Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256)
gp: K = bnfinit(y^32 - 44*y^30 + 868*y^28 - 9906*y^26 + 69988*y^24 - 299086*y^22 + 650567*y^20 - 58759*y^18 - 2384362*y^16 + 1399292*y^14 + 4696121*y^12 + 4850223*y^10 + 3856809*y^8 + 1033776*y^6 + 130272*y^4 + 8000*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256)
\( x^{32} - 44 x^{30} + 868 x^{28} - 9906 x^{26} + 69988 x^{24} - 299086 x^{22} + 650567 x^{20} + \cdots + 256 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3810113501072511163514761705493816987484160000000000000000\)
\(\medspace = 2^{32}\cdot 5^{16}\cdot 7^{16}\cdot 29^{8}\cdot 769^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(63.01\) | 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: | $2\cdot 5^{1/2}7^{1/2}29^{1/2}769^{1/2}\approx 1766.9578376407287$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(29\), \(769\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{20}+\frac{1}{4}a^{18}-\frac{1}{2}a^{16}+\frac{1}{4}a^{12}+\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{21}-\frac{1}{4}a^{19}+\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{18}-\frac{1}{2}a^{16}+\frac{1}{4}a^{14}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{23}+\frac{1}{8}a^{19}-\frac{1}{4}a^{17}+\frac{1}{8}a^{15}-\frac{1}{2}a^{11}-\frac{3}{8}a^{9}-\frac{1}{4}a^{7}-\frac{3}{8}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{24}-\frac{1}{8}a^{22}-\frac{1}{16}a^{20}-\frac{3}{8}a^{18}+\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{3}{8}a^{12}+\frac{3}{16}a^{10}+\frac{1}{8}a^{8}+\frac{5}{16}a^{6}+\frac{5}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{25}-\frac{1}{16}a^{23}+\frac{3}{32}a^{21}-\frac{1}{16}a^{19}-\frac{7}{32}a^{17}+\frac{7}{16}a^{15}-\frac{1}{16}a^{13}+\frac{7}{32}a^{11}+\frac{3}{16}a^{9}+\frac{13}{32}a^{7}-\frac{11}{32}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{704}a^{26}-\frac{5}{352}a^{24}-\frac{37}{704}a^{22}-\frac{9}{352}a^{20}+\frac{11}{64}a^{18}-\frac{141}{352}a^{16}-\frac{53}{352}a^{14}+\frac{255}{704}a^{12}-\frac{5}{352}a^{10}-\frac{19}{704}a^{8}+\frac{333}{704}a^{6}-\frac{19}{88}a^{4}-\frac{1}{2}a^{2}+\frac{1}{11}$, $\frac{1}{704}a^{27}-\frac{5}{352}a^{25}-\frac{37}{704}a^{23}-\frac{9}{352}a^{21}+\frac{11}{64}a^{19}-\frac{141}{352}a^{17}-\frac{53}{352}a^{15}+\frac{255}{704}a^{13}-\frac{5}{352}a^{11}-\frac{19}{704}a^{9}+\frac{333}{704}a^{7}-\frac{19}{88}a^{5}-\frac{1}{2}a^{3}+\frac{1}{11}a$, $\frac{1}{2760884544}a^{28}+\frac{6531}{83663168}a^{26}+\frac{20614853}{2760884544}a^{24}-\frac{289997579}{2760884544}a^{22}+\frac{135500339}{2760884544}a^{20}+\frac{549003283}{2760884544}a^{18}+\frac{1628589}{5228948}a^{16}-\frac{286838857}{920294848}a^{14}+\frac{406442321}{2760884544}a^{12}+\frac{83255829}{920294848}a^{10}-\frac{406799}{1151328}a^{8}-\frac{938400211}{2760884544}a^{6}-\frac{175436311}{690221136}a^{4}-\frac{4543205}{14379607}a^{2}-\frac{21170986}{43138821}$, $\frac{1}{5521769088}a^{29}+\frac{6531}{167326336}a^{27}+\frac{20614853}{5521769088}a^{25}-\frac{289997579}{5521769088}a^{23}+\frac{135500339}{5521769088}a^{21}+\frac{549003283}{5521769088}a^{19}-\frac{3600359}{10457896}a^{17}+\frac{633455991}{1840589696}a^{15}+\frac{406442321}{5521769088}a^{13}-\frac{837039019}{1840589696}a^{11}-\frac{406799}{2302656}a^{9}+\frac{1822484333}{5521769088}a^{7}+\frac{514784825}{1380442272}a^{5}+\frac{4918201}{14379607}a^{3}-\frac{10585493}{43138821}a$, $\frac{1}{31\!\cdots\!16}a^{30}-\frac{28\!\cdots\!65}{31\!\cdots\!16}a^{28}+\frac{32\!\cdots\!99}{47\!\cdots\!48}a^{26}-\frac{86\!\cdots\!79}{31\!\cdots\!16}a^{24}+\frac{80\!\cdots\!91}{31\!\cdots\!16}a^{22}-\frac{18\!\cdots\!45}{31\!\cdots\!16}a^{20}-\frac{45\!\cdots\!05}{99\!\cdots\!88}a^{18}-\frac{35\!\cdots\!77}{10\!\cdots\!72}a^{16}-\frac{12\!\cdots\!11}{31\!\cdots\!16}a^{14}+\frac{11\!\cdots\!35}{31\!\cdots\!16}a^{12}-\frac{39\!\cdots\!03}{15\!\cdots\!08}a^{10}+\frac{59\!\cdots\!77}{31\!\cdots\!16}a^{8}+\frac{12\!\cdots\!65}{26\!\cdots\!68}a^{6}-\frac{71\!\cdots\!19}{73\!\cdots\!32}a^{4}+\frac{12\!\cdots\!71}{49\!\cdots\!44}a^{2}-\frac{46\!\cdots\!22}{11\!\cdots\!01}$, $\frac{1}{31\!\cdots\!16}a^{31}-\frac{28\!\cdots\!65}{31\!\cdots\!16}a^{29}+\frac{32\!\cdots\!99}{47\!\cdots\!48}a^{27}+\frac{12\!\cdots\!09}{31\!\cdots\!16}a^{25}-\frac{11\!\cdots\!85}{31\!\cdots\!16}a^{23}+\frac{11\!\cdots\!19}{31\!\cdots\!16}a^{21}-\frac{48\!\cdots\!33}{19\!\cdots\!76}a^{19}-\frac{56\!\cdots\!13}{10\!\cdots\!72}a^{17}+\frac{17\!\cdots\!21}{31\!\cdots\!16}a^{15}-\frac{86\!\cdots\!41}{31\!\cdots\!16}a^{13}+\frac{74\!\cdots\!09}{15\!\cdots\!08}a^{11}-\frac{39\!\cdots\!03}{31\!\cdots\!16}a^{9}+\frac{97\!\cdots\!43}{26\!\cdots\!68}a^{7}-\frac{26\!\cdots\!15}{59\!\cdots\!56}a^{5}-\frac{12\!\cdots\!81}{24\!\cdots\!22}a^{3}+\frac{19\!\cdots\!57}{22\!\cdots\!02}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{86}$, which has order $344$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $344$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{634554891604445}{252385030615695648} a^{31} + \frac{111994819583001397}{1009540122462782592} a^{29} - \frac{2216971415029647181}{1009540122462782592} a^{27} + \frac{770188158841584957}{30592124923114624} a^{25} - \frac{60254171376939197953}{336513374154260864} a^{23} + \frac{260413099862384172681}{336513374154260864} a^{21} - \frac{1746183179604137886257}{1009540122462782592} a^{19} + \frac{14922714673047409721}{42064171769282608} a^{17} + \frac{6022317377962114263713}{1009540122462782592} a^{15} - \frac{4298027343599361230263}{1009540122462782592} a^{13} - \frac{11438841379219429885733}{1009540122462782592} a^{11} - \frac{1810475673198738923991}{168256687077130432} a^{9} - \frac{8356626132771034850363}{1009540122462782592} a^{7} - \frac{188411543681098809905}{126192515307847824} a^{5} - \frac{4848592845816736309}{63096257653923912} a^{3} + \frac{10213335764275055}{15774064413480978} a \)
(order $4$)
| 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: |
$\frac{18\!\cdots\!17}{23\!\cdots\!72}a^{31}-\frac{40\!\cdots\!21}{11\!\cdots\!36}a^{29}+\frac{80\!\cdots\!53}{11\!\cdots\!36}a^{27}-\frac{46\!\cdots\!41}{57\!\cdots\!68}a^{25}+\frac{66\!\cdots\!75}{11\!\cdots\!36}a^{23}-\frac{14\!\cdots\!07}{57\!\cdots\!68}a^{21}+\frac{13\!\cdots\!45}{23\!\cdots\!72}a^{19}-\frac{11\!\cdots\!77}{76\!\cdots\!24}a^{17}-\frac{54\!\cdots\!69}{28\!\cdots\!84}a^{15}+\frac{17\!\cdots\!55}{11\!\cdots\!36}a^{13}+\frac{80\!\cdots\!19}{23\!\cdots\!72}a^{11}+\frac{72\!\cdots\!03}{23\!\cdots\!72}a^{9}+\frac{17\!\cdots\!41}{76\!\cdots\!24}a^{7}+\frac{13\!\cdots\!95}{57\!\cdots\!68}a^{5}-\frac{48\!\cdots\!11}{14\!\cdots\!92}a^{3}-\frac{23\!\cdots\!31}{32\!\cdots\!18}a$, $\frac{38\!\cdots\!53}{64\!\cdots\!44}a^{30}-\frac{16\!\cdots\!87}{64\!\cdots\!44}a^{28}+\frac{52\!\cdots\!87}{10\!\cdots\!71}a^{26}-\frac{38\!\cdots\!43}{64\!\cdots\!44}a^{24}+\frac{68\!\cdots\!35}{16\!\cdots\!36}a^{22}-\frac{11\!\cdots\!97}{64\!\cdots\!44}a^{20}+\frac{25\!\cdots\!43}{64\!\cdots\!44}a^{18}-\frac{14\!\cdots\!79}{21\!\cdots\!48}a^{16}-\frac{91\!\cdots\!87}{64\!\cdots\!44}a^{14}+\frac{15\!\cdots\!21}{16\!\cdots\!36}a^{12}+\frac{87\!\cdots\!03}{32\!\cdots\!72}a^{10}+\frac{77\!\cdots\!71}{29\!\cdots\!52}a^{8}+\frac{44\!\cdots\!77}{21\!\cdots\!48}a^{6}+\frac{68\!\cdots\!41}{16\!\cdots\!36}a^{4}+\frac{31\!\cdots\!46}{10\!\cdots\!71}a^{2}+\frac{73\!\cdots\!56}{91\!\cdots\!61}$, $\frac{76\!\cdots\!01}{39\!\cdots\!52}a^{31}-\frac{22\!\cdots\!03}{26\!\cdots\!68}a^{29}+\frac{33\!\cdots\!77}{19\!\cdots\!76}a^{27}-\frac{15\!\cdots\!25}{79\!\cdots\!04}a^{25}+\frac{27\!\cdots\!53}{19\!\cdots\!76}a^{23}-\frac{43\!\cdots\!05}{72\!\cdots\!64}a^{21}+\frac{17\!\cdots\!49}{13\!\cdots\!84}a^{19}-\frac{39\!\cdots\!77}{13\!\cdots\!84}a^{17}-\frac{33\!\cdots\!49}{72\!\cdots\!64}a^{15}+\frac{22\!\cdots\!65}{66\!\cdots\!92}a^{13}+\frac{68\!\cdots\!15}{79\!\cdots\!04}a^{11}+\frac{64\!\cdots\!35}{79\!\cdots\!04}a^{9}+\frac{22\!\cdots\!01}{36\!\cdots\!32}a^{7}+\frac{13\!\cdots\!91}{13\!\cdots\!84}a^{5}+\frac{27\!\cdots\!91}{99\!\cdots\!88}a^{3}-\frac{18\!\cdots\!39}{41\!\cdots\!37}a$, $\frac{13\!\cdots\!37}{59\!\cdots\!56}a^{31}+\frac{14\!\cdots\!77}{64\!\cdots\!12}a^{30}-\frac{20\!\cdots\!85}{19\!\cdots\!52}a^{29}-\frac{19\!\cdots\!45}{19\!\cdots\!64}a^{28}+\frac{12\!\cdots\!45}{59\!\cdots\!56}a^{27}+\frac{12\!\cdots\!91}{64\!\cdots\!12}a^{26}-\frac{17\!\cdots\!17}{73\!\cdots\!32}a^{25}-\frac{14\!\cdots\!15}{64\!\cdots\!12}a^{24}+\frac{96\!\cdots\!77}{59\!\cdots\!56}a^{23}+\frac{10\!\cdots\!57}{64\!\cdots\!12}a^{22}-\frac{10\!\cdots\!61}{14\!\cdots\!64}a^{21}-\frac{45\!\cdots\!49}{64\!\cdots\!12}a^{20}+\frac{14\!\cdots\!25}{98\!\cdots\!76}a^{19}+\frac{15\!\cdots\!21}{97\!\cdots\!32}a^{18}+\frac{12\!\cdots\!69}{89\!\cdots\!16}a^{17}-\frac{53\!\cdots\!11}{19\!\cdots\!64}a^{16}-\frac{33\!\cdots\!67}{59\!\cdots\!56}a^{15}-\frac{35\!\cdots\!03}{64\!\cdots\!12}a^{14}+\frac{55\!\cdots\!83}{19\!\cdots\!52}a^{13}+\frac{79\!\cdots\!63}{21\!\cdots\!04}a^{12}+\frac{61\!\cdots\!01}{53\!\cdots\!96}a^{11}+\frac{33\!\cdots\!15}{32\!\cdots\!56}a^{10}+\frac{72\!\cdots\!81}{59\!\cdots\!56}a^{9}+\frac{65\!\cdots\!03}{64\!\cdots\!12}a^{8}+\frac{57\!\cdots\!37}{59\!\cdots\!56}a^{7}+\frac{25\!\cdots\!79}{32\!\cdots\!56}a^{6}+\frac{58\!\cdots\!23}{19\!\cdots\!52}a^{5}+\frac{22\!\cdots\!77}{13\!\cdots\!44}a^{4}+\frac{12\!\cdots\!01}{36\!\cdots\!66}a^{3}+\frac{28\!\cdots\!79}{20\!\cdots\!16}a^{2}+\frac{11\!\cdots\!32}{61\!\cdots\!11}a+\frac{75\!\cdots\!56}{16\!\cdots\!43}$, $\frac{13\!\cdots\!37}{59\!\cdots\!56}a^{31}-\frac{44\!\cdots\!95}{22\!\cdots\!48}a^{30}-\frac{20\!\cdots\!85}{19\!\cdots\!52}a^{29}+\frac{15\!\cdots\!45}{17\!\cdots\!84}a^{28}+\frac{12\!\cdots\!45}{59\!\cdots\!56}a^{27}-\frac{28\!\cdots\!31}{16\!\cdots\!44}a^{26}-\frac{17\!\cdots\!17}{73\!\cdots\!32}a^{25}+\frac{32\!\cdots\!45}{16\!\cdots\!44}a^{24}+\frac{96\!\cdots\!77}{59\!\cdots\!56}a^{23}-\frac{25\!\cdots\!35}{17\!\cdots\!84}a^{22}-\frac{10\!\cdots\!61}{14\!\cdots\!64}a^{21}+\frac{10\!\cdots\!45}{17\!\cdots\!84}a^{20}+\frac{14\!\cdots\!25}{98\!\cdots\!76}a^{19}-\frac{24\!\cdots\!97}{17\!\cdots\!84}a^{18}+\frac{12\!\cdots\!69}{89\!\cdots\!16}a^{17}+\frac{20\!\cdots\!39}{88\!\cdots\!92}a^{16}-\frac{33\!\cdots\!67}{59\!\cdots\!56}a^{15}+\frac{85\!\cdots\!45}{17\!\cdots\!84}a^{14}+\frac{55\!\cdots\!83}{19\!\cdots\!52}a^{13}-\frac{57\!\cdots\!43}{17\!\cdots\!84}a^{12}+\frac{61\!\cdots\!01}{53\!\cdots\!96}a^{11}-\frac{16\!\cdots\!07}{17\!\cdots\!84}a^{10}+\frac{72\!\cdots\!81}{59\!\cdots\!56}a^{9}-\frac{79\!\cdots\!39}{88\!\cdots\!92}a^{8}+\frac{57\!\cdots\!37}{59\!\cdots\!56}a^{7}-\frac{12\!\cdots\!45}{17\!\cdots\!84}a^{6}+\frac{58\!\cdots\!23}{19\!\cdots\!52}a^{5}-\frac{13\!\cdots\!75}{88\!\cdots\!92}a^{4}+\frac{12\!\cdots\!01}{36\!\cdots\!66}a^{3}-\frac{29\!\cdots\!19}{22\!\cdots\!48}a^{2}+\frac{10\!\cdots\!21}{61\!\cdots\!11}a-\frac{24\!\cdots\!32}{55\!\cdots\!87}$, $\frac{44\!\cdots\!63}{31\!\cdots\!16}a^{31}+\frac{99\!\cdots\!47}{12\!\cdots\!24}a^{30}-\frac{66\!\cdots\!85}{10\!\cdots\!72}a^{29}-\frac{44\!\cdots\!11}{12\!\cdots\!24}a^{28}+\frac{39\!\cdots\!15}{31\!\cdots\!16}a^{27}+\frac{87\!\cdots\!47}{12\!\cdots\!24}a^{26}-\frac{46\!\cdots\!85}{31\!\cdots\!16}a^{25}-\frac{99\!\cdots\!53}{12\!\cdots\!24}a^{24}+\frac{33\!\cdots\!33}{31\!\cdots\!16}a^{23}+\frac{70\!\cdots\!13}{12\!\cdots\!24}a^{22}-\frac{14\!\cdots\!15}{31\!\cdots\!16}a^{21}-\frac{30\!\cdots\!95}{12\!\cdots\!24}a^{20}+\frac{28\!\cdots\!33}{26\!\cdots\!68}a^{19}+\frac{85\!\cdots\!77}{16\!\cdots\!28}a^{18}-\frac{49\!\cdots\!31}{10\!\cdots\!72}a^{17}-\frac{42\!\cdots\!43}{42\!\cdots\!08}a^{16}-\frac{10\!\cdots\!69}{31\!\cdots\!16}a^{15}-\frac{23\!\cdots\!65}{12\!\cdots\!24}a^{14}+\frac{35\!\cdots\!07}{10\!\cdots\!72}a^{13}+\frac{14\!\cdots\!23}{11\!\cdots\!84}a^{12}+\frac{90\!\cdots\!75}{15\!\cdots\!08}a^{11}+\frac{20\!\cdots\!89}{58\!\cdots\!92}a^{10}+\frac{13\!\cdots\!47}{31\!\cdots\!16}a^{9}+\frac{44\!\cdots\!31}{12\!\cdots\!24}a^{8}+\frac{12\!\cdots\!67}{39\!\cdots\!52}a^{7}+\frac{28\!\cdots\!57}{10\!\cdots\!52}a^{6}-\frac{51\!\cdots\!39}{13\!\cdots\!84}a^{5}+\frac{40\!\cdots\!93}{73\!\cdots\!24}a^{4}-\frac{36\!\cdots\!87}{24\!\cdots\!22}a^{3}+\frac{62\!\cdots\!61}{10\!\cdots\!58}a^{2}-\frac{75\!\cdots\!15}{82\!\cdots\!74}a+\frac{77\!\cdots\!06}{50\!\cdots\!29}$, $\frac{64\!\cdots\!13}{10\!\cdots\!72}a^{31}-\frac{84\!\cdots\!89}{31\!\cdots\!16}a^{29}+\frac{55\!\cdots\!27}{10\!\cdots\!72}a^{27}-\frac{19\!\cdots\!55}{31\!\cdots\!16}a^{25}+\frac{13\!\cdots\!67}{31\!\cdots\!16}a^{23}-\frac{57\!\cdots\!77}{31\!\cdots\!16}a^{21}+\frac{61\!\cdots\!69}{15\!\cdots\!08}a^{19}-\frac{17\!\cdots\!35}{10\!\cdots\!72}a^{17}-\frac{15\!\cdots\!97}{10\!\cdots\!72}a^{15}+\frac{25\!\cdots\!83}{31\!\cdots\!16}a^{13}+\frac{76\!\cdots\!71}{26\!\cdots\!68}a^{11}+\frac{96\!\cdots\!15}{31\!\cdots\!16}a^{9}+\frac{38\!\cdots\!41}{15\!\cdots\!08}a^{7}+\frac{68\!\cdots\!43}{99\!\cdots\!88}a^{5}+\frac{24\!\cdots\!77}{33\!\cdots\!96}a^{3}+\frac{31\!\cdots\!14}{11\!\cdots\!01}a+1$, $\frac{12\!\cdots\!37}{15\!\cdots\!08}a^{31}+\frac{56\!\cdots\!69}{32\!\cdots\!56}a^{30}-\frac{10\!\cdots\!19}{39\!\cdots\!52}a^{29}-\frac{24\!\cdots\!81}{32\!\cdots\!56}a^{28}+\frac{15\!\cdots\!45}{39\!\cdots\!52}a^{27}+\frac{49\!\cdots\!29}{32\!\cdots\!56}a^{26}-\frac{17\!\cdots\!85}{79\!\cdots\!04}a^{25}-\frac{18\!\cdots\!41}{10\!\cdots\!52}a^{24}-\frac{32\!\cdots\!91}{39\!\cdots\!52}a^{23}+\frac{12\!\cdots\!87}{97\!\cdots\!32}a^{22}+\frac{16\!\cdots\!61}{79\!\cdots\!04}a^{21}-\frac{57\!\cdots\!71}{10\!\cdots\!52}a^{20}-\frac{21\!\cdots\!01}{15\!\cdots\!08}a^{19}+\frac{23\!\cdots\!81}{20\!\cdots\!16}a^{18}+\frac{21\!\cdots\!27}{52\!\cdots\!36}a^{17}-\frac{17\!\cdots\!57}{10\!\cdots\!52}a^{16}-\frac{20\!\cdots\!01}{79\!\cdots\!04}a^{15}-\frac{13\!\cdots\!43}{32\!\cdots\!56}a^{14}-\frac{53\!\cdots\!29}{39\!\cdots\!52}a^{13}+\frac{86\!\cdots\!11}{32\!\cdots\!56}a^{12}+\frac{21\!\cdots\!49}{15\!\cdots\!08}a^{11}+\frac{13\!\cdots\!51}{16\!\cdots\!28}a^{10}+\frac{49\!\cdots\!71}{15\!\cdots\!08}a^{9}+\frac{87\!\cdots\!39}{10\!\cdots\!52}a^{8}+\frac{15\!\cdots\!79}{52\!\cdots\!36}a^{7}+\frac{51\!\cdots\!11}{80\!\cdots\!64}a^{6}+\frac{52\!\cdots\!23}{24\!\cdots\!22}a^{5}+\frac{60\!\cdots\!35}{40\!\cdots\!32}a^{4}+\frac{38\!\cdots\!71}{99\!\cdots\!88}a^{3}+\frac{15\!\cdots\!39}{10\!\cdots\!58}a^{2}+\frac{51\!\cdots\!59}{22\!\cdots\!02}a+\frac{21\!\cdots\!62}{50\!\cdots\!29}$, $\frac{13\!\cdots\!41}{79\!\cdots\!04}a^{31}-\frac{21\!\cdots\!75}{64\!\cdots\!12}a^{30}-\frac{12\!\cdots\!03}{15\!\cdots\!08}a^{29}+\frac{31\!\cdots\!81}{21\!\cdots\!04}a^{28}+\frac{23\!\cdots\!37}{15\!\cdots\!08}a^{27}-\frac{18\!\cdots\!97}{64\!\cdots\!12}a^{26}-\frac{24\!\cdots\!09}{14\!\cdots\!28}a^{25}+\frac{20\!\cdots\!17}{64\!\cdots\!12}a^{24}+\frac{19\!\cdots\!99}{15\!\cdots\!08}a^{23}-\frac{14\!\cdots\!39}{64\!\cdots\!12}a^{22}-\frac{84\!\cdots\!41}{15\!\cdots\!08}a^{21}+\frac{61\!\cdots\!75}{64\!\cdots\!12}a^{20}+\frac{19\!\cdots\!43}{15\!\cdots\!08}a^{19}-\frac{21\!\cdots\!45}{10\!\cdots\!52}a^{18}-\frac{75\!\cdots\!51}{26\!\cdots\!68}a^{17}-\frac{27\!\cdots\!01}{21\!\cdots\!04}a^{16}-\frac{64\!\cdots\!55}{15\!\cdots\!08}a^{15}+\frac{51\!\cdots\!85}{64\!\cdots\!12}a^{14}+\frac{47\!\cdots\!91}{15\!\cdots\!08}a^{13}-\frac{80\!\cdots\!73}{21\!\cdots\!04}a^{12}+\frac{12\!\cdots\!73}{15\!\cdots\!08}a^{11}-\frac{52\!\cdots\!65}{32\!\cdots\!56}a^{10}+\frac{57\!\cdots\!03}{79\!\cdots\!04}a^{9}-\frac{11\!\cdots\!85}{64\!\cdots\!12}a^{8}+\frac{29\!\cdots\!19}{52\!\cdots\!36}a^{7}-\frac{45\!\cdots\!41}{32\!\cdots\!56}a^{6}+\frac{38\!\cdots\!45}{39\!\cdots\!52}a^{5}-\frac{28\!\cdots\!39}{67\!\cdots\!72}a^{4}+\frac{48\!\cdots\!93}{49\!\cdots\!44}a^{3}-\frac{72\!\cdots\!51}{20\!\cdots\!16}a^{2}+\frac{13\!\cdots\!25}{24\!\cdots\!22}a-\frac{18\!\cdots\!14}{16\!\cdots\!43}$, $\frac{89\!\cdots\!49}{15\!\cdots\!08}a^{31}+\frac{26\!\cdots\!62}{28\!\cdots\!17}a^{30}-\frac{13\!\cdots\!27}{52\!\cdots\!36}a^{29}-\frac{11\!\cdots\!87}{28\!\cdots\!17}a^{28}+\frac{71\!\cdots\!83}{14\!\cdots\!28}a^{27}+\frac{20\!\cdots\!13}{25\!\cdots\!47}a^{26}-\frac{89\!\cdots\!63}{15\!\cdots\!08}a^{25}-\frac{23\!\cdots\!37}{25\!\cdots\!47}a^{24}+\frac{63\!\cdots\!27}{15\!\cdots\!08}a^{23}+\frac{18\!\cdots\!03}{28\!\cdots\!17}a^{22}-\frac{27\!\cdots\!41}{15\!\cdots\!08}a^{21}-\frac{80\!\cdots\!36}{28\!\cdots\!17}a^{20}+\frac{24\!\cdots\!27}{66\!\cdots\!92}a^{19}+\frac{17\!\cdots\!98}{28\!\cdots\!17}a^{18}-\frac{26\!\cdots\!61}{52\!\cdots\!36}a^{17}-\frac{31\!\cdots\!93}{28\!\cdots\!17}a^{16}-\frac{21\!\cdots\!71}{15\!\cdots\!08}a^{15}-\frac{62\!\cdots\!43}{28\!\cdots\!17}a^{14}+\frac{45\!\cdots\!53}{52\!\cdots\!36}a^{13}+\frac{42\!\cdots\!27}{28\!\cdots\!17}a^{12}+\frac{20\!\cdots\!03}{79\!\cdots\!04}a^{11}+\frac{12\!\cdots\!45}{28\!\cdots\!17}a^{10}+\frac{41\!\cdots\!29}{15\!\cdots\!08}a^{9}+\frac{11\!\cdots\!45}{28\!\cdots\!17}a^{8}+\frac{80\!\cdots\!75}{39\!\cdots\!52}a^{7}+\frac{92\!\cdots\!40}{28\!\cdots\!17}a^{6}+\frac{38\!\cdots\!41}{82\!\cdots\!74}a^{5}+\frac{19\!\cdots\!40}{28\!\cdots\!17}a^{4}+\frac{47\!\cdots\!35}{12\!\cdots\!11}a^{3}+\frac{21\!\cdots\!64}{28\!\cdots\!17}a^{2}+\frac{38\!\cdots\!08}{41\!\cdots\!37}a+\frac{13\!\cdots\!75}{28\!\cdots\!17}$, $\frac{18\!\cdots\!33}{26\!\cdots\!84}a^{31}+\frac{16\!\cdots\!61}{10\!\cdots\!72}a^{30}-\frac{87\!\cdots\!01}{28\!\cdots\!56}a^{29}-\frac{73\!\cdots\!17}{10\!\cdots\!72}a^{28}+\frac{18\!\cdots\!79}{31\!\cdots\!16}a^{27}+\frac{14\!\cdots\!21}{10\!\cdots\!72}a^{26}-\frac{21\!\cdots\!57}{31\!\cdots\!16}a^{25}-\frac{15\!\cdots\!21}{96\!\cdots\!52}a^{24}+\frac{15\!\cdots\!21}{31\!\cdots\!16}a^{23}+\frac{12\!\cdots\!27}{10\!\cdots\!72}a^{22}-\frac{66\!\cdots\!07}{31\!\cdots\!16}a^{21}-\frac{52\!\cdots\!17}{10\!\cdots\!72}a^{20}+\frac{42\!\cdots\!21}{90\!\cdots\!08}a^{19}+\frac{29\!\cdots\!35}{26\!\cdots\!68}a^{18}-\frac{93\!\cdots\!95}{10\!\cdots\!72}a^{17}-\frac{33\!\cdots\!07}{10\!\cdots\!72}a^{16}-\frac{51\!\cdots\!73}{31\!\cdots\!16}a^{15}-\frac{39\!\cdots\!95}{10\!\cdots\!72}a^{14}+\frac{35\!\cdots\!13}{31\!\cdots\!16}a^{13}+\frac{31\!\cdots\!39}{10\!\cdots\!72}a^{12}+\frac{49\!\cdots\!27}{15\!\cdots\!08}a^{11}+\frac{36\!\cdots\!93}{52\!\cdots\!36}a^{10}+\frac{87\!\cdots\!71}{29\!\cdots\!24}a^{9}+\frac{65\!\cdots\!29}{10\!\cdots\!72}a^{8}+\frac{61\!\cdots\!69}{26\!\cdots\!68}a^{7}+\frac{62\!\cdots\!41}{13\!\cdots\!84}a^{6}+\frac{17\!\cdots\!13}{39\!\cdots\!52}a^{5}+\frac{38\!\cdots\!87}{66\!\cdots\!92}a^{4}+\frac{29\!\cdots\!01}{90\!\cdots\!08}a^{3}+\frac{16\!\cdots\!31}{41\!\cdots\!37}a^{2}+\frac{29\!\cdots\!63}{24\!\cdots\!22}a+\frac{40\!\cdots\!74}{37\!\cdots\!67}$, $\frac{95\!\cdots\!25}{26\!\cdots\!68}a^{31}+\frac{11\!\cdots\!89}{25\!\cdots\!76}a^{30}-\frac{83\!\cdots\!81}{52\!\cdots\!36}a^{29}-\frac{50\!\cdots\!97}{25\!\cdots\!76}a^{28}+\frac{16\!\cdots\!33}{52\!\cdots\!36}a^{27}+\frac{98\!\cdots\!45}{25\!\cdots\!76}a^{26}-\frac{18\!\cdots\!09}{52\!\cdots\!36}a^{25}-\frac{37\!\cdots\!13}{86\!\cdots\!92}a^{24}+\frac{13\!\cdots\!15}{52\!\cdots\!36}a^{23}+\frac{26\!\cdots\!81}{86\!\cdots\!92}a^{22}-\frac{57\!\cdots\!11}{52\!\cdots\!36}a^{21}-\frac{11\!\cdots\!75}{86\!\cdots\!92}a^{20}+\frac{12\!\cdots\!27}{52\!\cdots\!36}a^{19}+\frac{19\!\cdots\!95}{64\!\cdots\!44}a^{18}-\frac{58\!\cdots\!49}{26\!\cdots\!68}a^{17}-\frac{42\!\cdots\!05}{86\!\cdots\!92}a^{16}-\frac{45\!\cdots\!81}{52\!\cdots\!36}a^{15}-\frac{27\!\cdots\!83}{25\!\cdots\!76}a^{14}+\frac{27\!\cdots\!99}{52\!\cdots\!36}a^{13}+\frac{18\!\cdots\!79}{25\!\cdots\!76}a^{12}+\frac{89\!\cdots\!63}{52\!\cdots\!36}a^{11}+\frac{25\!\cdots\!05}{12\!\cdots\!88}a^{10}+\frac{22\!\cdots\!03}{13\!\cdots\!84}a^{9}+\frac{15\!\cdots\!21}{78\!\cdots\!72}a^{8}+\frac{72\!\cdots\!81}{52\!\cdots\!36}a^{7}+\frac{48\!\cdots\!01}{32\!\cdots\!72}a^{6}+\frac{58\!\cdots\!23}{16\!\cdots\!48}a^{5}+\frac{25\!\cdots\!53}{80\!\cdots\!68}a^{4}+\frac{12\!\cdots\!61}{33\!\cdots\!96}a^{3}+\frac{23\!\cdots\!86}{10\!\cdots\!71}a^{2}+\frac{11\!\cdots\!87}{82\!\cdots\!74}a+\frac{22\!\cdots\!54}{91\!\cdots\!61}$, $\frac{89\!\cdots\!07}{15\!\cdots\!08}a^{31}-\frac{11\!\cdots\!89}{25\!\cdots\!76}a^{30}-\frac{39\!\cdots\!99}{15\!\cdots\!08}a^{29}+\frac{50\!\cdots\!97}{25\!\cdots\!76}a^{28}+\frac{78\!\cdots\!63}{15\!\cdots\!08}a^{27}-\frac{98\!\cdots\!45}{25\!\cdots\!76}a^{26}-\frac{30\!\cdots\!83}{52\!\cdots\!36}a^{25}+\frac{37\!\cdots\!13}{86\!\cdots\!92}a^{24}+\frac{19\!\cdots\!09}{48\!\cdots\!76}a^{23}-\frac{26\!\cdots\!81}{86\!\cdots\!92}a^{22}-\frac{94\!\cdots\!49}{52\!\cdots\!36}a^{21}+\frac{11\!\cdots\!75}{86\!\cdots\!92}a^{20}+\frac{10\!\cdots\!93}{24\!\cdots\!22}a^{19}-\frac{19\!\cdots\!95}{64\!\cdots\!44}a^{18}-\frac{60\!\cdots\!59}{52\!\cdots\!36}a^{17}+\frac{42\!\cdots\!05}{86\!\cdots\!92}a^{16}-\frac{31\!\cdots\!35}{23\!\cdots\!24}a^{15}+\frac{27\!\cdots\!83}{25\!\cdots\!76}a^{14}+\frac{16\!\cdots\!93}{15\!\cdots\!08}a^{13}-\frac{18\!\cdots\!79}{25\!\cdots\!76}a^{12}+\frac{19\!\cdots\!95}{79\!\cdots\!04}a^{11}-\frac{25\!\cdots\!05}{12\!\cdots\!88}a^{10}+\frac{10\!\cdots\!71}{48\!\cdots\!76}a^{9}-\frac{15\!\cdots\!21}{78\!\cdots\!72}a^{8}+\frac{17\!\cdots\!67}{99\!\cdots\!88}a^{7}-\frac{48\!\cdots\!01}{32\!\cdots\!72}a^{6}+\frac{88\!\cdots\!95}{39\!\cdots\!52}a^{5}-\frac{25\!\cdots\!53}{80\!\cdots\!68}a^{4}+\frac{97\!\cdots\!29}{49\!\cdots\!44}a^{3}-\frac{23\!\cdots\!86}{10\!\cdots\!71}a^{2}+\frac{59\!\cdots\!39}{12\!\cdots\!11}a-\frac{22\!\cdots\!54}{91\!\cdots\!61}$, $\frac{64\!\cdots\!13}{10\!\cdots\!72}a^{31}-\frac{84\!\cdots\!89}{31\!\cdots\!16}a^{29}+\frac{55\!\cdots\!27}{10\!\cdots\!72}a^{27}-\frac{19\!\cdots\!55}{31\!\cdots\!16}a^{25}+\frac{13\!\cdots\!67}{31\!\cdots\!16}a^{23}-\frac{57\!\cdots\!77}{31\!\cdots\!16}a^{21}+\frac{61\!\cdots\!69}{15\!\cdots\!08}a^{19}-\frac{17\!\cdots\!35}{10\!\cdots\!72}a^{17}-\frac{15\!\cdots\!97}{10\!\cdots\!72}a^{15}+\frac{25\!\cdots\!83}{31\!\cdots\!16}a^{13}+\frac{76\!\cdots\!71}{26\!\cdots\!68}a^{11}+\frac{96\!\cdots\!15}{31\!\cdots\!16}a^{9}+\frac{38\!\cdots\!41}{15\!\cdots\!08}a^{7}+\frac{68\!\cdots\!43}{99\!\cdots\!88}a^{5}+\frac{24\!\cdots\!77}{33\!\cdots\!96}a^{3}+\frac{31\!\cdots\!14}{11\!\cdots\!01}a-1$, $\frac{45\!\cdots\!57}{79\!\cdots\!04}a^{31}-\frac{55\!\cdots\!65}{31\!\cdots\!16}a^{30}-\frac{18\!\cdots\!09}{72\!\cdots\!64}a^{29}+\frac{81\!\cdots\!87}{10\!\cdots\!72}a^{28}+\frac{39\!\cdots\!59}{79\!\cdots\!04}a^{27}-\frac{48\!\cdots\!69}{31\!\cdots\!16}a^{26}-\frac{45\!\cdots\!33}{79\!\cdots\!04}a^{25}+\frac{54\!\cdots\!55}{31\!\cdots\!16}a^{24}+\frac{32\!\cdots\!45}{79\!\cdots\!04}a^{23}-\frac{35\!\cdots\!95}{29\!\cdots\!24}a^{22}-\frac{14\!\cdots\!55}{79\!\cdots\!04}a^{21}+\frac{16\!\cdots\!25}{31\!\cdots\!16}a^{20}+\frac{14\!\cdots\!19}{36\!\cdots\!32}a^{19}-\frac{29\!\cdots\!85}{26\!\cdots\!68}a^{18}-\frac{22\!\cdots\!91}{24\!\cdots\!88}a^{17}+\frac{20\!\cdots\!97}{10\!\cdots\!72}a^{16}-\frac{10\!\cdots\!33}{79\!\cdots\!04}a^{15}+\frac{13\!\cdots\!71}{31\!\cdots\!16}a^{14}+\frac{80\!\cdots\!85}{79\!\cdots\!04}a^{13}-\frac{23\!\cdots\!17}{10\!\cdots\!72}a^{12}+\frac{50\!\cdots\!11}{19\!\cdots\!76}a^{11}-\frac{20\!\cdots\!15}{23\!\cdots\!24}a^{10}+\frac{18\!\cdots\!17}{79\!\cdots\!04}a^{9}-\frac{26\!\cdots\!99}{28\!\cdots\!56}a^{8}+\frac{23\!\cdots\!03}{13\!\cdots\!84}a^{7}-\frac{28\!\cdots\!83}{39\!\cdots\!52}a^{6}+\frac{13\!\cdots\!47}{49\!\cdots\!44}a^{5}-\frac{32\!\cdots\!65}{15\!\cdots\!68}a^{4}-\frac{47\!\cdots\!75}{99\!\cdots\!88}a^{3}-\frac{13\!\cdots\!45}{49\!\cdots\!44}a^{2}-\frac{28\!\cdots\!37}{12\!\cdots\!11}a-\frac{50\!\cdots\!08}{41\!\cdots\!37}$
| 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: | \( 33723689221921.4 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 33723689221921.4 \cdot 344}{4\cdot\sqrt{3810113501072511163514761705493816987484160000000000000000}}\cr\approx \mathstrut & 0.277231431063130 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256)
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^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256, 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^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256);
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^32 - 44*x^30 + 868*x^28 - 9906*x^26 + 69988*x^24 - 299086*x^22 + 650567*x^20 - 58759*x^18 - 2384362*x^16 + 1399292*x^14 + 4696121*x^12 + 4850223*x^10 + 3856809*x^8 + 1033776*x^6 + 130272*x^4 + 8000*x^2 + 256);
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
$D_4^2:C_2^3$ (as 32T12882):
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 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
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]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{8}$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(769\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |