Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1)
gp: K = bnfinit(y^27 - 4*y^26 + 14*y^25 - 10*y^24 + 37*y^22 - 3*y^21 - y^20 + 139*y^19 - 26*y^18 + 76*y^17 + 144*y^16 - 117*y^15 - 101*y^14 + 12*y^13 - 340*y^12 + y^11 + 205*y^10 + 307*y^9 + 774*y^8 + 909*y^7 + 892*y^6 + 864*y^5 + 521*y^4 + 327*y^3 + 86*y^2 + 71*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1)
\( x^{27} - 4 x^{26} + 14 x^{25} - 10 x^{24} + 37 x^{22} - 3 x^{21} - x^{20} + 139 x^{19} - 26 x^{18} + 76 x^{17} + 144 x^{16} - 117 x^{15} - 101 x^{14} + 12 x^{13} - 340 x^{12} + x^{11} + 205 x^{10} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(-14905779350658261917360296447434047234191\)
\(\medspace = -\,1231^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.75\) | 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))
| |
| Ramified primes: |
\(1231\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1231}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{12}-\frac{2}{9}a^{4}$, $\frac{1}{27}a^{21}-\frac{1}{27}a^{19}+\frac{1}{27}a^{18}+\frac{1}{27}a^{17}+\frac{1}{27}a^{16}-\frac{1}{9}a^{15}-\frac{2}{27}a^{13}-\frac{1}{9}a^{12}+\frac{2}{27}a^{11}+\frac{1}{27}a^{10}+\frac{4}{27}a^{9}+\frac{4}{27}a^{8}-\frac{2}{9}a^{7}+\frac{1}{3}a^{6}-\frac{8}{27}a^{5}+\frac{4}{9}a^{4}-\frac{10}{27}a^{3}+\frac{7}{27}a^{2}+\frac{13}{27}a+\frac{4}{27}$, $\frac{1}{27}a^{22}-\frac{1}{27}a^{20}+\frac{1}{27}a^{19}+\frac{1}{27}a^{18}+\frac{1}{27}a^{17}-\frac{2}{27}a^{14}-\frac{1}{9}a^{13}+\frac{2}{27}a^{12}+\frac{1}{27}a^{11}+\frac{4}{27}a^{10}+\frac{4}{27}a^{9}-\frac{1}{9}a^{8}+\frac{1}{3}a^{7}-\frac{8}{27}a^{6}+\frac{4}{9}a^{5}-\frac{10}{27}a^{4}+\frac{7}{27}a^{3}+\frac{13}{27}a^{2}+\frac{4}{27}a-\frac{2}{9}$, $\frac{1}{81}a^{23}+\frac{1}{81}a^{22}+\frac{1}{81}a^{21}+\frac{1}{27}a^{20}+\frac{4}{81}a^{18}+\frac{2}{81}a^{16}-\frac{8}{81}a^{15}-\frac{5}{81}a^{14}+\frac{4}{81}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{10}{81}a^{10}-\frac{1}{27}a^{9}+\frac{5}{81}a^{8}-\frac{38}{81}a^{7}-\frac{5}{81}a^{6}+\frac{4}{81}a^{5}+\frac{11}{27}a^{4}-\frac{4}{9}a^{3}-\frac{23}{81}a^{2}+\frac{4}{27}a+\frac{38}{81}$, $\frac{1}{135999}a^{24}+\frac{206}{135999}a^{23}+\frac{1538}{135999}a^{22}-\frac{1697}{135999}a^{21}-\frac{2246}{45333}a^{20}+\frac{3988}{135999}a^{19}-\frac{3137}{135999}a^{18}-\frac{742}{135999}a^{17}-\frac{121}{15111}a^{16}+\frac{128}{135999}a^{15}+\frac{11759}{135999}a^{14}-\frac{6080}{135999}a^{13}-\frac{1181}{15111}a^{12}-\frac{16958}{135999}a^{11}+\frac{2437}{135999}a^{10}+\frac{13640}{135999}a^{9}+\frac{53}{15111}a^{8}-\frac{51472}{135999}a^{7}-\frac{157}{1863}a^{6}-\frac{485}{135999}a^{5}-\frac{14119}{45333}a^{4}+\frac{29305}{135999}a^{3}+\frac{5695}{135999}a^{2}-\frac{52588}{135999}a-\frac{56035}{135999}$, $\frac{1}{21623841}a^{25}+\frac{58}{21623841}a^{24}+\frac{93617}{21623841}a^{23}-\frac{41273}{21623841}a^{22}+\frac{205801}{21623841}a^{21}-\frac{509888}{21623841}a^{20}+\frac{70070}{2402649}a^{19}-\frac{217667}{7207947}a^{18}+\frac{647686}{21623841}a^{17}+\frac{2937}{88987}a^{16}-\frac{3541480}{21623841}a^{15}+\frac{2405755}{21623841}a^{14}+\frac{100081}{21623841}a^{13}-\frac{84328}{940167}a^{12}+\frac{16042}{313389}a^{11}+\frac{835195}{7207947}a^{10}-\frac{5197}{407997}a^{9}-\frac{701096}{7207947}a^{8}-\frac{2521333}{21623841}a^{7}+\frac{9580327}{21623841}a^{6}-\frac{7559657}{21623841}a^{5}-\frac{9563372}{21623841}a^{4}-\frac{2451215}{7207947}a^{3}+\frac{2971090}{7207947}a^{2}-\frac{1035866}{7207947}a-\frac{9106297}{21623841}$, $\frac{1}{717044643038151}a^{26}+\frac{15099116}{717044643038151}a^{25}-\frac{445493129}{239014881012717}a^{24}-\frac{525974495980}{239014881012717}a^{23}-\frac{6273715611433}{717044643038151}a^{22}+\frac{7554204232715}{717044643038151}a^{21}+\frac{38591987442955}{717044643038151}a^{20}+\frac{32820573286}{611291255787}a^{19}-\frac{37328321562947}{717044643038151}a^{18}+\frac{357275968862}{8639092084797}a^{17}-\frac{376548944054}{42179096649303}a^{16}-\frac{1200727106698}{26557209001413}a^{15}-\frac{1565854272901}{9822529356687}a^{14}-\frac{71411258304379}{717044643038151}a^{13}+\frac{53009882666989}{717044643038151}a^{12}+\frac{12441069640637}{79671627004239}a^{11}+\frac{9892065981073}{717044643038151}a^{10}+\frac{28139744111119}{717044643038151}a^{9}+\frac{5385579118484}{717044643038151}a^{8}-\frac{10536512001476}{239014881012717}a^{7}-\frac{276749630863258}{717044643038151}a^{6}-\frac{98538605571637}{717044643038151}a^{5}+\frac{208887455610868}{717044643038151}a^{4}-\frac{45066103184386}{239014881012717}a^{3}+\frac{87820955548717}{239014881012717}a^{2}-\frac{344841463323733}{717044643038151}a+\frac{42099399472001}{717044643038151}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (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: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$\frac{409678074001}{717044643038151}a^{26}-\frac{432331809635}{31175854045137}a^{25}+\frac{15974627535236}{239014881012717}a^{24}-\frac{18420093224269}{79671627004239}a^{23}+\frac{251786574910409}{717044643038151}a^{22}-\frac{224934277119883}{717044643038151}a^{21}-\frac{159910226355986}{717044643038151}a^{20}+\frac{7203133391330}{14059698883101}a^{19}-\frac{400166684898851}{717044643038151}a^{18}-\frac{11279923546894}{8639092084797}a^{17}+\frac{97117569696331}{42179096649303}a^{16}-\frac{775359303538325}{239014881012717}a^{15}+\frac{359716734682451}{717044643038151}a^{14}+\frac{19\!\cdots\!89}{717044643038151}a^{13}-\frac{18\!\cdots\!53}{717044643038151}a^{12}+\frac{102833200896734}{79671627004239}a^{11}+\frac{163582751191214}{31175854045137}a^{10}-\frac{46\!\cdots\!13}{717044643038151}a^{9}+\frac{19\!\cdots\!37}{717044643038151}a^{8}-\frac{535608052586218}{239014881012717}a^{7}-\frac{66\!\cdots\!87}{717044643038151}a^{6}-\frac{821807328323647}{717044643038151}a^{5}-\frac{41\!\cdots\!12}{717044643038151}a^{4}-\frac{14\!\cdots\!95}{239014881012717}a^{3}+\frac{156379040572100}{79671627004239}a^{2}-\frac{20\!\cdots\!14}{717044643038151}a+\frac{946211382680999}{717044643038151}$, $\frac{3671227302868}{239014881012717}a^{26}-\frac{16060633004918}{239014881012717}a^{25}+\frac{6362790613421}{26557209001413}a^{24}-\frac{19304302504823}{79671627004239}a^{23}+\frac{6908525069242}{79671627004239}a^{22}+\frac{127225734441353}{239014881012717}a^{21}-\frac{798948181871}{2950801000157}a^{20}+\frac{2204732282522}{14059698883101}a^{19}+\frac{471399151823789}{239014881012717}a^{18}-\frac{3490691469776}{2879697361599}a^{17}+\frac{24392460593131}{14059698883101}a^{16}+\frac{116426995692827}{79671627004239}a^{15}-\frac{213542499668639}{79671627004239}a^{14}-\frac{7578945204055}{239014881012717}a^{13}+\frac{1718085047788}{79671627004239}a^{12}-\frac{55025018331874}{10391951348379}a^{11}+\frac{677530081652087}{239014881012717}a^{10}+\frac{539588210357714}{239014881012717}a^{9}+\frac{736397959808312}{239014881012717}a^{8}+\frac{312797434459778}{26557209001413}a^{7}+\frac{724176337794118}{79671627004239}a^{6}+\frac{22\!\cdots\!25}{239014881012717}a^{5}+\frac{811943262312827}{79671627004239}a^{4}+\frac{925047236563564}{239014881012717}a^{3}+\frac{728007802546519}{239014881012717}a^{2}+\frac{161582083708288}{239014881012717}a+\frac{137068124541119}{239014881012717}$, $\frac{34913949297793}{717044643038151}a^{26}-\frac{42439859550863}{239014881012717}a^{25}+\frac{441284105617303}{717044643038151}a^{24}-\frac{192261621565921}{717044643038151}a^{23}-\frac{22868792615911}{239014881012717}a^{22}+\frac{392556666059993}{239014881012717}a^{21}+\frac{357286816355936}{717044643038151}a^{20}+\frac{1575686503147}{14059698883101}a^{19}+\frac{45\!\cdots\!09}{717044643038151}a^{18}+\frac{7267605190249}{8639092084797}a^{17}+\frac{162266077902175}{42179096649303}a^{16}+\frac{48\!\cdots\!75}{717044643038151}a^{15}-\frac{97437443439637}{26557209001413}a^{14}-\frac{15\!\cdots\!59}{239014881012717}a^{13}-\frac{23\!\cdots\!71}{717044643038151}a^{12}-\frac{47535922216186}{2950801000157}a^{11}-\frac{20\!\cdots\!49}{717044643038151}a^{10}+\frac{66\!\cdots\!01}{717044643038151}a^{9}+\frac{15\!\cdots\!70}{717044643038151}a^{8}+\frac{32\!\cdots\!00}{717044643038151}a^{7}+\frac{13\!\cdots\!17}{239014881012717}a^{6}+\frac{13\!\cdots\!41}{239014881012717}a^{5}+\frac{36\!\cdots\!54}{717044643038151}a^{4}+\frac{71\!\cdots\!40}{239014881012717}a^{3}+\frac{40\!\cdots\!26}{239014881012717}a^{2}+\frac{18\!\cdots\!44}{717044643038151}a+\frac{823183542136870}{717044643038151}$, $\frac{5360115432233}{717044643038151}a^{26}-\frac{6251824790168}{239014881012717}a^{25}+\frac{1243886994190}{13529144208267}a^{24}-\frac{27201566156543}{717044643038151}a^{23}+\frac{4148528679559}{239014881012717}a^{22}+\frac{44775603059597}{239014881012717}a^{21}+\frac{60619161582145}{717044643038151}a^{20}+\frac{1309123396429}{4686566294367}a^{19}+\frac{454637627875004}{717044643038151}a^{18}+\frac{1862464628642}{8639092084797}a^{17}+\frac{49713693560156}{42179096649303}a^{16}+\frac{340999150356193}{717044643038151}a^{15}-\frac{6073284498952}{8852403000471}a^{14}+\frac{23611795529615}{239014881012717}a^{13}-\frac{14\!\cdots\!50}{717044643038151}a^{12}-\frac{539342105955895}{239014881012717}a^{11}+\frac{608182244995610}{717044643038151}a^{10}-\frac{115500472185482}{717044643038151}a^{9}+\frac{24\!\cdots\!43}{717044643038151}a^{8}+\frac{64\!\cdots\!93}{717044643038151}a^{7}+\frac{17\!\cdots\!91}{239014881012717}a^{6}+\frac{22\!\cdots\!01}{239014881012717}a^{5}+\frac{63\!\cdots\!96}{717044643038151}a^{4}+\frac{10\!\cdots\!09}{239014881012717}a^{3}+\frac{270067063185835}{79671627004239}a^{2}+\frac{654870716996854}{717044643038151}a+\frac{413606949187670}{717044643038151}$, $\frac{8849104875941}{717044643038151}a^{26}-\frac{28584332957233}{717044643038151}a^{25}+\frac{81030822904573}{717044643038151}a^{24}+\frac{64549519391051}{717044643038151}a^{23}-\frac{251384942201116}{717044643038151}a^{22}+\frac{351555699890240}{717044643038151}a^{21}+\frac{162035090434121}{239014881012717}a^{20}-\frac{13887216198451}{14059698883101}a^{19}+\frac{38047853387725}{31175854045137}a^{18}+\frac{2013919780315}{959899120533}a^{17}-\frac{1279298133634}{577795844511}a^{16}+\frac{809872948684574}{717044643038151}a^{15}+\frac{11\!\cdots\!53}{717044643038151}a^{14}-\frac{41\!\cdots\!87}{717044643038151}a^{13}-\frac{49020727706599}{79671627004239}a^{12}+\frac{612104442516559}{239014881012717}a^{11}-\frac{32\!\cdots\!79}{717044643038151}a^{10}+\frac{44466874321318}{10391951348379}a^{9}+\frac{76\!\cdots\!50}{717044643038151}a^{8}+\frac{37\!\cdots\!40}{717044643038151}a^{7}+\frac{51\!\cdots\!40}{717044643038151}a^{6}+\frac{62\!\cdots\!28}{717044643038151}a^{5}+\frac{667868289674779}{239014881012717}a^{4}-\frac{368151959628149}{239014881012717}a^{3}-\frac{216169207622705}{79671627004239}a^{2}-\frac{700639394215289}{717044643038151}a-\frac{28491500627923}{26557209001413}$, $\frac{996295636388}{239014881012717}a^{26}-\frac{1137402128054}{239014881012717}a^{25}+\frac{1094656092740}{239014881012717}a^{24}+\frac{11576947424920}{79671627004239}a^{23}-\frac{46025567134181}{239014881012717}a^{22}+\frac{44438033338595}{239014881012717}a^{21}+\frac{85514940787346}{239014881012717}a^{20}-\frac{938951142197}{14059698883101}a^{19}+\frac{25878110751910}{79671627004239}a^{18}+\frac{1281110986453}{959899120533}a^{17}-\frac{6576251487155}{14059698883101}a^{16}+\frac{96647792733350}{79671627004239}a^{15}-\frac{21506320620209}{239014881012717}a^{14}-\frac{419693141541808}{239014881012717}a^{13}-\frac{325384732162120}{239014881012717}a^{12}-\frac{95000521419889}{239014881012717}a^{11}-\frac{73530543340774}{26557209001413}a^{10}+\frac{221473115473774}{79671627004239}a^{9}+\frac{963523754098430}{239014881012717}a^{8}+\frac{61437447985196}{8852403000471}a^{7}+\frac{19\!\cdots\!75}{239014881012717}a^{6}+\frac{19\!\cdots\!90}{239014881012717}a^{5}+\frac{13\!\cdots\!74}{239014881012717}a^{4}+\frac{32388434694337}{10391951348379}a^{3}+\frac{22977728801653}{239014881012717}a^{2}-\frac{233686061693377}{239014881012717}a-\frac{59535538560955}{79671627004239}$, $\frac{11580232358185}{717044643038151}a^{26}-\frac{51468033172363}{717044643038151}a^{25}+\frac{62656768047559}{239014881012717}a^{24}-\frac{2423018643860}{8852403000471}a^{23}+\frac{67466167704143}{717044643038151}a^{22}+\frac{7709085831797}{9822529356687}a^{21}-\frac{376266244059470}{717044643038151}a^{20}+\frac{196956346379}{4686566294367}a^{19}+\frac{21\!\cdots\!46}{717044643038151}a^{18}-\frac{12852726707005}{8639092084797}a^{17}+\frac{59080829470627}{42179096649303}a^{16}+\frac{920032365750580}{239014881012717}a^{15}-\frac{22\!\cdots\!44}{717044643038151}a^{14}-\frac{771952918499959}{717044643038151}a^{13}+\frac{14\!\cdots\!82}{717044643038151}a^{12}-\frac{20\!\cdots\!50}{239014881012717}a^{11}-\frac{556329401875196}{717044643038151}a^{10}+\frac{154613307247571}{31175854045137}a^{9}+\frac{17\!\cdots\!13}{717044643038151}a^{8}+\frac{29\!\cdots\!68}{239014881012717}a^{7}+\frac{11\!\cdots\!37}{717044643038151}a^{6}+\frac{11\!\cdots\!78}{717044643038151}a^{5}+\frac{11\!\cdots\!23}{717044643038151}a^{4}+\frac{22\!\cdots\!03}{239014881012717}a^{3}+\frac{21545577702926}{4509714736089}a^{2}+\frac{981801060615920}{717044643038151}a+\frac{649409067897197}{717044643038151}$, $\frac{3473993142259}{79671627004239}a^{26}-\frac{118718592005480}{717044643038151}a^{25}+\frac{407142667725049}{717044643038151}a^{24}-\frac{218531696141032}{717044643038151}a^{23}-\frac{74536923822128}{717044643038151}a^{22}+\frac{10\!\cdots\!60}{717044643038151}a^{21}+\frac{244784938946185}{717044643038151}a^{20}-\frac{1818055740997}{14059698883101}a^{19}+\frac{12\!\cdots\!82}{239014881012717}a^{18}-\frac{968865324610}{8639092084797}a^{17}+\frac{12280955891558}{4686566294367}a^{16}+\frac{35\!\cdots\!28}{717044643038151}a^{15}-\frac{30\!\cdots\!52}{717044643038151}a^{14}-\frac{45\!\cdots\!11}{717044643038151}a^{13}-\frac{968627551598456}{717044643038151}a^{12}-\frac{103472904813044}{8852403000471}a^{11}+\frac{7346887250473}{10391951348379}a^{10}+\frac{74\!\cdots\!88}{717044643038151}a^{9}+\frac{41\!\cdots\!27}{239014881012717}a^{8}+\frac{24\!\cdots\!26}{717044643038151}a^{7}+\frac{27\!\cdots\!87}{717044643038151}a^{6}+\frac{24\!\cdots\!92}{717044643038151}a^{5}+\frac{20\!\cdots\!22}{717044643038151}a^{4}+\frac{33\!\cdots\!60}{239014881012717}a^{3}+\frac{17\!\cdots\!36}{239014881012717}a^{2}+\frac{2613870458575}{26557209001413}a+\frac{659182473179159}{717044643038151}$, $\frac{12773761978436}{717044643038151}a^{26}-\frac{18407744452598}{239014881012717}a^{25}+\frac{189962787577439}{717044643038151}a^{24}-\frac{162559132947572}{717044643038151}a^{23}-\frac{12664463004296}{239014881012717}a^{22}+\frac{59889537271316}{79671627004239}a^{21}-\frac{180391132715711}{717044643038151}a^{20}-\frac{4218393746669}{14059698883101}a^{19}+\frac{18\!\cdots\!60}{717044643038151}a^{18}-\frac{10217183140927}{8639092084797}a^{17}+\frac{23222249680712}{42179096649303}a^{16}+\frac{16\!\cdots\!60}{717044643038151}a^{15}-\frac{235925290235653}{79671627004239}a^{14}-\frac{174057930475997}{79671627004239}a^{13}+\frac{11\!\cdots\!16}{717044643038151}a^{12}-\frac{11\!\cdots\!94}{239014881012717}a^{11}+\frac{943230560978504}{717044643038151}a^{10}+\frac{36\!\cdots\!82}{717044643038151}a^{9}+\frac{36\!\cdots\!11}{717044643038151}a^{8}+\frac{64\!\cdots\!43}{717044643038151}a^{7}+\frac{24\!\cdots\!75}{239014881012717}a^{6}+\frac{24505383160772}{3463983782793}a^{5}+\frac{33\!\cdots\!08}{717044643038151}a^{4}+\frac{6504389521735}{8852403000471}a^{3}-\frac{41551730698337}{239014881012717}a^{2}-\frac{12\!\cdots\!27}{717044643038151}a+\frac{553633620534470}{717044643038151}$, $\frac{33754202481797}{717044643038151}a^{26}-\frac{135856146805507}{717044643038151}a^{25}+\frac{470892133363393}{717044643038151}a^{24}-\frac{327086088411295}{717044643038151}a^{23}-\frac{69458806810711}{717044643038151}a^{22}+\frac{13\!\cdots\!81}{717044643038151}a^{21}-\frac{47932935559955}{239014881012717}a^{20}-\frac{4270694984647}{14059698883101}a^{19}+\frac{47\!\cdots\!88}{717044643038151}a^{18}-\frac{4017895688042}{2879697361599}a^{17}+\frac{113849577889949}{42179096649303}a^{16}+\frac{52\!\cdots\!47}{717044643038151}a^{15}-\frac{44\!\cdots\!09}{717044643038151}a^{14}-\frac{38\!\cdots\!08}{717044643038151}a^{13}+\frac{496613557167599}{239014881012717}a^{12}-\frac{38\!\cdots\!61}{239014881012717}a^{11}+\frac{56447382256382}{717044643038151}a^{10}+\frac{28\!\cdots\!40}{239014881012717}a^{9}+\frac{93\!\cdots\!41}{717044643038151}a^{8}+\frac{24\!\cdots\!85}{717044643038151}a^{7}+\frac{29\!\cdots\!33}{717044643038151}a^{6}+\frac{26\!\cdots\!31}{717044643038151}a^{5}+\frac{320800637989409}{8852403000471}a^{4}+\frac{48\!\cdots\!57}{239014881012717}a^{3}+\frac{27\!\cdots\!63}{239014881012717}a^{2}+\frac{16\!\cdots\!82}{717044643038151}a+\frac{179990203623193}{79671627004239}$, $\frac{137267459419}{3366406774827}a^{26}-\frac{552099110215}{3366406774827}a^{25}+\frac{1893619331366}{3366406774827}a^{24}-\frac{1240320336880}{3366406774827}a^{23}-\frac{596147137343}{3366406774827}a^{22}+\frac{5699633956664}{3366406774827}a^{21}-\frac{730096946522}{3366406774827}a^{20}-\frac{4326724870}{8609735997}a^{19}+\frac{19992155083762}{3366406774827}a^{18}-\frac{52633020284}{40559117769}a^{17}+\frac{264178388006}{198023927931}a^{16}+\frac{23271123536996}{3366406774827}a^{15}-\frac{20721660998120}{3366406774827}a^{14}-\frac{19863694308415}{3366406774827}a^{13}+\frac{10695919229524}{3366406774827}a^{12}-\frac{44907471811081}{3366406774827}a^{11}-\frac{1912886487548}{3366406774827}a^{10}+\frac{45699782924840}{3366406774827}a^{9}+\frac{37781755385188}{3366406774827}a^{8}+\frac{94972502622050}{3366406774827}a^{7}+\frac{114574315168564}{3366406774827}a^{6}+\frac{88622580200264}{3366406774827}a^{5}+\frac{84944761321849}{3366406774827}a^{4}+\frac{41582366616944}{3366406774827}a^{3}+\frac{1758492206933}{374045197203}a^{2}+\frac{632828920715}{1122135591609}a+\frac{3344363716976}{3366406774827}$, $\frac{473204178482}{717044643038151}a^{26}+\frac{2575092220771}{717044643038151}a^{25}-\frac{2993320791814}{239014881012717}a^{24}+\frac{15937247940551}{239014881012717}a^{23}-\frac{12057764543150}{717044643038151}a^{22}-\frac{8568186042695}{717044643038151}a^{21}+\frac{141469231242236}{717044643038151}a^{20}+\frac{464278894609}{4686566294367}a^{19}+\frac{47569004176439}{717044643038151}a^{18}+\frac{6302136317521}{8639092084797}a^{17}+\frac{8928240876140}{42179096649303}a^{16}+\frac{98868832812550}{239014881012717}a^{15}+\frac{444005377180813}{717044643038151}a^{14}-\frac{486247301412122}{717044643038151}a^{13}-\frac{834197317164805}{717044643038151}a^{12}-\frac{142491504303803}{239014881012717}a^{11}-\frac{13\!\cdots\!60}{717044643038151}a^{10}+\frac{212827623712076}{717044643038151}a^{9}+\frac{16\!\cdots\!67}{717044643038151}a^{8}+\frac{90524397632155}{26557209001413}a^{7}+\frac{45\!\cdots\!41}{717044643038151}a^{6}+\frac{51\!\cdots\!37}{717044643038151}a^{5}+\frac{33\!\cdots\!07}{717044643038151}a^{4}+\frac{11\!\cdots\!46}{239014881012717}a^{3}+\frac{395061446602948}{239014881012717}a^{2}-\frac{143825746713899}{717044643038151}a-\frac{104332904770739}{717044643038151}$, $\frac{12145560773719}{239014881012717}a^{26}-\frac{48977926472491}{239014881012717}a^{25}+\frac{1091757573829}{1503238245363}a^{24}-\frac{45152227765175}{79671627004239}a^{23}+\frac{33892196896631}{239014881012717}a^{22}+\frac{421786573875881}{239014881012717}a^{21}-\frac{47402406423506}{239014881012717}a^{20}+\frac{1396112590400}{4686566294367}a^{19}+\frac{16\!\cdots\!53}{239014881012717}a^{18}-\frac{4276793518204}{2879697361599}a^{17}+\frac{70927341224851}{14059698883101}a^{16}+\frac{529714639298069}{79671627004239}a^{15}-\frac{13\!\cdots\!67}{239014881012717}a^{14}-\frac{12155968404616}{3274176452229}a^{13}-\frac{204078586735154}{239014881012717}a^{12}-\frac{464647138696888}{26557209001413}a^{11}+\frac{46089978139306}{239014881012717}a^{10}+\frac{19\!\cdots\!10}{239014881012717}a^{9}+\frac{37\!\cdots\!00}{239014881012717}a^{8}+\frac{10\!\cdots\!46}{26557209001413}a^{7}+\frac{11\!\cdots\!14}{239014881012717}a^{6}+\frac{12\!\cdots\!67}{239014881012717}a^{5}+\frac{11\!\cdots\!23}{239014881012717}a^{4}+\frac{24\!\cdots\!11}{79671627004239}a^{3}+\frac{15\!\cdots\!00}{79671627004239}a^{2}+\frac{14\!\cdots\!15}{239014881012717}a+\frac{10\!\cdots\!18}{239014881012717}$
| 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: | \( 5105816638.575763 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 5105816638.575763 \cdot 1}{2\cdot\sqrt{14905779350658261917360296447434047234191}}\cr\approx \mathstrut & 0.994777969725537 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
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 54 |
| The 15 conjugacy class representatives for $D_{27}$ |
| Character table for $D_{27}$ |
Intermediate fields
| 3.1.1231.1, 9.1.2296318960321.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $27$ | ${\href{/padicField/3.2.0.1}{2} }^{13}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.9.0.1}{9} }^{3}$ | $27$ | ${\href{/padicField/11.3.0.1}{3} }^{9}$ | $27$ | ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $27$ | ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $27$ | $27$ | $27$ | ${\href{/padicField/41.9.0.1}{9} }^{3}$ | $27$ | ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(1231\)
| $\Q_{1231}$ | $x$ | $1$ | $1$ | $0$ | 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
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.