frac{95\!\cdots\!95}{22\!\cdots\!89}a^{2}+\frac{68747044528373}{22\!\cdots\!89}a+\frac{475209078752560}{756135719087463}$, $\frac{27822020293987}{20\!\cdots\!01}a^{26}-\frac{27764790354674}{68\!\cdots\!67}a^{25}+\frac{81088112771617}{20\!\cdots\!01}a^{24}+\frac{37004046248534}{15\!\cdots\!77}a^{23}+\frac{820709891176957}{68\!\cdots\!67}a^{22}+\frac{683170446412301}{68\!\cdots\!67}a^{21}+\frac{40\!\cdots\!04}{20\!\cdots\!01}a^{20}-\frac{382241772540359}{68\!\cdots\!67}a^{19}-\frac{11072947329368}{10\!\cdots\!79}a^{18}-\frac{41\!\cdots\!70}{68\!\cdots\!67}a^{17}-\frac{17\!\cdots\!20}{20\!\cdots\!01}a^{16}-\frac{89\!\cdots\!11}{68\!\cdots\!67}a^{15}-\frac{10\!\cdots\!67}{20\!\cdots\!01}a^{14}-\frac{18\!\cdots\!69}{68\!\cdots\!67}a^{13}+\frac{37\!\cdots\!47}{20\!\cdots\!01}a^{12}+\frac{34\!\cdots\!04}{20\!\cdots\!01}a^{11}+\frac{60\!\cdots\!26}{15\!\cdots\!77}a^{10}+\frac{59\!\cdots\!80}{20\!\cdots\!01}a^{9}+\frac{47\!\cdots\!78}{20\!\cdots\!01}a^{8}-\frac{173156734627714}{887637583276587}a^{7}-\frac{10\!\cdots\!85}{20\!\cdots\!01}a^{6}-\frac{28\!\cdots\!76}{68\!\cdots\!67}a^{5}-\frac{88\!\cdots\!44}{20\!\cdots\!01}a^{4}-\frac{56\!\cdots\!14}{20\!\cdots\!01}a^{3}+\frac{683060061184}{1473948770151}a^{2}+\frac{14\!\cdots\!28}{18\!\cdots\!91}a-\frac{93\!\cdots\!50}{20\!\cdots\!01}$, $\frac{58564248358766}{20\!\cdots\!01}a^{26}-\frac{2254866640222}{358169551146693}a^{25}+\frac{68172095315663}{20\!\cdots\!01}a^{24}+\frac{57300136201624}{10\!\cdots\!79}a^{23}+\frac{20\!\cdots\!13}{68\!\cdots\!67}a^{22}+\frac{30\!\cdots\!12}{68\!\cdots\!67}a^{21}+\frac{16\!\cdots\!39}{20\!\cdots\!01}a^{20}+\frac{43\!\cdots\!31}{68\!\cdots\!67}a^{19}+\frac{13\!\cdots\!95}{20\!\cdots\!01}a^{18}-\frac{37\!\cdots\!07}{68\!\cdots\!67}a^{17}-\frac{37\!\cdots\!95}{18\!\cdots\!91}a^{16}-\frac{22\!\cdots\!25}{523478574752859}a^{15}-\frac{97\!\cdots\!49}{20\!\cdots\!01}a^{14}-\frac{35\!\cdots\!50}{68\!\cdots\!67}a^{13}-\frac{32\!\cdots\!34}{20\!\cdots\!01}a^{12}+\frac{26\!\cdots\!93}{20\!\cdots\!01}a^{11}+\frac{17\!\cdots\!29}{20\!\cdots\!01}a^{10}+\frac{26\!\cdots\!88}{20\!\cdots\!01}a^{9}+\frac{37\!\cdots\!60}{20\!\cdots\!01}a^{8}+\frac{33\!\cdots\!88}{20\!\cdots\!01}a^{7}+\frac{24\!\cdots\!22}{15\!\cdots\!77}a^{6}+\frac{47\!\cdots\!56}{68\!\cdots\!67}a^{5}+\frac{61\!\cdots\!70}{10\!\cdots\!79}a^{4}-\frac{17\!\cdots\!35}{18\!\cdots\!91}a^{3}+\frac{33\!\cdots\!32}{22\!\cdots\!89}a^{2}-\frac{98\!\cdots\!97}{20\!\cdots\!01}a+\frac{10\!\cdots\!78}{18\!\cdots\!91}$, $\frac{88582025534281}{20\!\cdots\!01}a^{26}-\frac{74238784344932}{68\!\cdots\!67}a^{25}+\frac{172698206292724}{20\!\cdots\!01}a^{24}+\frac{142091487938323}{18\!\cdots\!91}a^{23}+\frac{28\!\cdots\!06}{68\!\cdots\!67}a^{22}+\frac{36\!\cdots\!73}{68\!\cdots\!67}a^{21}+\frac{20\!\cdots\!39}{20\!\cdots\!01}a^{20}+\frac{130581606425747}{295879194425529}a^{19}+\frac{85\!\cdots\!75}{20\!\cdots\!01}a^{18}-\frac{10\!\cdots\!74}{618656497435197}a^{17}-\frac{68\!\cdots\!54}{20\!\cdots\!01}a^{16}-\frac{42\!\cdots\!97}{68\!\cdots\!67}a^{15}-\frac{10\!\cdots\!53}{20\!\cdots\!01}a^{14}-\frac{28\!\cdots\!94}{618656497435197}a^{13}+\frac{68\!\cdots\!94}{20\!\cdots\!01}a^{12}+\frac{10\!\cdots\!91}{15\!\cdots\!77}a^{11}+\frac{34\!\cdots\!97}{20\!\cdots\!01}a^{10}+\frac{34\!\cdots\!30}{20\!\cdots\!01}a^{9}+\frac{37\!\cdots\!39}{18\!\cdots\!91}a^{8}+\frac{20\!\cdots\!63}{20\!\cdots\!01}a^{7}+\frac{12\!\cdots\!39}{20\!\cdots\!01}a^{6}-\frac{62\!\cdots\!84}{618656497435197}a^{5}-\frac{45\!\cdots\!01}{887637583276587}a^{4}-\frac{29\!\cdots\!78}{20\!\cdots\!01}a^{3}+\frac{80052878759101}{22\!\cdots\!89}a^{2}-\frac{74\!\cdots\!22}{20\!\cdots\!01}a+\frac{12\!\cdots\!69}{18\!\cdots\!91}$, $\frac{136685056043092}{20\!\cdots\!01}a^{26}-\frac{17572110248207}{68\!\cdots\!67}a^{25}-\frac{16357008858370}{887637583276587}a^{24}+\frac{27\!\cdots\!10}{20\!\cdots\!01}a^{23}+\frac{62\!\cdots\!92}{68\!\cdots\!67}a^{22}+\frac{15\!\cdots\!78}{68\!\cdots\!67}a^{21}+\frac{33\!\cdots\!13}{887637583276587}a^{20}+\frac{28\!\cdots\!53}{618656497435197}a^{19}+\frac{70\!\cdots\!10}{20\!\cdots\!01}a^{18}+\frac{652269777456733}{68\!\cdots\!67}a^{17}-\frac{17\!\cdots\!63}{20\!\cdots\!01}a^{16}-\frac{13\!\cdots\!28}{68\!\cdots\!67}a^{15}-\frac{26\!\cdots\!88}{887637583276587}a^{14}-\frac{790851492350006}{26898108584139}a^{13}-\frac{37\!\cdots\!91}{20\!\cdots\!01}a^{12}+\frac{14\!\cdots\!62}{20\!\cdots\!01}a^{11}+\frac{73\!\cdots\!62}{20\!\cdots\!01}a^{10}+\frac{14\!\cdots\!36}{20\!\cdots\!01}a^{9}+\frac{91\!\cdots\!07}{97682604858189}a^{8}+\frac{20\!\cdots\!22}{20\!\cdots\!01}a^{7}+\frac{15\!\cdots\!51}{20\!\cdots\!01}a^{6}+\frac{31\!\cdots\!78}{68\!\cdots\!67}a^{5}+\frac{24\!\cdots\!74}{20\!\cdots\!01}a^{4}+\frac{45\!\cdots\!20}{20\!\cdots\!01}a^{3}-\frac{64\!\cdots\!35}{756135719087463}a^{2}+\frac{64\!\cdots\!23}{15\!\cdots\!77}a-\frac{15\!\cdots\!43}{20\!\cdots\!01}$, $\frac{1190431927606}{658569819850371}a^{26}-\frac{1667683162679}{219523273283457}a^{25}+\frac{6118238275642}{658569819850371}a^{24}+\frac{20009666604209}{658569819850371}a^{23}+\frac{26299865386057}{219523273283457}a^{22}-\frac{23833287769348}{219523273283457}a^{21}-\frac{101848292364448}{658569819850371}a^{20}-\frac{172042335248444}{219523273283457}a^{19}-\frac{384720312101249}{658569819850371}a^{18}-\frac{283682947544255}{219523273283457}a^{17}-\frac{392326998918835}{658569819850371}a^{16}+\frac{22820140751800}{219523273283457}a^{15}+\frac{21\!\cdots\!32}{658569819850371}a^{14}+\frac{883619086990249}{219523273283457}a^{13}+\frac{45\!\cdots\!38}{658569819850371}a^{12}+\frac{22\!\cdots\!91}{658569819850371}a^{11}+\frac{21\!\cdots\!31}{658569819850371}a^{10}-\frac{26\!\cdots\!68}{658569819850371}a^{9}-\frac{50\!\cdots\!56}{658569819850371}a^{8}-\frac{10\!\cdots\!21}{658569819850371}a^{7}-\frac{99\!\cdots\!38}{658569819850371}a^{6}-\frac{208544014474453}{11553856488603}a^{5}-\frac{37\!\cdots\!20}{658569819850371}a^{4}-\frac{53\!\cdots\!90}{658569819850371}a^{3}+\frac{1128107523728}{404278587999}a^{2}-\frac{90193066828768}{59869983622761}a+\frac{230225759673137}{658569819850371}$ 74313215719169/6805221471787167*a^26 - 17827932524626/756135719087463*a^25 + 87533871614216/6805221471787167*a^24 + 103499837832202/523478574752859*a^23 + 2576517283284287/2268407157262389*a^22 + 19013803111712/10853622762021*a^21 + 21408035506752520/6805221471787167*a^20 + 5202103578528100/2268407157262389*a^19 + 1333396076225338/618656497435197*a^18 - 531609339815269/174492858250953*a^17 - 59760312671900330/6805221471787167*a^16 - 13625613471242413/756135719087463*a^15 - 127232850667937018/6805221471787167*a^14 - 14246807523743809/756135719087463*a^13 - 6655326540358997/6805221471787167*a^12 + 77939139482623985/6805221471787167*a^11 + 278623320725996912/6805221471787167*a^10 + 367425980142662968/6805221471787167*a^9 + 470458722877797977/6805221471787167*a^8 + 365325410300467382/6805221471787167*a^7 + 297857543822264039/6805221471787167*a^6 + 3985447639398661/756135719087463*a^5 + 36850377508371472/6805221471787167*a^4 - 132062240024681723/6805221471787167*a^3 + 951872595849139/2268407157262389*a^2 - 32342175447899047/6805221471787167*a + 8662488247915873/6805221471787167, 270904564493110/20415664415361501*a^26 - 215643172835603/6805221471787167*a^25 + 340132787248849/20415664415361501*a^24 + 461064442144315/1855969492305591*a^23 + 9027465400023679/6805221471787167*a^22 + 502678572436633/295879194425529*a^21 + 56217262349815913/20415664415361501*a^20 + 713527010185163/618656497435197*a^19 + 24202495624542448/20415664415361501*a^18 - 28499522414973230/6805221471787167*a^17 - 9847525631593744/1074508653440079*a^16 - 115755283156159112/6805221471787167*a^15 - 266725943245816027/20415664415361501*a^14 - 75967194965101052/6805221471787167*a^13 + 140808276706358255/20415664415361501*a^12 + 249788583635167891/20415664415361501*a^11 + 32962680730673822/887637583276587*a^10 + 881577303462566561/20415664415361501*a^9 + 1103733835473150355/20415664415361501*a^8 + 646597676061324601/20415664415361501*a^7 + 550336984107226270/20415664415361501*a^6 - 1321615065513529/523478574752859*a^5 + 334614624361138097/20415664415361501*a^4 - 182273889312144013/20415664415361501*a^3 + 894439006827653/98626398141843*a^2 - 4614118333999115/1074508653440079*a + 21241559777049992/20415664415361501, 60292762415432/20415664415361501*a^26 - 568365644311/523478574752859*a^25 - 155091094045684/20415664415361501*a^24 + 2705692856591/46717767540873*a^23 + 2770894353788477/6805221471787167*a^22 + 537095260805341/523478574752859*a^21 + 2697782457546247/1570435724258577*a^20 + 14447638285167089/6805221471787167*a^19 + 3321200363156323/1855969492305591*a^18 + 1773535927533473/6805221471787167*a^17 - 6577107210917779/1855969492305591*a^16 - 61413610410019117/6805221471787167*a^15 - 275387139163923698/20415664415361501*a^14 - 5095777969672963/358169551146693*a^13 - 196061423133192566/20415664415361501*a^12 + 24643312494529688/20415664415361501*a^11 + 314553491769285251/20415664415361501*a^10 + 650099710078236052/20415664415361501*a^9 + 82319302781821519/1855969492305591*a^8 + 968779316762045777/20415664415361501*a^7 + 816243034579825178/20415664415361501*a^6 + 176333742472100758/6805221471787167*a^5 + 214921782694566793/20415664415361501*a^4 + 2520716357559406/1074508653440079*a^3 - 4687652287904078/2268407157262389*a^2 - 156658438335856/1074508653440079*a + 16830177362574343/20415664415361501, 79432750765169/6805221471787167*a^26 - 55396093475908/2268407157262389*a^25 + 99604340046698/6805221471787167*a^24 + 128075923733177/618656497435197*a^23 + 253746663834130/206218832478399*a^22 + 4542916491887185/2268407157262389*a^21 + 25574442446748058/6805221471787167*a^20 + 7084375734602018/2268407157262389*a^19 + 2031425607888307/618656497435197*a^18 - 399823497909461/174492858250953*a^17 - 59434497821428454/6805221471787167*a^16 - 15133006132400575/756135719087463*a^15 - 157101091602126170/6805221471787167*a^14 - 58214057357566579/2268407157262389*a^13 - 53105239369897697/6805221471787167*a^12 + 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6138384955981870/1074508653440079*a^4 - 1740916914352435/1855969492305591*a^3 + 3331903798405432/2268407157262389*a^2 - 9813501717489997/20415664415361501*a + 1018652168364278/1855969492305591, 88582025534281/20415664415361501*a^26 - 74238784344932/6805221471787167*a^25 + 172698206292724/20415664415361501*a^24 + 142091487938323/1855969492305591*a^23 + 2878834365530206/6805221471787167*a^22 + 3649800005446073/6805221471787167*a^21 + 20521952665575539/20415664415361501*a^20 + 130581606425747/295879194425529*a^19 + 8547615316048375/20415664415361501*a^18 - 1093946869813774/618656497435197*a^17 - 68609885441375554/20415664415361501*a^16 - 42350522673616397/6805221471787167*a^15 - 102266912155324153/20415664415361501*a^14 - 2863599899581894/618656497435197*a^13 + 68871719801966894/20415664415361501*a^12 + 10423957716715891/1570435724258577*a^11 + 344978559451334797/20415664415361501*a^10 + 349573709994549530/20415664415361501*a^9 + 37555990393151939/1855969492305591*a^8 + 205983099369868063/20415664415361501*a^7 + 123378990121065739/20415664415361501*a^6 - 6236600953711784/618656497435197*a^5 - 4576761020683301/887637583276587*a^4 - 296393064744970378/20415664415361501*a^3 + 80052878759101/2268407157262389*a^2 - 74533200471607022/20415664415361501*a + 1243961723832469/1855969492305591, 136685056043092/20415664415361501*a^26 - 17572110248207/6805221471787167*a^25 - 16357008858370/887637583276587*a^24 + 2751625799428010/20415664415361501*a^23 + 6233654539375492/6805221471787167*a^22 + 15598777422677678/6805221471787167*a^21 + 3311161349022313/887637583276587*a^20 + 2812340678684453/618656497435197*a^19 + 70765746995989510/20415664415361501*a^18 + 652269777456733/6805221471787167*a^17 - 175283081137665463/20415664415361501*a^16 - 136398529261800428/6805221471787167*a^15 - 26028067236609188/887637583276587*a^14 - 790851492350006/26898108584139*a^13 - 371141090354563891/20415664415361501*a^12 + 144331256627807962/20415664415361501*a^11 + 738036173656369462/20415664415361501*a^10 + 1457872359351778736/20415664415361501*a^9 + 9143406081476207/97682604858189*a^8 + 2013198198300349822/20415664415361501*a^7 + 1598355970823326051/20415664415361501*a^6 + 316057855013228678/6805221471787167*a^5 + 247751666764131974/20415664415361501*a^4 + 45880189845123620/20415664415361501*a^3 - 6430605572561435/756135719087463*a^2 + 6489326441124823/1570435724258577*a - 1585941313224043/20415664415361501, 1190431927606/658569819850371*a^26 - 1667683162679/219523273283457*a^25 + 6118238275642/658569819850371*a^24 + 20009666604209/658569819850371*a^23 + 26299865386057/219523273283457*a^22 - 23833287769348/219523273283457*a^21 - 101848292364448/658569819850371*a^20 - 172042335248444/219523273283457*a^19 - 384720312101249/658569819850371*a^18 - 283682947544255/219523273283457*a^17 - 392326998918835/658569819850371*a^16 + 22820140751800/219523273283457*a^15 + 2167846589178332/658569819850371*a^14 + 883619086990249/219523273283457*a^13 + 4577266356016538/658569819850371*a^12 + 2209176680209591/658569819850371*a^11 + 2166733099880131/658569819850371*a^10 - 2647970089174468/658569819850371*a^9 - 5085426834491456/658569819850371*a^8 - 10856012552681921/658569819850371*a^7 - 9968536848029138/658569819850371*a^6 - 208544014474453/11553856488603*a^5 - 3725382214325920/658569819850371*a^4 - 5347458848368690/658569819850371*a^3 + 1128107523728/404278587999*a^2 - 90193066828768/59869983622761*a + 230225759673137/658569819850371 (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:		\( 25801652404.76311 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 25801652404.76311 \cdot 4}{2\cdot\sqrt{3639553781467035763182087002112051895733239}}\cr\approx \mathstrut & 1.28683163686933 \end{aligned}\] (assuming GRH)

Galois group

$D_{27}$ (as 27T8):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(1879\) 1879	$\Q_{1879}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations