Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681)
gp: K = bnfinit(y^28 - 5*y^27 + 47*y^26 - 152*y^25 + 1049*y^24 - 3001*y^23 + 14492*y^22 - 30908*y^21 + 115818*y^20 - 204097*y^19 + 649914*y^18 - 933216*y^17 + 2508376*y^16 - 2978786*y^15 + 7071411*y^14 - 6804920*y^13 + 13577493*y^12 - 10099944*y^11 + 17701828*y^10 - 9450995*y^9 + 11808392*y^8 - 1974799*y^7 + 3396106*y^6 - 287718*y^5 + 726277*y^4 + 16080*y^3 + 60245*y^2 - 6888*y + 1681, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681)
\( x^{28} - 5 x^{27} + 47 x^{26} - 152 x^{25} + 1049 x^{24} - 3001 x^{23} + 14492 x^{22} - 30908 x^{21} + \cdots + 1681 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3654513548980364725264702136425345220781793212890625\)
\(\medspace = 3^{14}\cdot 5^{14}\cdot 29^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.43\) | 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: | $3^{1/2}5^{1/2}29^{6/7}\approx 69.4271963583542$ | ||
| Ramified primes: |
\(3\), \(5\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(435=3\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(194,·)$, $\chi_{435}(326,·)$, $\chi_{435}(199,·)$, $\chi_{435}(136,·)$, $\chi_{435}(74,·)$, $\chi_{435}(139,·)$, $\chi_{435}(94,·)$, $\chi_{435}(16,·)$, $\chi_{435}(401,·)$, $\chi_{435}(146,·)$, $\chi_{435}(344,·)$, $\chi_{435}(281,·)$, $\chi_{435}(239,·)$, $\chi_{435}(284,·)$, $\chi_{435}(349,·)$, $\chi_{435}(286,·)$, $\chi_{435}(161,·)$, $\chi_{435}(226,·)$, $\chi_{435}(169,·)$, $\chi_{435}(364,·)$, $\chi_{435}(431,·)$, $\chi_{435}(49,·)$, $\chi_{435}(371,·)$, $\chi_{435}(181,·)$, $\chi_{435}(314,·)$, $\chi_{435}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{41}a^{23}-\frac{10}{41}a^{22}-\frac{13}{41}a^{21}+\frac{3}{41}a^{20}+\frac{9}{41}a^{19}+\frac{4}{41}a^{18}-\frac{9}{41}a^{17}+\frac{6}{41}a^{16}-\frac{11}{41}a^{15}+\frac{4}{41}a^{14}-\frac{10}{41}a^{13}-\frac{9}{41}a^{12}-\frac{12}{41}a^{11}-\frac{10}{41}a^{10}-\frac{10}{41}a^{9}-\frac{1}{41}a^{8}+\frac{12}{41}a^{7}-\frac{14}{41}a^{6}+\frac{10}{41}a^{5}-\frac{5}{41}a^{4}-\frac{17}{41}a^{3}+\frac{16}{41}a^{2}+\frac{20}{41}a$, $\frac{1}{41123}a^{24}+\frac{381}{41123}a^{23}-\frac{18929}{41123}a^{22}+\frac{250}{41123}a^{21}-\frac{5460}{41123}a^{20}-\frac{17838}{41123}a^{19}+\frac{6147}{41123}a^{18}+\frac{10550}{41123}a^{17}-\frac{9063}{41123}a^{16}-\frac{10119}{41123}a^{15}+\frac{2743}{41123}a^{14}+\frac{2518}{41123}a^{13}-\frac{17430}{41123}a^{12}+\frac{9443}{41123}a^{11}+\frac{6904}{41123}a^{10}-\frac{915}{2419}a^{9}+\frac{18399}{41123}a^{8}-\frac{19389}{41123}a^{7}+\frac{15569}{41123}a^{6}+\frac{1568}{41123}a^{5}+\frac{10328}{41123}a^{4}-\frac{2777}{41123}a^{3}+\frac{14886}{41123}a^{2}+\frac{234}{697}a+\frac{122}{1003}$, $\frac{1}{41123}a^{25}+\frac{402}{41123}a^{23}+\frac{922}{2419}a^{22}-\frac{18464}{41123}a^{21}+\frac{6272}{41123}a^{20}+\frac{17130}{41123}a^{19}+\frac{12554}{41123}a^{18}+\frac{1441}{41123}a^{17}-\frac{11448}{41123}a^{16}-\frac{440}{2419}a^{15}-\frac{14490}{41123}a^{14}+\frac{10164}{41123}a^{13}-\frac{11653}{41123}a^{12}-\frac{13178}{41123}a^{11}-\frac{14107}{41123}a^{10}-\frac{17981}{41123}a^{9}+\frac{2625}{41123}a^{8}+\frac{638}{41123}a^{7}-\frac{8509}{41123}a^{6}-\frac{11358}{41123}a^{5}+\frac{10063}{41123}a^{4}+\frac{3725}{41123}a^{3}+\frac{17214}{41123}a^{2}+\frac{8660}{41123}a-\frac{344}{1003}$, $\frac{1}{11639412797}a^{26}+\frac{90482}{11639412797}a^{25}+\frac{113690}{11639412797}a^{24}+\frac{44768126}{11639412797}a^{23}+\frac{1385250353}{11639412797}a^{22}+\frac{31643050}{684671341}a^{21}+\frac{3239853883}{11639412797}a^{20}-\frac{2829787740}{11639412797}a^{19}+\frac{2110761710}{11639412797}a^{18}-\frac{3933425257}{11639412797}a^{17}-\frac{1900308217}{11639412797}a^{16}-\frac{2058964356}{11639412797}a^{15}+\frac{2505722651}{11639412797}a^{14}-\frac{4636511259}{11639412797}a^{13}+\frac{1514246237}{11639412797}a^{12}-\frac{1104984636}{11639412797}a^{11}+\frac{4887975349}{11639412797}a^{10}-\frac{28103078}{11639412797}a^{9}+\frac{3754209872}{11639412797}a^{8}+\frac{4805896816}{11639412797}a^{7}-\frac{3642556465}{11639412797}a^{6}-\frac{295968095}{684671341}a^{5}-\frac{5346240212}{11639412797}a^{4}+\frac{4597992423}{11639412797}a^{3}-\frac{2457595563}{11639412797}a^{2}+\frac{523477101}{11639412797}a+\frac{57322540}{283888117}$, $\frac{1}{83\!\cdots\!89}a^{27}-\frac{34\!\cdots\!72}{83\!\cdots\!89}a^{26}-\frac{51\!\cdots\!48}{83\!\cdots\!89}a^{25}-\frac{42\!\cdots\!63}{20\!\cdots\!29}a^{24}-\frac{79\!\cdots\!31}{83\!\cdots\!89}a^{23}+\frac{32\!\cdots\!44}{83\!\cdots\!89}a^{22}+\frac{66\!\cdots\!91}{14\!\cdots\!71}a^{21}+\frac{33\!\cdots\!63}{83\!\cdots\!89}a^{20}-\frac{14\!\cdots\!81}{83\!\cdots\!89}a^{19}+\frac{30\!\cdots\!93}{83\!\cdots\!89}a^{18}-\frac{29\!\cdots\!83}{83\!\cdots\!89}a^{17}-\frac{27\!\cdots\!55}{83\!\cdots\!89}a^{16}-\frac{37\!\cdots\!91}{83\!\cdots\!89}a^{15}+\frac{31\!\cdots\!94}{83\!\cdots\!89}a^{14}-\frac{13\!\cdots\!19}{83\!\cdots\!89}a^{13}-\frac{25\!\cdots\!93}{83\!\cdots\!89}a^{12}+\frac{32\!\cdots\!04}{83\!\cdots\!89}a^{11}-\frac{56\!\cdots\!19}{83\!\cdots\!89}a^{10}-\frac{77\!\cdots\!79}{49\!\cdots\!17}a^{9}-\frac{31\!\cdots\!28}{83\!\cdots\!89}a^{8}+\frac{10\!\cdots\!25}{83\!\cdots\!89}a^{7}-\frac{39\!\cdots\!42}{83\!\cdots\!89}a^{6}+\frac{58\!\cdots\!94}{83\!\cdots\!89}a^{5}-\frac{15\!\cdots\!64}{83\!\cdots\!89}a^{4}-\frac{12\!\cdots\!90}{83\!\cdots\!89}a^{3}-\frac{89\!\cdots\!23}{83\!\cdots\!89}a^{2}-\frac{17\!\cdots\!36}{83\!\cdots\!89}a+\frac{24\!\cdots\!42}{20\!\cdots\!29}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{4}\times C_{4}\times C_{452}$, which has order $7232$ (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: |
\( -\frac{2944727907733851313844166695580586247643057933080334230251436957902}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{27} + \frac{14271014244497695456942016188160138110021276736171669922386030024919}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{26} - \frac{135978308529120404887408203190974516593584323418471359924636877640401}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{25} + \frac{425536117233239771730783713724313289079093355274780523470454240434385}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{24} - \frac{3012765312874440015416700186619905078007069272443493500857940345723134}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{23} + \frac{8338710311987213613917630675882671769621993807103639362273825095004280}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{22} - \frac{41151506115984017970877109756340043753316327168979014317153599803844452}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{21} + \frac{83992996979847503273447678914499035181237397705166314022201055558762189}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{20} - \frac{324804460454186693343653891617518807381958859045684362397963682147893897}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{19} + \frac{543906411037652109715607979798839794708718513271825997000468514034362298}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{18} - \frac{1803644658886089076090483947500570091181814785162840151605644744386981660}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{17} + \frac{2423517944561405299035290118522252965963277892766509164949168857041016231}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{16} - \frac{6865459408811625500018690377645442917201354846624570074057046078496714509}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{15} + \frac{7500461609322319923928028421126120383311543812442659654129090193085504130}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{14} - \frac{19101125488377777174193915609101763151345060775211909510523620519838738384}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{13} + \frac{16415555309730099806856068763615671098455285285566516110596563061975971242}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{12} - \frac{35866484987036329775270117766943400957791820891405149091575556794667568989}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{11} + \frac{22663749541833966543233969669138216258227987804229724710839773761551577003}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{10} - \frac{45628561583377037475183550196108885274738643778225967079513540388962845611}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{9} + \frac{18511702236365329041937840439579219312109272307870612947743476562606785588}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{8} - \frac{28075173717772617821653071636187454557705495418108951279776174741925465456}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{7} - \frac{649273056571248661482193461080844355860706103452234065796108746994178172}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{6} - \frac{7680157428893372518999554886997769133836112457788756459639815166016630161}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{5} - \frac{753277665347945426832902039256201334326984014324432946402408520415558682}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{4} - \frac{1653599691071112203219362968187998293682957001362444431775939969844656890}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{3} - \frac{452085557506791951951668125272014802832983106016857957449153709218711097}{29429157677826837274001452072245634479959326651655814586196951567585917} a^{2} - \frac{112584997216705782453532935320254040117798848782008067201574713607840477}{29429157677826837274001452072245634479959326651655814586196951567585917} a + \frac{260942716696870531257097801519003430910692667546008670278151888027091}{717784333605532616439059806640137426340471381747702794785291501648437} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{35\!\cdots\!71}{49\!\cdots\!17}a^{27}-\frac{16\!\cdots\!54}{49\!\cdots\!17}a^{26}+\frac{16\!\cdots\!04}{49\!\cdots\!17}a^{25}-\frac{49\!\cdots\!16}{49\!\cdots\!17}a^{24}+\frac{35\!\cdots\!65}{49\!\cdots\!17}a^{23}-\frac{95\!\cdots\!85}{49\!\cdots\!17}a^{22}+\frac{48\!\cdots\!06}{49\!\cdots\!17}a^{21}-\frac{94\!\cdots\!67}{49\!\cdots\!17}a^{20}+\frac{37\!\cdots\!73}{49\!\cdots\!17}a^{19}-\frac{60\!\cdots\!42}{49\!\cdots\!17}a^{18}+\frac{20\!\cdots\!94}{49\!\cdots\!17}a^{17}-\frac{26\!\cdots\!91}{49\!\cdots\!17}a^{16}+\frac{79\!\cdots\!67}{49\!\cdots\!17}a^{15}-\frac{79\!\cdots\!39}{49\!\cdots\!17}a^{14}+\frac{22\!\cdots\!25}{49\!\cdots\!17}a^{13}-\frac{16\!\cdots\!07}{49\!\cdots\!17}a^{12}+\frac{41\!\cdots\!79}{49\!\cdots\!17}a^{11}-\frac{21\!\cdots\!84}{49\!\cdots\!17}a^{10}+\frac{12\!\cdots\!84}{11\!\cdots\!37}a^{9}-\frac{15\!\cdots\!72}{49\!\cdots\!17}a^{8}+\frac{32\!\cdots\!42}{49\!\cdots\!17}a^{7}+\frac{46\!\cdots\!18}{49\!\cdots\!17}a^{6}+\frac{10\!\cdots\!91}{49\!\cdots\!17}a^{5}+\frac{19\!\cdots\!75}{49\!\cdots\!17}a^{4}+\frac{24\!\cdots\!16}{49\!\cdots\!17}a^{3}+\frac{75\!\cdots\!04}{49\!\cdots\!17}a^{2}+\frac{37\!\cdots\!21}{49\!\cdots\!17}a+\frac{73\!\cdots\!39}{11\!\cdots\!37}$, $\frac{22\!\cdots\!19}{95\!\cdots\!57}a^{27}-\frac{12\!\cdots\!98}{95\!\cdots\!57}a^{26}+\frac{11\!\cdots\!52}{95\!\cdots\!57}a^{25}-\frac{23\!\cdots\!48}{55\!\cdots\!21}a^{24}+\frac{25\!\cdots\!55}{95\!\cdots\!57}a^{23}-\frac{81\!\cdots\!86}{95\!\cdots\!57}a^{22}+\frac{36\!\cdots\!08}{95\!\cdots\!57}a^{21}-\frac{88\!\cdots\!06}{95\!\cdots\!57}a^{20}+\frac{30\!\cdots\!69}{95\!\cdots\!57}a^{19}-\frac{61\!\cdots\!73}{95\!\cdots\!57}a^{18}+\frac{17\!\cdots\!37}{95\!\cdots\!57}a^{17}-\frac{17\!\cdots\!38}{55\!\cdots\!21}a^{16}+\frac{69\!\cdots\!69}{95\!\cdots\!57}a^{15}-\frac{10\!\cdots\!80}{95\!\cdots\!57}a^{14}+\frac{20\!\cdots\!19}{95\!\cdots\!57}a^{13}-\frac{14\!\cdots\!72}{55\!\cdots\!21}a^{12}+\frac{23\!\cdots\!34}{55\!\cdots\!21}a^{11}-\frac{40\!\cdots\!49}{95\!\cdots\!57}a^{10}+\frac{54\!\cdots\!94}{95\!\cdots\!57}a^{9}-\frac{43\!\cdots\!02}{95\!\cdots\!57}a^{8}+\frac{39\!\cdots\!36}{95\!\cdots\!57}a^{7}-\frac{18\!\cdots\!22}{95\!\cdots\!57}a^{6}+\frac{98\!\cdots\!23}{95\!\cdots\!57}a^{5}-\frac{32\!\cdots\!18}{95\!\cdots\!57}a^{4}+\frac{14\!\cdots\!29}{95\!\cdots\!57}a^{3}-\frac{41\!\cdots\!80}{95\!\cdots\!57}a^{2}+\frac{14\!\cdots\!39}{23\!\cdots\!77}a-\frac{53\!\cdots\!73}{23\!\cdots\!77}$, $\frac{18\!\cdots\!48}{95\!\cdots\!57}a^{27}-\frac{38\!\cdots\!78}{95\!\cdots\!57}a^{26}+\frac{58\!\cdots\!77}{95\!\cdots\!57}a^{25}-\frac{84\!\cdots\!28}{55\!\cdots\!21}a^{24}+\frac{10\!\cdots\!63}{95\!\cdots\!57}a^{23}+\frac{49\!\cdots\!83}{95\!\cdots\!57}a^{22}+\frac{82\!\cdots\!78}{95\!\cdots\!57}a^{21}+\frac{28\!\cdots\!42}{95\!\cdots\!57}a^{20}+\frac{14\!\cdots\!23}{95\!\cdots\!57}a^{19}+\frac{31\!\cdots\!20}{95\!\cdots\!57}a^{18}-\frac{15\!\cdots\!14}{95\!\cdots\!57}a^{17}+\frac{13\!\cdots\!26}{55\!\cdots\!21}a^{16}-\frac{18\!\cdots\!11}{95\!\cdots\!57}a^{15}+\frac{99\!\cdots\!76}{95\!\cdots\!57}a^{14}-\frac{81\!\cdots\!74}{95\!\cdots\!57}a^{13}+\frac{18\!\cdots\!02}{55\!\cdots\!21}a^{12}-\frac{14\!\cdots\!51}{55\!\cdots\!21}a^{11}+\frac{65\!\cdots\!42}{95\!\cdots\!57}a^{10}-\frac{45\!\cdots\!51}{95\!\cdots\!57}a^{9}+\frac{90\!\cdots\!01}{95\!\cdots\!57}a^{8}-\frac{56\!\cdots\!14}{95\!\cdots\!57}a^{7}+\frac{64\!\cdots\!43}{95\!\cdots\!57}a^{6}-\frac{13\!\cdots\!61}{95\!\cdots\!57}a^{5}+\frac{70\!\cdots\!53}{95\!\cdots\!57}a^{4}+\frac{14\!\cdots\!45}{95\!\cdots\!57}a^{3}+\frac{79\!\cdots\!21}{95\!\cdots\!57}a^{2}-\frac{26\!\cdots\!92}{23\!\cdots\!77}a-\frac{56\!\cdots\!12}{23\!\cdots\!77}$, $\frac{45\!\cdots\!14}{83\!\cdots\!89}a^{27}-\frac{22\!\cdots\!52}{83\!\cdots\!89}a^{26}+\frac{21\!\cdots\!42}{83\!\cdots\!89}a^{25}-\frac{69\!\cdots\!57}{83\!\cdots\!89}a^{24}+\frac{47\!\cdots\!11}{83\!\cdots\!89}a^{23}-\frac{13\!\cdots\!17}{83\!\cdots\!89}a^{22}+\frac{66\!\cdots\!55}{83\!\cdots\!89}a^{21}-\frac{13\!\cdots\!64}{83\!\cdots\!89}a^{20}+\frac{52\!\cdots\!32}{83\!\cdots\!89}a^{19}-\frac{92\!\cdots\!93}{83\!\cdots\!89}a^{18}+\frac{29\!\cdots\!90}{83\!\cdots\!89}a^{17}-\frac{42\!\cdots\!86}{83\!\cdots\!89}a^{16}+\frac{11\!\cdots\!40}{83\!\cdots\!89}a^{15}-\frac{13\!\cdots\!56}{83\!\cdots\!89}a^{14}+\frac{32\!\cdots\!46}{83\!\cdots\!89}a^{13}-\frac{30\!\cdots\!88}{83\!\cdots\!89}a^{12}+\frac{61\!\cdots\!78}{83\!\cdots\!89}a^{11}-\frac{45\!\cdots\!53}{83\!\cdots\!89}a^{10}+\frac{81\!\cdots\!98}{83\!\cdots\!89}a^{9}-\frac{41\!\cdots\!72}{83\!\cdots\!89}a^{8}+\frac{54\!\cdots\!53}{83\!\cdots\!89}a^{7}-\frac{86\!\cdots\!00}{83\!\cdots\!89}a^{6}+\frac{16\!\cdots\!52}{83\!\cdots\!89}a^{5}-\frac{12\!\cdots\!91}{83\!\cdots\!89}a^{4}+\frac{33\!\cdots\!71}{83\!\cdots\!89}a^{3}+\frac{38\!\cdots\!07}{23\!\cdots\!61}a^{2}+\frac{27\!\cdots\!97}{83\!\cdots\!89}a-\frac{76\!\cdots\!28}{20\!\cdots\!29}$, $\frac{69\!\cdots\!48}{49\!\cdots\!17}a^{27}-\frac{34\!\cdots\!90}{49\!\cdots\!17}a^{26}+\frac{32\!\cdots\!55}{49\!\cdots\!17}a^{25}-\frac{10\!\cdots\!33}{49\!\cdots\!17}a^{24}+\frac{72\!\cdots\!47}{49\!\cdots\!17}a^{23}-\frac{20\!\cdots\!98}{49\!\cdots\!17}a^{22}+\frac{10\!\cdots\!70}{49\!\cdots\!17}a^{21}-\frac{21\!\cdots\!79}{49\!\cdots\!17}a^{20}+\frac{80\!\cdots\!57}{49\!\cdots\!17}a^{19}-\frac{14\!\cdots\!96}{49\!\cdots\!17}a^{18}+\frac{44\!\cdots\!51}{49\!\cdots\!17}a^{17}-\frac{64\!\cdots\!23}{49\!\cdots\!17}a^{16}+\frac{42\!\cdots\!93}{11\!\cdots\!37}a^{15}-\frac{20\!\cdots\!55}{49\!\cdots\!17}a^{14}+\frac{48\!\cdots\!44}{49\!\cdots\!17}a^{13}-\frac{46\!\cdots\!30}{49\!\cdots\!17}a^{12}+\frac{92\!\cdots\!52}{49\!\cdots\!17}a^{11}-\frac{68\!\cdots\!97}{49\!\cdots\!17}a^{10}+\frac{12\!\cdots\!41}{49\!\cdots\!17}a^{9}-\frac{62\!\cdots\!47}{49\!\cdots\!17}a^{8}+\frac{79\!\cdots\!16}{49\!\cdots\!17}a^{7}-\frac{11\!\cdots\!22}{49\!\cdots\!17}a^{6}+\frac{22\!\cdots\!05}{49\!\cdots\!17}a^{5}-\frac{14\!\cdots\!76}{49\!\cdots\!17}a^{4}+\frac{48\!\cdots\!00}{49\!\cdots\!17}a^{3}+\frac{83\!\cdots\!67}{14\!\cdots\!33}a^{2}+\frac{39\!\cdots\!41}{49\!\cdots\!17}a-\frac{18\!\cdots\!59}{20\!\cdots\!43}$, $\frac{26\!\cdots\!19}{83\!\cdots\!89}a^{27}-\frac{13\!\cdots\!87}{83\!\cdots\!89}a^{26}+\frac{12\!\cdots\!43}{83\!\cdots\!89}a^{25}-\frac{41\!\cdots\!28}{83\!\cdots\!89}a^{24}+\frac{28\!\cdots\!04}{83\!\cdots\!89}a^{23}-\frac{81\!\cdots\!48}{83\!\cdots\!89}a^{22}+\frac{38\!\cdots\!27}{83\!\cdots\!89}a^{21}-\frac{85\!\cdots\!52}{83\!\cdots\!89}a^{20}+\frac{31\!\cdots\!89}{83\!\cdots\!89}a^{19}-\frac{56\!\cdots\!83}{83\!\cdots\!89}a^{18}+\frac{17\!\cdots\!87}{83\!\cdots\!89}a^{17}-\frac{25\!\cdots\!36}{83\!\cdots\!89}a^{16}+\frac{67\!\cdots\!74}{83\!\cdots\!89}a^{15}-\frac{83\!\cdots\!62}{83\!\cdots\!89}a^{14}+\frac{18\!\cdots\!88}{83\!\cdots\!89}a^{13}-\frac{46\!\cdots\!09}{20\!\cdots\!29}a^{12}+\frac{35\!\cdots\!47}{83\!\cdots\!89}a^{11}-\frac{28\!\cdots\!63}{83\!\cdots\!89}a^{10}+\frac{45\!\cdots\!96}{83\!\cdots\!89}a^{9}-\frac{25\!\cdots\!78}{83\!\cdots\!89}a^{8}+\frac{28\!\cdots\!54}{83\!\cdots\!89}a^{7}-\frac{38\!\cdots\!22}{83\!\cdots\!89}a^{6}+\frac{55\!\cdots\!50}{83\!\cdots\!89}a^{5}+\frac{53\!\cdots\!39}{83\!\cdots\!89}a^{4}+\frac{10\!\cdots\!15}{83\!\cdots\!89}a^{3}+\frac{18\!\cdots\!91}{83\!\cdots\!89}a^{2}-\frac{12\!\cdots\!82}{83\!\cdots\!89}a+\frac{21\!\cdots\!66}{20\!\cdots\!29}$, $\frac{27\!\cdots\!51}{95\!\cdots\!57}a^{27}-\frac{14\!\cdots\!39}{95\!\cdots\!57}a^{26}+\frac{13\!\cdots\!32}{95\!\cdots\!57}a^{25}-\frac{26\!\cdots\!68}{55\!\cdots\!21}a^{24}+\frac{29\!\cdots\!96}{95\!\cdots\!57}a^{23}-\frac{88\!\cdots\!26}{95\!\cdots\!57}a^{22}+\frac{41\!\cdots\!50}{95\!\cdots\!57}a^{21}-\frac{94\!\cdots\!21}{95\!\cdots\!57}a^{20}+\frac{33\!\cdots\!26}{95\!\cdots\!57}a^{19}-\frac{63\!\cdots\!18}{95\!\cdots\!57}a^{18}+\frac{18\!\cdots\!96}{95\!\cdots\!57}a^{17}-\frac{17\!\cdots\!65}{55\!\cdots\!21}a^{16}+\frac{74\!\cdots\!77}{95\!\cdots\!57}a^{15}-\frac{98\!\cdots\!55}{95\!\cdots\!57}a^{14}+\frac{21\!\cdots\!93}{95\!\cdots\!57}a^{13}-\frac{13\!\cdots\!13}{55\!\cdots\!21}a^{12}+\frac{59\!\cdots\!15}{13\!\cdots\!81}a^{11}-\frac{36\!\cdots\!43}{95\!\cdots\!57}a^{10}+\frac{54\!\cdots\!53}{95\!\cdots\!57}a^{9}-\frac{37\!\cdots\!99}{95\!\cdots\!57}a^{8}+\frac{37\!\cdots\!06}{95\!\cdots\!57}a^{7}-\frac{13\!\cdots\!32}{95\!\cdots\!57}a^{6}+\frac{92\!\cdots\!65}{95\!\cdots\!57}a^{5}-\frac{27\!\cdots\!35}{95\!\cdots\!57}a^{4}+\frac{15\!\cdots\!88}{95\!\cdots\!57}a^{3}-\frac{36\!\cdots\!15}{95\!\cdots\!57}a^{2}+\frac{13\!\cdots\!01}{23\!\cdots\!77}a-\frac{86\!\cdots\!16}{23\!\cdots\!77}$, $\frac{74\!\cdots\!48}{83\!\cdots\!89}a^{27}-\frac{33\!\cdots\!89}{83\!\cdots\!89}a^{26}+\frac{33\!\cdots\!38}{83\!\cdots\!89}a^{25}-\frac{23\!\cdots\!71}{20\!\cdots\!29}a^{24}+\frac{71\!\cdots\!66}{83\!\cdots\!89}a^{23}-\frac{10\!\cdots\!24}{49\!\cdots\!17}a^{22}+\frac{95\!\cdots\!52}{83\!\cdots\!89}a^{21}-\frac{17\!\cdots\!39}{83\!\cdots\!89}a^{20}+\frac{73\!\cdots\!38}{83\!\cdots\!89}a^{19}-\frac{10\!\cdots\!68}{83\!\cdots\!89}a^{18}+\frac{39\!\cdots\!54}{83\!\cdots\!89}a^{17}-\frac{43\!\cdots\!79}{83\!\cdots\!89}a^{16}+\frac{84\!\cdots\!98}{49\!\cdots\!17}a^{15}-\frac{12\!\cdots\!32}{83\!\cdots\!89}a^{14}+\frac{38\!\cdots\!25}{83\!\cdots\!89}a^{13}-\frac{21\!\cdots\!67}{83\!\cdots\!89}a^{12}+\frac{66\!\cdots\!58}{83\!\cdots\!89}a^{11}-\frac{17\!\cdots\!56}{83\!\cdots\!89}a^{10}+\frac{74\!\cdots\!10}{83\!\cdots\!89}a^{9}+\frac{69\!\cdots\!62}{83\!\cdots\!89}a^{8}+\frac{27\!\cdots\!37}{83\!\cdots\!89}a^{7}+\frac{41\!\cdots\!80}{83\!\cdots\!89}a^{6}+\frac{57\!\cdots\!49}{83\!\cdots\!89}a^{5}+\frac{11\!\cdots\!23}{83\!\cdots\!89}a^{4}+\frac{94\!\cdots\!53}{83\!\cdots\!89}a^{3}+\frac{66\!\cdots\!88}{23\!\cdots\!61}a^{2}-\frac{13\!\cdots\!88}{83\!\cdots\!89}a+\frac{83\!\cdots\!40}{20\!\cdots\!29}$, $\frac{76\!\cdots\!80}{83\!\cdots\!89}a^{27}-\frac{36\!\cdots\!00}{83\!\cdots\!89}a^{26}+\frac{35\!\cdots\!36}{83\!\cdots\!89}a^{25}-\frac{26\!\cdots\!47}{20\!\cdots\!29}a^{24}+\frac{78\!\cdots\!17}{83\!\cdots\!89}a^{23}-\frac{36\!\cdots\!94}{14\!\cdots\!71}a^{22}+\frac{10\!\cdots\!02}{83\!\cdots\!89}a^{21}-\frac{21\!\cdots\!57}{83\!\cdots\!89}a^{20}+\frac{84\!\cdots\!31}{83\!\cdots\!89}a^{19}-\frac{13\!\cdots\!09}{83\!\cdots\!89}a^{18}+\frac{46\!\cdots\!98}{83\!\cdots\!89}a^{17}-\frac{61\!\cdots\!12}{83\!\cdots\!89}a^{16}+\frac{17\!\cdots\!19}{83\!\cdots\!89}a^{15}-\frac{19\!\cdots\!33}{83\!\cdots\!89}a^{14}+\frac{49\!\cdots\!62}{83\!\cdots\!89}a^{13}-\frac{41\!\cdots\!43}{83\!\cdots\!89}a^{12}+\frac{93\!\cdots\!38}{83\!\cdots\!89}a^{11}-\frac{56\!\cdots\!20}{83\!\cdots\!89}a^{10}+\frac{12\!\cdots\!26}{83\!\cdots\!89}a^{9}-\frac{44\!\cdots\!86}{83\!\cdots\!89}a^{8}+\frac{75\!\cdots\!77}{83\!\cdots\!89}a^{7}+\frac{46\!\cdots\!48}{83\!\cdots\!89}a^{6}+\frac{21\!\cdots\!80}{83\!\cdots\!89}a^{5}+\frac{42\!\cdots\!01}{83\!\cdots\!89}a^{4}+\frac{48\!\cdots\!51}{83\!\cdots\!89}a^{3}+\frac{13\!\cdots\!88}{83\!\cdots\!89}a^{2}+\frac{47\!\cdots\!59}{83\!\cdots\!89}a+\frac{13\!\cdots\!02}{20\!\cdots\!29}$, $\frac{37\!\cdots\!68}{83\!\cdots\!89}a^{27}+\frac{10\!\cdots\!76}{83\!\cdots\!89}a^{26}+\frac{20\!\cdots\!28}{83\!\cdots\!89}a^{25}+\frac{85\!\cdots\!15}{83\!\cdots\!89}a^{24}-\frac{87\!\cdots\!94}{83\!\cdots\!89}a^{23}+\frac{20\!\cdots\!88}{83\!\cdots\!89}a^{22}-\frac{41\!\cdots\!11}{83\!\cdots\!89}a^{21}+\frac{32\!\cdots\!28}{83\!\cdots\!89}a^{20}-\frac{56\!\cdots\!94}{83\!\cdots\!89}a^{19}+\frac{27\!\cdots\!28}{83\!\cdots\!89}a^{18}-\frac{25\!\cdots\!94}{49\!\cdots\!17}a^{17}+\frac{16\!\cdots\!64}{83\!\cdots\!89}a^{16}-\frac{21\!\cdots\!36}{83\!\cdots\!89}a^{15}+\frac{65\!\cdots\!20}{83\!\cdots\!89}a^{14}-\frac{74\!\cdots\!58}{83\!\cdots\!89}a^{13}+\frac{18\!\cdots\!72}{83\!\cdots\!89}a^{12}-\frac{18\!\cdots\!24}{83\!\cdots\!89}a^{11}+\frac{36\!\cdots\!70}{83\!\cdots\!89}a^{10}-\frac{29\!\cdots\!07}{83\!\cdots\!89}a^{9}+\frac{46\!\cdots\!72}{83\!\cdots\!89}a^{8}-\frac{30\!\cdots\!68}{83\!\cdots\!89}a^{7}+\frac{29\!\cdots\!25}{83\!\cdots\!89}a^{6}-\frac{74\!\cdots\!18}{83\!\cdots\!89}a^{5}+\frac{35\!\cdots\!68}{83\!\cdots\!89}a^{4}-\frac{16\!\cdots\!91}{83\!\cdots\!89}a^{3}+\frac{41\!\cdots\!80}{83\!\cdots\!89}a^{2}-\frac{13\!\cdots\!10}{20\!\cdots\!29}a-\frac{44\!\cdots\!25}{11\!\cdots\!37}$, $\frac{77\!\cdots\!87}{83\!\cdots\!89}a^{27}-\frac{23\!\cdots\!41}{49\!\cdots\!17}a^{26}+\frac{37\!\cdots\!46}{83\!\cdots\!89}a^{25}-\frac{12\!\cdots\!25}{83\!\cdots\!89}a^{24}+\frac{83\!\cdots\!17}{83\!\cdots\!89}a^{23}-\frac{24\!\cdots\!25}{83\!\cdots\!89}a^{22}+\frac{11\!\cdots\!38}{83\!\cdots\!89}a^{21}-\frac{26\!\cdots\!90}{83\!\cdots\!89}a^{20}+\frac{93\!\cdots\!11}{83\!\cdots\!89}a^{19}-\frac{17\!\cdots\!52}{83\!\cdots\!89}a^{18}+\frac{52\!\cdots\!30}{83\!\cdots\!89}a^{17}-\frac{81\!\cdots\!23}{83\!\cdots\!89}a^{16}+\frac{20\!\cdots\!66}{83\!\cdots\!89}a^{15}-\frac{26\!\cdots\!75}{83\!\cdots\!89}a^{14}+\frac{57\!\cdots\!41}{83\!\cdots\!89}a^{13}-\frac{61\!\cdots\!35}{83\!\cdots\!89}a^{12}+\frac{64\!\cdots\!98}{49\!\cdots\!17}a^{11}-\frac{93\!\cdots\!22}{83\!\cdots\!89}a^{10}+\frac{13\!\cdots\!63}{83\!\cdots\!89}a^{9}-\frac{89\!\cdots\!48}{83\!\cdots\!89}a^{8}+\frac{87\!\cdots\!54}{83\!\cdots\!89}a^{7}-\frac{18\!\cdots\!86}{83\!\cdots\!89}a^{6}+\frac{13\!\cdots\!98}{83\!\cdots\!89}a^{5}+\frac{18\!\cdots\!04}{83\!\cdots\!89}a^{4}+\frac{20\!\cdots\!62}{83\!\cdots\!89}a^{3}+\frac{15\!\cdots\!13}{83\!\cdots\!89}a^{2}+\frac{18\!\cdots\!49}{83\!\cdots\!89}a+\frac{29\!\cdots\!51}{20\!\cdots\!29}$, $\frac{17\!\cdots\!68}{83\!\cdots\!89}a^{27}-\frac{82\!\cdots\!80}{83\!\cdots\!89}a^{26}+\frac{78\!\cdots\!11}{83\!\cdots\!89}a^{25}-\frac{24\!\cdots\!30}{83\!\cdots\!89}a^{24}+\frac{17\!\cdots\!99}{83\!\cdots\!89}a^{23}-\frac{47\!\cdots\!78}{83\!\cdots\!89}a^{22}+\frac{23\!\cdots\!65}{83\!\cdots\!89}a^{21}-\frac{46\!\cdots\!88}{83\!\cdots\!89}a^{20}+\frac{18\!\cdots\!37}{83\!\cdots\!89}a^{19}-\frac{29\!\cdots\!74}{83\!\cdots\!89}a^{18}+\frac{10\!\cdots\!25}{83\!\cdots\!89}a^{17}-\frac{12\!\cdots\!40}{83\!\cdots\!89}a^{16}+\frac{38\!\cdots\!93}{83\!\cdots\!89}a^{15}-\frac{38\!\cdots\!57}{83\!\cdots\!89}a^{14}+\frac{10\!\cdots\!44}{83\!\cdots\!89}a^{13}-\frac{81\!\cdots\!25}{83\!\cdots\!89}a^{12}+\frac{19\!\cdots\!30}{83\!\cdots\!89}a^{11}-\frac{10\!\cdots\!16}{83\!\cdots\!89}a^{10}+\frac{23\!\cdots\!30}{83\!\cdots\!89}a^{9}-\frac{66\!\cdots\!72}{83\!\cdots\!89}a^{8}+\frac{12\!\cdots\!57}{83\!\cdots\!89}a^{7}+\frac{35\!\cdots\!28}{83\!\cdots\!89}a^{6}+\frac{28\!\cdots\!07}{83\!\cdots\!89}a^{5}+\frac{12\!\cdots\!92}{83\!\cdots\!89}a^{4}+\frac{70\!\cdots\!57}{83\!\cdots\!89}a^{3}+\frac{83\!\cdots\!68}{23\!\cdots\!61}a^{2}+\frac{32\!\cdots\!36}{83\!\cdots\!89}a-\frac{33\!\cdots\!46}{20\!\cdots\!29}$, $\frac{80\!\cdots\!29}{83\!\cdots\!89}a^{27}-\frac{30\!\cdots\!01}{49\!\cdots\!17}a^{26}+\frac{44\!\cdots\!46}{83\!\cdots\!89}a^{25}-\frac{18\!\cdots\!93}{83\!\cdots\!89}a^{24}+\frac{10\!\cdots\!72}{83\!\cdots\!89}a^{23}-\frac{38\!\cdots\!90}{83\!\cdots\!89}a^{22}+\frac{16\!\cdots\!49}{83\!\cdots\!89}a^{21}-\frac{45\!\cdots\!18}{83\!\cdots\!89}a^{20}+\frac{14\!\cdots\!55}{83\!\cdots\!89}a^{19}-\frac{33\!\cdots\!20}{83\!\cdots\!89}a^{18}+\frac{87\!\cdots\!37}{83\!\cdots\!89}a^{17}-\frac{17\!\cdots\!92}{83\!\cdots\!89}a^{16}+\frac{37\!\cdots\!37}{83\!\cdots\!89}a^{15}-\frac{63\!\cdots\!32}{83\!\cdots\!89}a^{14}+\frac{11\!\cdots\!42}{83\!\cdots\!89}a^{13}-\frac{16\!\cdots\!95}{83\!\cdots\!89}a^{12}+\frac{25\!\cdots\!04}{83\!\cdots\!89}a^{11}-\frac{31\!\cdots\!12}{83\!\cdots\!89}a^{10}+\frac{38\!\cdots\!35}{83\!\cdots\!89}a^{9}-\frac{39\!\cdots\!10}{83\!\cdots\!89}a^{8}+\frac{36\!\cdots\!56}{83\!\cdots\!89}a^{7}-\frac{25\!\cdots\!15}{83\!\cdots\!89}a^{6}+\frac{14\!\cdots\!37}{83\!\cdots\!89}a^{5}-\frac{61\!\cdots\!48}{83\!\cdots\!89}a^{4}+\frac{25\!\cdots\!05}{83\!\cdots\!89}a^{3}-\frac{28\!\cdots\!77}{23\!\cdots\!61}a^{2}+\frac{30\!\cdots\!03}{83\!\cdots\!89}a-\frac{61\!\cdots\!23}{11\!\cdots\!37}$
| 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: | \( 34681517373.86067 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 34681517373.86067 \cdot 7232}{6\cdot\sqrt{3654513548980364725264702136425345220781793212890625}}\cr\approx \mathstrut & 0.103349558629174 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681)
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^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681, 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^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681);
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^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\times C_{14}$ (as 28T2):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.14.0.1}{14} }^{2}$ | R | R | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | ${\href{/padicField/19.7.0.1}{7} }^{4}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/31.7.0.1}{7} }^{4}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $28$ | $2$ | $14$ | $14$ | |||
|
\(5\)
| Deg $28$ | $2$ | $14$ | $14$ | |||
|
\(29\)
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |