Normalized defining polynomial

\( x^{27} - 9 x^{25} - 9 x^{24} + 18 x^{23} + 72 x^{22} + 60 x^{21} - 126 x^{20} - 252 x^{19} - 162 x^{18} + \cdots - 25 \) x^27 - 9*x^25 - 9*x^24 + 18*x^23 + 72*x^22 + 60*x^21 - 126*x^20 - 252*x^19 - 162*x^18 + 108*x^17 + 126*x^16 - 216*x^15 + 450*x^14 + 1692*x^13 + 2670*x^12 + 1899*x^11 - 900*x^10 - 1461*x^9 - 3591*x^8 - 6354*x^7 - 990*x^6 + 4392*x^5 + 2448*x^4 - 435*x^3 - 720*x^2 - 225*x - 25

Invariants

Degree:		$27$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[3, 12]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(9527252910578083379673525288960000000000\) 9527252910578083379673525288960000000000 \(\medspace = 2^{24}\cdot 3^{54}\cdot 5^{10}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(30.25\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$2^{8/9}3^{37/18}5^{1/2}\approx 39.61106018062899$
Ramified primes:		\(2\), \(3\), \(5\) 2, 3, 5	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{7}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{8}$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{9}-\frac{2}{9}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{10}-\frac{2}{9}a$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{11}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{21}+\frac{1}{9}a^{12}-\frac{2}{9}a^{3}$, $\frac{1}{9}a^{22}+\frac{1}{9}a^{13}-\frac{2}{9}a^{4}$, $\frac{1}{135}a^{23}-\frac{2}{135}a^{22}+\frac{4}{135}a^{21}+\frac{1}{27}a^{20}-\frac{1}{135}a^{19}-\frac{1}{135}a^{18}+\frac{2}{15}a^{17}+\frac{1}{15}a^{16}-\frac{2}{15}a^{15}-\frac{4}{27}a^{14}-\frac{14}{135}a^{13}+\frac{19}{135}a^{12}-\frac{2}{27}a^{11}+\frac{11}{135}a^{10}+\frac{4}{27}a^{9}+\frac{1}{15}a^{8}-\frac{1}{15}a^{7}-\frac{4}{15}a^{6}+\frac{2}{27}a^{5}-\frac{4}{27}a^{4}-\frac{59}{135}a^{3}-\frac{67}{135}a^{2}-\frac{11}{27}a-\frac{2}{27}$, $\frac{1}{135}a^{24}-\frac{2}{135}a^{21}-\frac{2}{45}a^{20}-\frac{1}{45}a^{19}+\frac{1}{135}a^{18}-\frac{11}{135}a^{15}-\frac{1}{15}a^{14}-\frac{1}{15}a^{13}+\frac{13}{135}a^{12}+\frac{7}{45}a^{11}-\frac{1}{45}a^{10}-\frac{11}{135}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{28}{135}a^{6}-\frac{1}{3}a^{5}+\frac{4}{15}a^{4}-\frac{4}{27}a^{3}+\frac{22}{45}a^{2}+\frac{4}{9}a+\frac{11}{27}$, $\frac{1}{2565}a^{25}+\frac{1}{285}a^{24}-\frac{1}{855}a^{23}+\frac{49}{2565}a^{22}+\frac{2}{45}a^{21}-\frac{4}{855}a^{20}+\frac{142}{2565}a^{19}+\frac{1}{45}a^{18}-\frac{46}{285}a^{17}+\frac{7}{2565}a^{16}-\frac{46}{285}a^{15}+\frac{10}{171}a^{14}-\frac{116}{2565}a^{13}+\frac{137}{855}a^{12}+\frac{137}{855}a^{11}-\frac{41}{2565}a^{10}+\frac{23}{171}a^{9}-\frac{22}{57}a^{8}+\frac{1279}{2565}a^{7}+\frac{7}{19}a^{6}-\frac{193}{855}a^{5}-\frac{1076}{2565}a^{4}+\frac{14}{45}a^{3}-\frac{77}{171}a^{2}-\frac{4}{27}a+\frac{65}{171}$, $\frac{1}{24\!\cdots\!45}a^{26}+\frac{11\!\cdots\!63}{24\!\cdots\!45}a^{25}-\frac{27\!\cdots\!55}{85\!\cdots\!57}a^{24}+\frac{24\!\cdots\!83}{81\!\cdots\!15}a^{23}+\frac{12\!\cdots\!79}{27\!\cdots\!05}a^{22}+\frac{50\!\cdots\!48}{24\!\cdots\!45}a^{21}+\frac{84\!\cdots\!62}{24\!\cdots\!45}a^{20}-\frac{48\!\cdots\!33}{24\!\cdots\!45}a^{19}+\frac{82\!\cdots\!24}{24\!\cdots\!45}a^{18}+\frac{16\!\cdots\!97}{24\!\cdots\!45}a^{17}+\frac{50\!\cdots\!93}{12\!\cdots\!55}a^{16}-\frac{11\!\cdots\!13}{81\!\cdots\!15}a^{15}-\frac{97\!\cdots\!83}{81\!\cdots\!15}a^{14}+\frac{15\!\cdots\!74}{27\!\cdots\!05}a^{13}+\frac{35\!\cdots\!44}{24\!\cdots\!45}a^{12}+\frac{22\!\cdots\!54}{48\!\cdots\!49}a^{11}-\frac{29\!\cdots\!63}{24\!\cdots\!45}a^{10}+\frac{29\!\cdots\!48}{24\!\cdots\!45}a^{9}+\frac{50\!\cdots\!85}{48\!\cdots\!49}a^{8}-\frac{27\!\cdots\!39}{48\!\cdots\!49}a^{7}+\frac{15\!\cdots\!47}{16\!\cdots\!83}a^{6}-\frac{24\!\cdots\!24}{81\!\cdots\!15}a^{5}+\frac{74\!\cdots\!24}{27\!\cdots\!05}a^{4}-\frac{64\!\cdots\!81}{24\!\cdots\!45}a^{3}-\frac{28\!\cdots\!14}{21\!\cdots\!65}a^{2}+\frac{84\!\cdots\!64}{48\!\cdots\!49}a+\frac{10\!\cdots\!75}{48\!\cdots\!49}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/3*a^9 - 1/3, 1/3*a^10 - 1/3*a, 1/3*a^11 - 1/3*a^2, 1/3*a^12 - 1/3*a^3, 1/3*a^13 - 1/3*a^4, 1/3*a^14 - 1/3*a^5, 1/3*a^15 - 1/3*a^6, 1/3*a^16 - 1/3*a^7, 1/3*a^17 - 1/3*a^8, 1/9*a^18 + 1/9*a^9 - 2/9, 1/9*a^19 + 1/9*a^10 - 2/9*a, 1/9*a^20 + 1/9*a^11 - 2/9*a^2, 1/9*a^21 + 1/9*a^12 - 2/9*a^3, 1/9*a^22 + 1/9*a^13 - 2/9*a^4, 1/135*a^23 - 2/135*a^22 + 4/135*a^21 + 1/27*a^20 - 1/135*a^19 - 1/135*a^18 + 2/15*a^17 + 1/15*a^16 - 2/15*a^15 - 4/27*a^14 - 14/135*a^13 + 19/135*a^12 - 2/27*a^11 + 11/135*a^10 + 4/27*a^9 + 1/15*a^8 - 1/15*a^7 - 4/15*a^6 + 2/27*a^5 - 4/27*a^4 - 59/135*a^3 - 67/135*a^2 - 11/27*a - 2/27, 1/135*a^24 - 2/135*a^21 - 2/45*a^20 - 1/45*a^19 + 1/135*a^18 - 11/135*a^15 - 1/15*a^14 - 1/15*a^13 + 13/135*a^12 + 7/45*a^11 - 1/45*a^10 - 11/135*a^9 + 2/5*a^8 - 2/5*a^7 + 28/135*a^6 - 1/3*a^5 + 4/15*a^4 - 4/27*a^3 + 22/45*a^2 + 4/9*a + 11/27, 1/2565*a^25 + 1/285*a^24 - 1/855*a^23 + 49/2565*a^22 + 2/45*a^21 - 4/855*a^20 + 142/2565*a^19 + 1/45*a^18 - 46/285*a^17 + 7/2565*a^16 - 46/285*a^15 + 10/171*a^14 - 116/2565*a^13 + 137/855*a^12 + 137/855*a^11 - 41/2565*a^10 + 23/171*a^9 - 22/57*a^8 + 1279/2565*a^7 + 7/19*a^6 - 193/855*a^5 - 1076/2565*a^4 + 14/45*a^3 - 77/171*a^2 - 4/27*a + 65/171, 1/2431765333485025102704350745*a^26 + 115481999728063248070663/2431765333485025102704350745*a^25 - 27259890697303324327955/8532509942052719658611757*a^24 + 2469824690539502423687183/810588444495008367568116915*a^23 + 12981347345285291479315379/270196148165002789189372305*a^22 + 50348199259430885764810748/2431765333485025102704350745*a^21 + 84232419129979968287236262/2431765333485025102704350745*a^20 - 48684562739049828065942233/2431765333485025102704350745*a^19 + 82583837806881101618209324/2431765333485025102704350745*a^18 + 169811545003856175052718197/2431765333485025102704350745*a^17 + 5027754440826982190206393/127987649130790794879176355*a^16 - 119053274456571410021897713/810588444495008367568116915*a^15 - 97567007239788052080527383/810588444495008367568116915*a^14 + 15842733353086873876305974/270196148165002789189372305*a^13 + 352271582390997929293075544/2431765333485025102704350745*a^12 + 22737699020833920846979954/486353066697005020540870149*a^11 - 293680233619501093719992863/2431765333485025102704350745*a^10 + 299886607251603669645322948/2431765333485025102704350745*a^9 + 50695008631909918263410285/486353066697005020540870149*a^8 - 27399907750904966221813339/486353066697005020540870149*a^7 + 1596053531769878839463947/162117688899001673513623383*a^6 - 248439601890942936609839224/810588444495008367568116915*a^5 + 74176872318078586483023524/270196148165002789189372305*a^4 - 648476807849439078196242781/2431765333485025102704350745*a^3 - 2800986656809127088337514/21520047198982523032781865*a^2 + 84325508200091328043873264/486353066697005020540870149*a + 106147639565550597670012475/486353066697005020540870149

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$14$	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:		$\frac{27\!\cdots\!04}{16\!\cdots\!83}a^{26}-\frac{24\!\cdots\!74}{18\!\cdots\!87}a^{25}-\frac{35\!\cdots\!34}{24\!\cdots\!45}a^{24}-\frac{25\!\cdots\!87}{81\!\cdots\!15}a^{23}+\frac{29\!\cdots\!89}{81\!\cdots\!15}a^{22}+\frac{22\!\cdots\!09}{24\!\cdots\!45}a^{21}+\frac{14\!\cdots\!88}{81\!\cdots\!15}a^{20}-\frac{20\!\cdots\!14}{81\!\cdots\!15}a^{19}-\frac{55\!\cdots\!78}{24\!\cdots\!45}a^{18}-\frac{28\!\cdots\!01}{81\!\cdots\!15}a^{17}+\frac{71\!\cdots\!99}{27\!\cdots\!05}a^{16}+\frac{22\!\cdots\!27}{24\!\cdots\!45}a^{15}-\frac{34\!\cdots\!83}{81\!\cdots\!15}a^{14}+\frac{17\!\cdots\!31}{16\!\cdots\!83}a^{13}+\frac{49\!\cdots\!39}{24\!\cdots\!45}a^{12}+\frac{21\!\cdots\!37}{81\!\cdots\!15}a^{11}+\frac{50\!\cdots\!47}{81\!\cdots\!15}a^{10}-\frac{63\!\cdots\!61}{24\!\cdots\!45}a^{9}-\frac{82\!\cdots\!83}{16\!\cdots\!83}a^{8}-\frac{27\!\cdots\!11}{54\!\cdots\!61}a^{7}-\frac{15\!\cdots\!11}{24\!\cdots\!45}a^{6}+\frac{75\!\cdots\!46}{16\!\cdots\!83}a^{5}+\frac{41\!\cdots\!07}{81\!\cdots\!15}a^{4}-\frac{21\!\cdots\!36}{24\!\cdots\!45}a^{3}-\frac{90\!\cdots\!13}{71\!\cdots\!55}a^{2}-\frac{26\!\cdots\!83}{16\!\cdots\!83}a+\frac{15\!\cdots\!85}{48\!\cdots\!49}$, $\frac{28\!\cdots\!00}{16\!\cdots\!83}a^{26}+\frac{25\!\cdots\!29}{24\!\cdots\!45}a^{25}+\frac{37\!\cdots\!44}{24\!\cdots\!45}a^{24}+\frac{35\!\cdots\!44}{48\!\cdots\!49}a^{23}-\frac{28\!\cdots\!91}{81\!\cdots\!15}a^{22}-\frac{26\!\cdots\!27}{24\!\cdots\!45}a^{21}-\frac{11\!\cdots\!81}{24\!\cdots\!45}a^{20}+\frac{60\!\cdots\!47}{24\!\cdots\!45}a^{19}+\frac{24\!\cdots\!88}{81\!\cdots\!15}a^{18}+\frac{19\!\cdots\!14}{16\!\cdots\!83}a^{17}-\frac{62\!\cdots\!89}{24\!\cdots\!45}a^{16}-\frac{38\!\cdots\!43}{48\!\cdots\!49}a^{15}+\frac{10\!\cdots\!53}{24\!\cdots\!45}a^{14}-\frac{44\!\cdots\!27}{42\!\cdots\!85}a^{13}-\frac{58\!\cdots\!79}{24\!\cdots\!45}a^{12}-\frac{83\!\cdots\!03}{24\!\cdots\!45}a^{11}-\frac{35\!\cdots\!96}{24\!\cdots\!45}a^{10}+\frac{19\!\cdots\!84}{81\!\cdots\!15}a^{9}+\frac{20\!\cdots\!48}{16\!\cdots\!83}a^{8}+\frac{14\!\cdots\!31}{24\!\cdots\!45}a^{7}+\frac{19\!\cdots\!07}{24\!\cdots\!45}a^{6}-\frac{63\!\cdots\!31}{24\!\cdots\!45}a^{5}-\frac{50\!\cdots\!27}{81\!\cdots\!15}a^{4}-\frac{27\!\cdots\!81}{24\!\cdots\!45}a^{3}+\frac{26\!\cdots\!23}{21\!\cdots\!65}a^{2}+\frac{31\!\cdots\!20}{48\!\cdots\!49}a+\frac{15\!\cdots\!53}{16\!\cdots\!83}$, $\frac{19\!\cdots\!40}{54\!\cdots\!61}a^{26}+\frac{18\!\cdots\!49}{81\!\cdots\!15}a^{25}+\frac{24\!\cdots\!93}{81\!\cdots\!15}a^{24}+\frac{30\!\cdots\!69}{24\!\cdots\!45}a^{23}-\frac{17\!\cdots\!82}{24\!\cdots\!45}a^{22}-\frac{51\!\cdots\!14}{24\!\cdots\!45}a^{21}-\frac{38\!\cdots\!25}{48\!\cdots\!49}a^{20}+\frac{12\!\cdots\!36}{24\!\cdots\!45}a^{19}+\frac{28\!\cdots\!80}{48\!\cdots\!49}a^{18}+\frac{57\!\cdots\!93}{27\!\cdots\!05}a^{17}-\frac{42\!\cdots\!27}{81\!\cdots\!15}a^{16}-\frac{95\!\cdots\!48}{81\!\cdots\!15}a^{15}+\frac{20\!\cdots\!61}{24\!\cdots\!45}a^{14}-\frac{51\!\cdots\!06}{24\!\cdots\!45}a^{13}-\frac{22\!\cdots\!10}{48\!\cdots\!49}a^{12}-\frac{16\!\cdots\!02}{24\!\cdots\!45}a^{11}-\frac{66\!\cdots\!95}{25\!\cdots\!71}a^{10}+\frac{11\!\cdots\!04}{24\!\cdots\!45}a^{9}+\frac{58\!\cdots\!81}{27\!\cdots\!05}a^{8}+\frac{93\!\cdots\!38}{81\!\cdots\!15}a^{7}+\frac{12\!\cdots\!56}{81\!\cdots\!15}a^{6}-\frac{14\!\cdots\!58}{24\!\cdots\!45}a^{5}-\frac{28\!\cdots\!91}{24\!\cdots\!45}a^{4}-\frac{35\!\cdots\!14}{24\!\cdots\!45}a^{3}+\frac{52\!\cdots\!32}{21\!\cdots\!65}a^{2}+\frac{51\!\cdots\!05}{48\!\cdots\!49}a+\frac{64\!\cdots\!83}{48\!\cdots\!49}$, $\frac{81\!\cdots\!71}{81\!\cdots\!15}a^{26}+\frac{67\!\cdots\!63}{81\!\cdots\!15}a^{25}+\frac{67\!\cdots\!27}{81\!\cdots\!15}a^{24}+\frac{53\!\cdots\!76}{24\!\cdots\!45}a^{23}-\frac{94\!\cdots\!79}{48\!\cdots\!49}a^{22}-\frac{13\!\cdots\!62}{24\!\cdots\!45}a^{21}-\frac{37\!\cdots\!66}{24\!\cdots\!45}a^{20}+\frac{32\!\cdots\!26}{24\!\cdots\!45}a^{19}+\frac{34\!\cdots\!56}{24\!\cdots\!45}a^{18}+\frac{47\!\cdots\!36}{81\!\cdots\!15}a^{17}-\frac{11\!\cdots\!39}{81\!\cdots\!15}a^{16}-\frac{76\!\cdots\!16}{16\!\cdots\!83}a^{15}+\frac{51\!\cdots\!86}{24\!\cdots\!45}a^{14}-\frac{30\!\cdots\!92}{48\!\cdots\!49}a^{13}-\frac{28\!\cdots\!59}{24\!\cdots\!45}a^{12}-\frac{43\!\cdots\!78}{24\!\cdots\!45}a^{11}-\frac{13\!\cdots\!94}{24\!\cdots\!45}a^{10}+\frac{30\!\cdots\!12}{24\!\cdots\!45}a^{9}+\frac{30\!\cdots\!43}{81\!\cdots\!15}a^{8}+\frac{27\!\cdots\!86}{81\!\cdots\!15}a^{7}+\frac{59\!\cdots\!07}{16\!\cdots\!83}a^{6}-\frac{44\!\cdots\!86}{24\!\cdots\!45}a^{5}-\frac{61\!\cdots\!42}{24\!\cdots\!45}a^{4}-\frac{11\!\cdots\!51}{24\!\cdots\!45}a^{3}+\frac{10\!\cdots\!67}{21\!\cdots\!65}a^{2}+\frac{14\!\cdots\!18}{48\!\cdots\!49}a+\frac{27\!\cdots\!66}{48\!\cdots\!49}$, $\frac{48\!\cdots\!77}{24\!\cdots\!45}a^{26}+\frac{15\!\cdots\!32}{24\!\cdots\!45}a^{25}+\frac{42\!\cdots\!64}{24\!\cdots\!45}a^{24}+\frac{29\!\cdots\!01}{24\!\cdots\!45}a^{23}-\frac{96\!\cdots\!86}{24\!\cdots\!45}a^{22}-\frac{31\!\cdots\!41}{24\!\cdots\!45}a^{21}-\frac{18\!\cdots\!62}{24\!\cdots\!45}a^{20}+\frac{66\!\cdots\!08}{24\!\cdots\!45}a^{19}+\frac{10\!\cdots\!17}{24\!\cdots\!45}a^{18}+\frac{24\!\cdots\!87}{12\!\cdots\!55}a^{17}-\frac{66\!\cdots\!76}{24\!\cdots\!45}a^{16}-\frac{77\!\cdots\!67}{48\!\cdots\!49}a^{15}+\frac{23\!\cdots\!66}{48\!\cdots\!49}a^{14}-\frac{50\!\cdots\!13}{48\!\cdots\!49}a^{13}-\frac{73\!\cdots\!67}{24\!\cdots\!45}a^{12}-\frac{21\!\cdots\!98}{48\!\cdots\!49}a^{11}-\frac{11\!\cdots\!56}{48\!\cdots\!49}a^{10}+\frac{61\!\cdots\!47}{24\!\cdots\!45}a^{9}+\frac{52\!\cdots\!47}{25\!\cdots\!71}a^{8}+\frac{15\!\cdots\!92}{24\!\cdots\!45}a^{7}+\frac{25\!\cdots\!38}{24\!\cdots\!45}a^{6}-\frac{34\!\cdots\!23}{24\!\cdots\!45}a^{5}-\frac{20\!\cdots\!28}{24\!\cdots\!45}a^{4}-\frac{52\!\cdots\!56}{24\!\cdots\!45}a^{3}+\frac{66\!\cdots\!06}{43\!\cdots\!73}a^{2}+\frac{42\!\cdots\!67}{48\!\cdots\!49}a+\frac{66\!\cdots\!97}{48\!\cdots\!49}$, $\frac{13\!\cdots\!24}{81\!\cdots\!15}a^{26}-\frac{11\!\cdots\!48}{24\!\cdots\!45}a^{25}-\frac{35\!\cdots\!32}{24\!\cdots\!45}a^{24}-\frac{50\!\cdots\!95}{48\!\cdots\!49}a^{23}+\frac{27\!\cdots\!22}{81\!\cdots\!15}a^{22}+\frac{26\!\cdots\!77}{24\!\cdots\!45}a^{21}+\frac{15\!\cdots\!03}{24\!\cdots\!45}a^{20}-\frac{56\!\cdots\!71}{24\!\cdots\!45}a^{19}-\frac{28\!\cdots\!57}{81\!\cdots\!15}a^{18}-\frac{23\!\cdots\!85}{16\!\cdots\!83}a^{17}+\frac{55\!\cdots\!66}{24\!\cdots\!45}a^{16}+\frac{34\!\cdots\!44}{24\!\cdots\!45}a^{15}-\frac{10\!\cdots\!19}{24\!\cdots\!45}a^{14}+\frac{13\!\cdots\!23}{16\!\cdots\!83}a^{13}+\frac{62\!\cdots\!01}{24\!\cdots\!45}a^{12}+\frac{17\!\cdots\!14}{48\!\cdots\!49}a^{11}+\frac{48\!\cdots\!96}{24\!\cdots\!45}a^{10}-\frac{17\!\cdots\!76}{81\!\cdots\!15}a^{9}-\frac{14\!\cdots\!28}{81\!\cdots\!15}a^{8}-\frac{12\!\cdots\!57}{24\!\cdots\!45}a^{7}-\frac{21\!\cdots\!58}{24\!\cdots\!45}a^{6}+\frac{32\!\cdots\!02}{24\!\cdots\!45}a^{5}+\frac{11\!\cdots\!37}{16\!\cdots\!83}a^{4}+\frac{37\!\cdots\!91}{24\!\cdots\!45}a^{3}-\frac{59\!\cdots\!92}{43\!\cdots\!73}a^{2}-\frac{34\!\cdots\!35}{48\!\cdots\!49}a-\frac{18\!\cdots\!19}{16\!\cdots\!83}$, $\frac{44\!\cdots\!38}{81\!\cdots\!15}a^{26}-\frac{74\!\cdots\!78}{24\!\cdots\!45}a^{25}-\frac{43\!\cdots\!58}{90\!\cdots\!35}a^{24}-\frac{63\!\cdots\!16}{27\!\cdots\!05}a^{23}+\frac{27\!\cdots\!39}{24\!\cdots\!45}a^{22}+\frac{27\!\cdots\!83}{81\!\cdots\!15}a^{21}+\frac{13\!\cdots\!97}{90\!\cdots\!35}a^{20}-\frac{19\!\cdots\!24}{24\!\cdots\!45}a^{19}-\frac{52\!\cdots\!37}{54\!\cdots\!61}a^{18}-\frac{28\!\cdots\!04}{81\!\cdots\!15}a^{17}+\frac{19\!\cdots\!43}{24\!\cdots\!45}a^{16}+\frac{13\!\cdots\!82}{54\!\cdots\!61}a^{15}-\frac{36\!\cdots\!03}{27\!\cdots\!05}a^{14}+\frac{78\!\cdots\!33}{24\!\cdots\!45}a^{13}+\frac{61\!\cdots\!76}{81\!\cdots\!15}a^{12}+\frac{28\!\cdots\!76}{27\!\cdots\!05}a^{11}+\frac{11\!\cdots\!16}{24\!\cdots\!45}a^{10}-\frac{20\!\cdots\!18}{27\!\cdots\!05}a^{9}-\frac{32\!\cdots\!43}{81\!\cdots\!15}a^{8}-\frac{42\!\cdots\!21}{24\!\cdots\!45}a^{7}-\frac{68\!\cdots\!22}{27\!\cdots\!05}a^{6}+\frac{23\!\cdots\!63}{27\!\cdots\!05}a^{5}+\frac{48\!\cdots\!33}{24\!\cdots\!45}a^{4}+\frac{19\!\cdots\!47}{81\!\cdots\!15}a^{3}-\frac{97\!\cdots\!07}{23\!\cdots\!85}a^{2}-\frac{83\!\cdots\!19}{48\!\cdots\!49}a-\frac{12\!\cdots\!31}{54\!\cdots\!61}$, $\frac{22\!\cdots\!49}{90\!\cdots\!35}a^{26}-\frac{41\!\cdots\!36}{24\!\cdots\!45}a^{25}-\frac{11\!\cdots\!86}{54\!\cdots\!61}a^{24}-\frac{58\!\cdots\!53}{81\!\cdots\!15}a^{23}+\frac{25\!\cdots\!32}{48\!\cdots\!49}a^{22}+\frac{23\!\cdots\!01}{16\!\cdots\!83}a^{21}+\frac{33\!\cdots\!27}{81\!\cdots\!15}a^{20}-\frac{88\!\cdots\!63}{24\!\cdots\!45}a^{19}-\frac{34\!\cdots\!51}{90\!\cdots\!35}a^{18}-\frac{21\!\cdots\!13}{27\!\cdots\!05}a^{17}+\frac{94\!\cdots\!02}{24\!\cdots\!45}a^{16}+\frac{16\!\cdots\!99}{27\!\cdots\!05}a^{15}-\frac{51\!\cdots\!99}{81\!\cdots\!15}a^{14}+\frac{37\!\cdots\!22}{24\!\cdots\!45}a^{13}+\frac{26\!\cdots\!96}{81\!\cdots\!15}a^{12}+\frac{67\!\cdots\!96}{16\!\cdots\!83}a^{11}+\frac{62\!\cdots\!46}{48\!\cdots\!49}a^{10}-\frac{34\!\cdots\!78}{90\!\cdots\!35}a^{9}-\frac{34\!\cdots\!56}{27\!\cdots\!05}a^{8}-\frac{17\!\cdots\!01}{24\!\cdots\!45}a^{7}-\frac{92\!\cdots\!07}{90\!\cdots\!35}a^{6}+\frac{46\!\cdots\!63}{81\!\cdots\!15}a^{5}+\frac{21\!\cdots\!73}{24\!\cdots\!45}a^{4}-\frac{50\!\cdots\!08}{81\!\cdots\!15}a^{3}-\frac{15\!\cdots\!66}{71\!\cdots\!55}a^{2}-\frac{15\!\cdots\!13}{48\!\cdots\!49}a+\frac{99\!\cdots\!54}{18\!\cdots\!87}$, $\frac{84\!\cdots\!39}{24\!\cdots\!45}a^{26}+\frac{25\!\cdots\!81}{24\!\cdots\!45}a^{25}+\frac{77\!\cdots\!36}{24\!\cdots\!45}a^{24}+\frac{73\!\cdots\!47}{24\!\cdots\!45}a^{23}-\frac{16\!\cdots\!51}{24\!\cdots\!45}a^{22}-\frac{12\!\cdots\!87}{48\!\cdots\!49}a^{21}-\frac{47\!\cdots\!58}{24\!\cdots\!45}a^{20}+\frac{11\!\cdots\!49}{24\!\cdots\!45}a^{19}+\frac{21\!\cdots\!77}{24\!\cdots\!45}a^{18}+\frac{11\!\cdots\!01}{24\!\cdots\!45}a^{17}-\frac{10\!\cdots\!69}{24\!\cdots\!45}a^{16}-\frac{10\!\cdots\!06}{24\!\cdots\!45}a^{15}+\frac{40\!\cdots\!30}{48\!\cdots\!49}a^{14}-\frac{38\!\cdots\!41}{24\!\cdots\!45}a^{13}-\frac{14\!\cdots\!53}{24\!\cdots\!45}a^{12}-\frac{21\!\cdots\!48}{24\!\cdots\!45}a^{11}-\frac{14\!\cdots\!71}{24\!\cdots\!45}a^{10}+\frac{95\!\cdots\!69}{24\!\cdots\!45}a^{9}+\frac{12\!\cdots\!74}{24\!\cdots\!45}a^{8}+\frac{28\!\cdots\!68}{24\!\cdots\!45}a^{7}+\frac{10\!\cdots\!14}{48\!\cdots\!49}a^{6}+\frac{40\!\cdots\!11}{24\!\cdots\!45}a^{5}-\frac{41\!\cdots\!18}{24\!\cdots\!45}a^{4}-\frac{34\!\cdots\!88}{48\!\cdots\!49}a^{3}+\frac{65\!\cdots\!31}{21\!\cdots\!65}a^{2}+\frac{11\!\cdots\!37}{48\!\cdots\!49}a+\frac{19\!\cdots\!89}{48\!\cdots\!49}$, $\frac{30\!\cdots\!99}{48\!\cdots\!49}a^{26}+\frac{13\!\cdots\!67}{24\!\cdots\!45}a^{25}+\frac{12\!\cdots\!69}{24\!\cdots\!45}a^{24}+\frac{81\!\cdots\!03}{81\!\cdots\!15}a^{23}-\frac{10\!\cdots\!74}{81\!\cdots\!15}a^{22}-\frac{82\!\cdots\!44}{24\!\cdots\!45}a^{21}-\frac{30\!\cdots\!35}{48\!\cdots\!49}a^{20}+\frac{21\!\cdots\!41}{24\!\cdots\!45}a^{19}+\frac{13\!\cdots\!05}{16\!\cdots\!83}a^{18}+\frac{42\!\cdots\!57}{24\!\cdots\!45}a^{17}-\frac{22\!\cdots\!96}{24\!\cdots\!45}a^{16}+\frac{45\!\cdots\!31}{24\!\cdots\!45}a^{15}+\frac{63\!\cdots\!58}{42\!\cdots\!85}a^{14}-\frac{33\!\cdots\!52}{81\!\cdots\!15}a^{13}-\frac{18\!\cdots\!58}{25\!\cdots\!71}a^{12}-\frac{23\!\cdots\!67}{24\!\cdots\!45}a^{11}-\frac{11\!\cdots\!26}{48\!\cdots\!49}a^{10}+\frac{72\!\cdots\!13}{81\!\cdots\!15}a^{9}+\frac{33\!\cdots\!19}{24\!\cdots\!45}a^{8}+\frac{47\!\cdots\!04}{24\!\cdots\!45}a^{7}+\frac{54\!\cdots\!03}{24\!\cdots\!45}a^{6}-\frac{13\!\cdots\!11}{81\!\cdots\!15}a^{5}-\frac{13\!\cdots\!07}{81\!\cdots\!15}a^{4}+\frac{51\!\cdots\!36}{24\!\cdots\!45}a^{3}+\frac{70\!\cdots\!37}{21\!\cdots\!65}a^{2}+\frac{42\!\cdots\!93}{48\!\cdots\!49}a+\frac{18\!\cdots\!22}{16\!\cdots\!83}$, $\frac{18\!\cdots\!58}{24\!\cdots\!45}a^{26}+\frac{46\!\cdots\!40}{48\!\cdots\!49}a^{25}+\frac{29\!\cdots\!69}{48\!\cdots\!49}a^{24}-\frac{11\!\cdots\!49}{81\!\cdots\!15}a^{23}-\frac{44\!\cdots\!79}{27\!\cdots\!05}a^{22}-\frac{84\!\cdots\!53}{24\!\cdots\!45}a^{21}+\frac{26\!\cdots\!37}{24\!\cdots\!45}a^{20}+\frac{27\!\cdots\!68}{24\!\cdots\!45}a^{19}+\frac{43\!\cdots\!16}{81\!\cdots\!15}a^{18}-\frac{64\!\cdots\!96}{24\!\cdots\!45}a^{17}-\frac{63\!\cdots\!57}{48\!\cdots\!49}a^{16}+\frac{11\!\cdots\!56}{24\!\cdots\!45}a^{15}+\frac{14\!\cdots\!02}{81\!\cdots\!15}a^{14}-\frac{15\!\cdots\!09}{27\!\cdots\!05}a^{13}-\frac{16\!\cdots\!18}{24\!\cdots\!45}a^{12}-\frac{19\!\cdots\!02}{24\!\cdots\!45}a^{11}+\frac{13\!\cdots\!15}{48\!\cdots\!49}a^{10}+\frac{10\!\cdots\!49}{81\!\cdots\!15}a^{9}-\frac{54\!\cdots\!71}{24\!\cdots\!45}a^{8}+\frac{53\!\cdots\!92}{24\!\cdots\!45}a^{7}+\frac{83\!\cdots\!10}{48\!\cdots\!49}a^{6}-\frac{51\!\cdots\!08}{16\!\cdots\!83}a^{5}-\frac{13\!\cdots\!59}{90\!\cdots\!35}a^{4}+\frac{29\!\cdots\!42}{24\!\cdots\!45}a^{3}+\frac{10\!\cdots\!77}{21\!\cdots\!65}a^{2}-\frac{67\!\cdots\!17}{48\!\cdots\!49}a-\frac{94\!\cdots\!98}{16\!\cdots\!83}$, $\frac{70\!\cdots\!22}{24\!\cdots\!45}a^{26}+\frac{13\!\cdots\!61}{81\!\cdots\!15}a^{25}+\frac{59\!\cdots\!58}{24\!\cdots\!45}a^{24}+\frac{32\!\cdots\!08}{27\!\cdots\!05}a^{23}-\frac{27\!\cdots\!91}{48\!\cdots\!49}a^{22}-\frac{42\!\cdots\!31}{24\!\cdots\!45}a^{21}-\frac{19\!\cdots\!79}{24\!\cdots\!45}a^{20}+\frac{93\!\cdots\!66}{24\!\cdots\!45}a^{19}+\frac{80\!\cdots\!48}{16\!\cdots\!83}a^{18}+\frac{56\!\cdots\!41}{24\!\cdots\!45}a^{17}-\frac{31\!\cdots\!51}{81\!\cdots\!15}a^{16}-\frac{23\!\cdots\!81}{24\!\cdots\!45}a^{15}+\frac{11\!\cdots\!50}{18\!\cdots\!87}a^{14}-\frac{39\!\cdots\!17}{24\!\cdots\!45}a^{13}-\frac{18\!\cdots\!14}{48\!\cdots\!49}a^{12}-\frac{13\!\cdots\!39}{24\!\cdots\!45}a^{11}-\frac{64\!\cdots\!32}{24\!\cdots\!45}a^{10}+\frac{26\!\cdots\!39}{81\!\cdots\!15}a^{9}+\frac{48\!\cdots\!06}{24\!\cdots\!45}a^{8}+\frac{15\!\cdots\!00}{16\!\cdots\!83}a^{7}+\frac{31\!\cdots\!49}{24\!\cdots\!45}a^{6}-\frac{90\!\cdots\!48}{27\!\cdots\!05}a^{5}-\frac{22\!\cdots\!44}{24\!\cdots\!45}a^{4}-\frac{50\!\cdots\!84}{24\!\cdots\!45}a^{3}+\frac{35\!\cdots\!59}{21\!\cdots\!65}a^{2}+\frac{54\!\cdots\!42}{48\!\cdots\!49}a+\frac{36\!\cdots\!94}{16\!\cdots\!83}$, $\frac{85\!\cdots\!88}{24\!\cdots\!45}a^{26}+\frac{57\!\cdots\!73}{24\!\cdots\!45}a^{25}+\frac{76\!\cdots\!62}{24\!\cdots\!45}a^{24}+\frac{26\!\cdots\!12}{24\!\cdots\!45}a^{23}-\frac{39\!\cdots\!64}{48\!\cdots\!49}a^{22}-\frac{51\!\cdots\!29}{24\!\cdots\!45}a^{21}-\frac{40\!\cdots\!08}{81\!\cdots\!15}a^{20}+\frac{46\!\cdots\!27}{81\!\cdots\!15}a^{19}+\frac{49\!\cdots\!12}{81\!\cdots\!15}a^{18}+\frac{99\!\cdots\!41}{24\!\cdots\!45}a^{17}-\frac{18\!\cdots\!62}{24\!\cdots\!45}a^{16}-\frac{13\!\cdots\!42}{48\!\cdots\!49}a^{15}+\frac{22\!\cdots\!51}{24\!\cdots\!45}a^{14}-\frac{50\!\cdots\!76}{24\!\cdots\!45}a^{13}-\frac{11\!\cdots\!53}{24\!\cdots\!45}a^{12}-\frac{45\!\cdots\!18}{81\!\cdots\!15}a^{11}-\frac{77\!\cdots\!46}{16\!\cdots\!83}a^{10}+\frac{64\!\cdots\!54}{81\!\cdots\!15}a^{9}+\frac{10\!\cdots\!81}{24\!\cdots\!45}a^{8}+\frac{26\!\cdots\!72}{24\!\cdots\!45}a^{7}+\frac{34\!\cdots\!42}{24\!\cdots\!45}a^{6}-\frac{26\!\cdots\!17}{24\!\cdots\!45}a^{5}-\frac{44\!\cdots\!46}{24\!\cdots\!45}a^{4}-\frac{42\!\cdots\!16}{24\!\cdots\!45}a^{3}+\frac{70\!\cdots\!52}{14\!\cdots\!91}a^{2}+\frac{32\!\cdots\!01}{16\!\cdots\!83}a+\frac{38\!\cdots\!83}{16\!\cdots\!83}$, $\frac{10\!\cdots\!07}{81\!\cdots\!15}a^{26}-\frac{84\!\cdots\!04}{24\!\cdots\!45}a^{25}-\frac{28\!\cdots\!11}{24\!\cdots\!45}a^{24}-\frac{20\!\cdots\!39}{24\!\cdots\!45}a^{23}+\frac{20\!\cdots\!67}{81\!\cdots\!15}a^{22}+\frac{14\!\cdots\!82}{16\!\cdots\!83}a^{21}+\frac{13\!\cdots\!79}{24\!\cdots\!45}a^{20}-\frac{43\!\cdots\!34}{24\!\cdots\!45}a^{19}-\frac{67\!\cdots\!19}{24\!\cdots\!45}a^{18}-\frac{10\!\cdots\!64}{81\!\cdots\!15}a^{17}+\frac{42\!\cdots\!32}{24\!\cdots\!45}a^{16}+\frac{27\!\cdots\!09}{24\!\cdots\!45}a^{15}-\frac{77\!\cdots\!42}{24\!\cdots\!45}a^{14}+\frac{28\!\cdots\!63}{42\!\cdots\!85}a^{13}+\frac{16\!\cdots\!89}{81\!\cdots\!15}a^{12}+\frac{14\!\cdots\!71}{48\!\cdots\!49}a^{11}+\frac{41\!\cdots\!74}{24\!\cdots\!45}a^{10}-\frac{38\!\cdots\!74}{24\!\cdots\!45}a^{9}-\frac{11\!\cdots\!01}{81\!\cdots\!15}a^{8}-\frac{20\!\cdots\!35}{48\!\cdots\!49}a^{7}-\frac{34\!\cdots\!89}{48\!\cdots\!49}a^{6}+\frac{14\!\cdots\!16}{24\!\cdots\!45}a^{5}+\frac{88\!\cdots\!21}{16\!\cdots\!83}a^{4}+\frac{12\!\cdots\!73}{81\!\cdots\!15}a^{3}-\frac{23\!\cdots\!79}{21\!\cdots\!65}a^{2}-\frac{29\!\cdots\!94}{48\!\cdots\!49}a-\frac{26\!\cdots\!37}{48\!\cdots\!49}$ 27453685235893374924077704/162117688899001673513623383*a^26 - 2483752529225264218500374/18013076544333519279291487*a^25 - 3520366623403207568389387634/2431765333485025102704350745*a^24 - 259013385114664625039573987/810588444495008367568116915*a^23 + 2932223797770432297496116089/810588444495008367568116915*a^22 + 22725838964087624764493127709/2431765333485025102704350745*a^21 + 1451380084286318354967084088/810588444495008367568116915*a^20 - 20148817989447103813831281014/810588444495008367568116915*a^19 - 55935734837949894665420516378/2431765333485025102704350745*a^18 - 2802399495892401293687943601/810588444495008367568116915*a^17 + 7174887000950202513970668199/270196148165002789189372305*a^16 + 2267369718315814901280681127/2431765333485025102704350745*a^15 - 34901409856629684322031170283/810588444495008367568116915*a^14 + 17890612506032409024657470431/162117688899001673513623383*a^13 + 499891268334352956716901616339/2431765333485025102704350745*a^12 + 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10940956138226078370189255364/810588444495008367568116915*a^17 + 42659790452028639004711965632/2431765333485025102704350745*a^16 + 27402086259183070213117764209/2431765333485025102704350745*a^15 - 77607315614756965821411633242/2431765333485025102704350745*a^14 + 2878327298877627911219943463/42662549710263598293058785*a^13 + 164453797613242524663687053189/810588444495008367568116915*a^12 + 143010284930280205683866565271/486353066697005020540870149*a^11 + 413531874918819226114060330474/2431765333485025102704350745*a^10 - 389506323097488722891474352974/2431765333485025102704350745*a^9 - 113660836074381672687816651601/810588444495008367568116915*a^8 - 207151063553437220636010363635/486353066697005020540870149*a^7 - 347212051694029402362538608689/486353066697005020540870149*a^6 + 145194979597402181880539703616/2431765333485025102704350745*a^5 + 88834461412204669742899691621/162117688899001673513623383*a^4 + 128967320287547706239948519873/810588444495008367568116915*a^3 - 2362720238531070671483303579/21520047198982523032781865*a^2 - 29358220170184573011106471594/486353066697005020540870149*a - 2690000758568715307576342537/486353066697005020540870149 (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:		\( 7929164662.056825 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 7929164662.056825 \cdot 1}{2\cdot\sqrt{9527252910578083379673525288960000000000}}\cr\approx \mathstrut & 1.23016113293332 \end{aligned}\] (assuming GRH)

Galois group

$C_3^3:S_4$ (as 27T211):

Intermediate fields

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

Sibling fields

Frobenius cycle types

In the table, R denotes a ramified prime. 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
\(2\) 2		Deg $27$	$9$	$3$	$24$
\(3\) 3	3.9.18.2	$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$	$9$	$1$	$18$	$C_3^2:C_2$	$[3/2, 5/2]_{2}$
	3.9.18.2	$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$	$9$	$1$	$18$	$C_3^2:C_2$	$[3/2, 5/2]_{2}$
	3.9.18.2	$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$	$9$	$1$	$18$	$C_3^2:C_2$	$[3/2, 5/2]_{2}$
\(5\) 5	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.4.2.2	$x^{4} - 20 x^{2} + 50$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	5.4.2.2	$x^{4} - 20 x^{2} + 50$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	5.6.0.1	$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	5.12.6.2	$x^{12} + 25 x^{8} - 500 x^{6} + 625 x^{4} + 31250$	$2$	$6$	$6$	$C_{12}$	$[\ ]_{2}^{6}$