Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 256)
gp: K = bnfinit(y^32 + 12*y^28 + 442*y^24 + 4491*y^20 + 22291*y^16 - 93453*y^12 + 82801*y^8 - 8160*y^4 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 256)
\( x^{32} + 12x^{28} + 442x^{24} + 4491x^{20} + 22291x^{16} - 93453x^{12} + 82801x^{8} - 8160x^{4} + 256 \)
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: |
\(30472057165568685620821441442859880284160000000000000000\)
\(\medspace = 2^{64}\cdot 5^{16}\cdot 89^{8}\cdot 229^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.18\) | 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: | $2^{2}5^{1/2}89^{1/2}229^{1/2}\approx 1276.9025021512018$ | ||
| Ramified primes: |
\(2\), \(5\), \(89\), \(229\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{26}-\frac{1}{2}a^{18}-\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{27}-\frac{1}{2}a^{23}+\frac{1}{4}a^{19}+\frac{3}{8}a^{15}+\frac{3}{8}a^{11}+\frac{3}{8}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{78\!\cdots\!88}a^{28}-\frac{14\!\cdots\!13}{19\!\cdots\!72}a^{24}+\frac{12\!\cdots\!65}{39\!\cdots\!44}a^{20}-\frac{14\!\cdots\!21}{78\!\cdots\!88}a^{16}-\frac{90\!\cdots\!45}{46\!\cdots\!64}a^{12}-\frac{31\!\cdots\!93}{78\!\cdots\!88}a^{8}-\frac{26\!\cdots\!99}{78\!\cdots\!88}a^{4}-\frac{24\!\cdots\!95}{49\!\cdots\!68}$, $\frac{1}{15\!\cdots\!76}a^{29}-\frac{14\!\cdots\!13}{39\!\cdots\!44}a^{25}+\frac{12\!\cdots\!65}{78\!\cdots\!88}a^{21}-\frac{14\!\cdots\!21}{15\!\cdots\!76}a^{17}-\frac{90\!\cdots\!45}{92\!\cdots\!28}a^{13}+\frac{46\!\cdots\!95}{15\!\cdots\!76}a^{9}+\frac{75\!\cdots\!89}{15\!\cdots\!76}a^{5}-\frac{24\!\cdots\!95}{98\!\cdots\!36}a$, $\frac{1}{31\!\cdots\!52}a^{30}-\frac{14\!\cdots\!13}{78\!\cdots\!88}a^{26}-\frac{65\!\cdots\!23}{15\!\cdots\!76}a^{22}-\frac{14\!\cdots\!21}{31\!\cdots\!52}a^{18}-\frac{90\!\cdots\!45}{18\!\cdots\!56}a^{14}+\frac{46\!\cdots\!95}{31\!\cdots\!52}a^{10}+\frac{75\!\cdots\!89}{31\!\cdots\!52}a^{6}-\frac{24\!\cdots\!95}{19\!\cdots\!72}a^{2}$, $\frac{1}{62\!\cdots\!04}a^{31}-\frac{14\!\cdots\!13}{15\!\cdots\!76}a^{27}-\frac{65\!\cdots\!23}{31\!\cdots\!52}a^{23}+\frac{29\!\cdots\!31}{62\!\cdots\!04}a^{19}+\frac{18\!\cdots\!11}{36\!\cdots\!12}a^{15}-\frac{26\!\cdots\!57}{62\!\cdots\!04}a^{11}+\frac{75\!\cdots\!89}{62\!\cdots\!04}a^{7}+\frac{17\!\cdots\!77}{39\!\cdots\!44}a^{3}$
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
$C_{2}\times C_{104}$, which has order $208$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $208$ (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: |
\( \frac{2669561201987325}{2920292266102385152} a^{29} + \frac{8046149166390179}{730073066525596288} a^{25} + \frac{590935489992251145}{1460146133051192576} a^{21} + \frac{12056956665662026815}{2920292266102385152} a^{17} + \frac{3543499135064968935}{171781898006022656} a^{13} - \frac{245477040528652851145}{2920292266102385152} a^{9} + \frac{211068989979389538493}{2920292266102385152} a^{5} - \frac{906176670576128635}{182518266631399072} a \)
(order $8$)
| 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: |
$\frac{2054132947}{48143005715968}a^{28}+\frac{6577726413}{12035751428992}a^{24}+\frac{464186147719}{24071502857984}a^{20}+\frac{9963234376113}{48143005715968}a^{16}+\frac{3144094502537}{2831941512704}a^{12}-\frac{153439631944647}{48143005715968}a^{8}+\frac{16059510668307}{48143005715968}a^{4}+\frac{1827244120987}{3008937857248}$, $\frac{27\!\cdots\!37}{11\!\cdots\!08}a^{31}-\frac{26\!\cdots\!25}{29\!\cdots\!52}a^{29}+\frac{83\!\cdots\!79}{29\!\cdots\!52}a^{27}-\frac{80\!\cdots\!79}{73\!\cdots\!88}a^{25}+\frac{60\!\cdots\!01}{58\!\cdots\!04}a^{23}-\frac{59\!\cdots\!45}{14\!\cdots\!76}a^{21}+\frac{12\!\cdots\!47}{11\!\cdots\!08}a^{19}-\frac{12\!\cdots\!15}{29\!\cdots\!52}a^{17}+\frac{36\!\cdots\!75}{68\!\cdots\!24}a^{15}-\frac{35\!\cdots\!35}{17\!\cdots\!56}a^{13}-\frac{24\!\cdots\!41}{11\!\cdots\!08}a^{11}+\frac{24\!\cdots\!45}{29\!\cdots\!52}a^{9}+\frac{19\!\cdots\!65}{11\!\cdots\!08}a^{7}-\frac{21\!\cdots\!93}{29\!\cdots\!52}a^{5}-\frac{90\!\cdots\!59}{73\!\cdots\!88}a^{3}+\frac{90\!\cdots\!35}{18\!\cdots\!72}a-1$, $\frac{27\!\cdots\!37}{11\!\cdots\!08}a^{31}+\frac{10\!\cdots\!37}{71\!\cdots\!68}a^{30}+\frac{26\!\cdots\!25}{29\!\cdots\!52}a^{29}+\frac{83\!\cdots\!79}{29\!\cdots\!52}a^{27}+\frac{32\!\cdots\!99}{17\!\cdots\!92}a^{26}+\frac{80\!\cdots\!79}{73\!\cdots\!88}a^{25}+\frac{60\!\cdots\!01}{58\!\cdots\!04}a^{23}+\frac{24\!\cdots\!41}{35\!\cdots\!84}a^{22}+\frac{59\!\cdots\!45}{14\!\cdots\!76}a^{21}+\frac{12\!\cdots\!47}{11\!\cdots\!08}a^{19}+\frac{49\!\cdots\!15}{71\!\cdots\!68}a^{18}+\frac{12\!\cdots\!15}{29\!\cdots\!52}a^{17}+\frac{36\!\cdots\!75}{68\!\cdots\!24}a^{15}+\frac{14\!\cdots\!27}{42\!\cdots\!04}a^{14}+\frac{35\!\cdots\!35}{17\!\cdots\!56}a^{13}-\frac{24\!\cdots\!41}{11\!\cdots\!08}a^{11}-\frac{98\!\cdots\!77}{71\!\cdots\!68}a^{10}-\frac{24\!\cdots\!45}{29\!\cdots\!52}a^{9}+\frac{19\!\cdots\!65}{11\!\cdots\!08}a^{7}+\frac{81\!\cdots\!33}{71\!\cdots\!68}a^{6}+\frac{21\!\cdots\!93}{29\!\cdots\!52}a^{5}-\frac{90\!\cdots\!59}{73\!\cdots\!88}a^{3}-\frac{23\!\cdots\!35}{44\!\cdots\!48}a^{2}-\frac{90\!\cdots\!35}{18\!\cdots\!72}a$, $\frac{63\!\cdots\!23}{15\!\cdots\!76}a^{29}+\frac{19\!\cdots\!01}{39\!\cdots\!44}a^{25}+\frac{14\!\cdots\!03}{78\!\cdots\!88}a^{21}+\frac{28\!\cdots\!41}{15\!\cdots\!76}a^{17}+\frac{83\!\cdots\!93}{92\!\cdots\!28}a^{13}-\frac{58\!\cdots\!79}{15\!\cdots\!76}a^{9}+\frac{50\!\cdots\!91}{15\!\cdots\!76}a^{5}-\frac{21\!\cdots\!89}{98\!\cdots\!36}a$, $\frac{27\!\cdots\!05}{15\!\cdots\!76}a^{31}+\frac{83\!\cdots\!47}{39\!\cdots\!44}a^{27}+\frac{60\!\cdots\!45}{78\!\cdots\!88}a^{23}+\frac{12\!\cdots\!99}{15\!\cdots\!76}a^{19}+\frac{36\!\cdots\!91}{92\!\cdots\!28}a^{15}-\frac{25\!\cdots\!73}{15\!\cdots\!76}a^{11}+\frac{20\!\cdots\!33}{15\!\cdots\!76}a^{7}-\frac{91\!\cdots\!83}{98\!\cdots\!36}a^{3}$, $\frac{19\!\cdots\!65}{78\!\cdots\!88}a^{30}-\frac{38\!\cdots\!01}{39\!\cdots\!44}a^{28}+\frac{58\!\cdots\!99}{19\!\cdots\!72}a^{26}-\frac{11\!\cdots\!63}{98\!\cdots\!36}a^{24}+\frac{43\!\cdots\!01}{39\!\cdots\!44}a^{22}-\frac{86\!\cdots\!61}{19\!\cdots\!72}a^{20}+\frac{88\!\cdots\!15}{78\!\cdots\!88}a^{18}-\frac{17\!\cdots\!71}{39\!\cdots\!44}a^{16}+\frac{25\!\cdots\!95}{46\!\cdots\!64}a^{14}-\frac{51\!\cdots\!55}{23\!\cdots\!32}a^{12}-\frac{17\!\cdots\!29}{78\!\cdots\!88}a^{10}+\frac{36\!\cdots\!57}{39\!\cdots\!44}a^{8}+\frac{14\!\cdots\!13}{78\!\cdots\!88}a^{6}-\frac{30\!\cdots\!65}{39\!\cdots\!44}a^{4}-\frac{65\!\cdots\!59}{49\!\cdots\!68}a^{2}+\frac{64\!\cdots\!23}{24\!\cdots\!84}$, $\frac{13\!\cdots\!27}{15\!\cdots\!76}a^{30}-\frac{30\!\cdots\!51}{15\!\cdots\!76}a^{29}-\frac{13\!\cdots\!15}{39\!\cdots\!44}a^{28}+\frac{42\!\cdots\!25}{39\!\cdots\!44}a^{26}-\frac{91\!\cdots\!77}{39\!\cdots\!44}a^{25}-\frac{41\!\cdots\!33}{98\!\cdots\!36}a^{24}+\frac{30\!\cdots\!55}{78\!\cdots\!88}a^{22}-\frac{67\!\cdots\!63}{78\!\cdots\!88}a^{21}-\frac{30\!\cdots\!23}{19\!\cdots\!72}a^{20}+\frac{63\!\cdots\!21}{15\!\cdots\!76}a^{18}-\frac{13\!\cdots\!33}{15\!\cdots\!76}a^{17}-\frac{61\!\cdots\!29}{39\!\cdots\!44}a^{16}+\frac{18\!\cdots\!13}{92\!\cdots\!28}a^{14}-\frac{39\!\cdots\!49}{92\!\cdots\!28}a^{13}-\frac{17\!\cdots\!57}{23\!\cdots\!32}a^{12}-\frac{12\!\cdots\!43}{15\!\cdots\!76}a^{10}+\frac{29\!\cdots\!71}{15\!\cdots\!76}a^{9}+\frac{12\!\cdots\!67}{39\!\cdots\!44}a^{8}+\frac{10\!\cdots\!51}{15\!\cdots\!76}a^{6}-\frac{25\!\cdots\!27}{15\!\cdots\!76}a^{5}-\frac{11\!\cdots\!51}{39\!\cdots\!44}a^{4}-\frac{47\!\cdots\!73}{98\!\cdots\!36}a^{2}+\frac{10\!\cdots\!37}{98\!\cdots\!36}a+\frac{23\!\cdots\!77}{24\!\cdots\!84}$, $\frac{12\!\cdots\!63}{31\!\cdots\!52}a^{30}+\frac{26\!\cdots\!25}{29\!\cdots\!52}a^{29}-\frac{66\!\cdots\!25}{39\!\cdots\!44}a^{28}+\frac{38\!\cdots\!45}{78\!\cdots\!88}a^{26}+\frac{80\!\cdots\!79}{73\!\cdots\!88}a^{25}-\frac{20\!\cdots\!75}{98\!\cdots\!36}a^{24}+\frac{28\!\cdots\!59}{15\!\cdots\!76}a^{22}+\frac{59\!\cdots\!45}{14\!\cdots\!76}a^{21}-\frac{14\!\cdots\!69}{19\!\cdots\!72}a^{20}+\frac{58\!\cdots\!41}{31\!\cdots\!52}a^{18}+\frac{12\!\cdots\!15}{29\!\cdots\!52}a^{17}-\frac{30\!\cdots\!07}{39\!\cdots\!44}a^{16}+\frac{17\!\cdots\!57}{18\!\cdots\!56}a^{14}+\frac{35\!\cdots\!35}{17\!\cdots\!56}a^{13}-\frac{89\!\cdots\!59}{23\!\cdots\!32}a^{12}-\frac{11\!\cdots\!75}{31\!\cdots\!52}a^{10}-\frac{24\!\cdots\!45}{29\!\cdots\!52}a^{9}+\frac{61\!\cdots\!17}{39\!\cdots\!44}a^{8}+\frac{94\!\cdots\!51}{31\!\cdots\!52}a^{6}+\frac{21\!\cdots\!93}{29\!\cdots\!52}a^{5}-\frac{50\!\cdots\!69}{39\!\cdots\!44}a^{4}-\frac{42\!\cdots\!37}{19\!\cdots\!72}a^{2}-\frac{90\!\cdots\!35}{18\!\cdots\!72}a+\frac{10\!\cdots\!55}{24\!\cdots\!84}$, $\frac{60\!\cdots\!81}{62\!\cdots\!04}a^{31}+\frac{10\!\cdots\!37}{71\!\cdots\!68}a^{30}-\frac{85\!\cdots\!11}{15\!\cdots\!76}a^{29}-\frac{199155243598767}{73\!\cdots\!88}a^{28}+\frac{18\!\cdots\!55}{15\!\cdots\!76}a^{27}+\frac{32\!\cdots\!99}{17\!\cdots\!92}a^{26}-\frac{25\!\cdots\!09}{39\!\cdots\!44}a^{25}-\frac{609844244287857}{18\!\cdots\!72}a^{24}+\frac{13\!\cdots\!45}{31\!\cdots\!52}a^{23}+\frac{24\!\cdots\!41}{35\!\cdots\!84}a^{22}-\frac{18\!\cdots\!55}{78\!\cdots\!88}a^{21}-\frac{44\!\cdots\!15}{36\!\cdots\!44}a^{20}+\frac{27\!\cdots\!03}{62\!\cdots\!04}a^{19}+\frac{49\!\cdots\!15}{71\!\cdots\!68}a^{18}-\frac{38\!\cdots\!49}{15\!\cdots\!76}a^{17}-\frac{91\!\cdots\!13}{73\!\cdots\!88}a^{16}+\frac{80\!\cdots\!67}{36\!\cdots\!12}a^{15}+\frac{14\!\cdots\!27}{42\!\cdots\!04}a^{14}-\frac{11\!\cdots\!77}{92\!\cdots\!28}a^{13}-\frac{27\!\cdots\!85}{42\!\cdots\!64}a^{12}-\frac{56\!\cdots\!49}{62\!\cdots\!04}a^{11}-\frac{98\!\cdots\!77}{71\!\cdots\!68}a^{10}+\frac{81\!\cdots\!39}{15\!\cdots\!76}a^{9}+\frac{17\!\cdots\!67}{73\!\cdots\!88}a^{8}+\frac{49\!\cdots\!29}{62\!\cdots\!04}a^{7}+\frac{81\!\cdots\!33}{71\!\cdots\!68}a^{6}-\frac{70\!\cdots\!31}{15\!\cdots\!76}a^{5}-\frac{13\!\cdots\!11}{73\!\cdots\!88}a^{4}-\frac{30\!\cdots\!59}{39\!\cdots\!44}a^{3}-\frac{23\!\cdots\!35}{44\!\cdots\!48}a^{2}+\frac{30\!\cdots\!81}{98\!\cdots\!36}a+\frac{27\!\cdots\!37}{45\!\cdots\!68}$, $\frac{230022050113}{97825682235776}a^{31}-\frac{10\!\cdots\!37}{71\!\cdots\!68}a^{30}+\frac{85\!\cdots\!11}{15\!\cdots\!76}a^{29}-\frac{18\!\cdots\!15}{78\!\cdots\!88}a^{28}+\frac{692408863911}{24456420558944}a^{27}-\frac{32\!\cdots\!99}{17\!\cdots\!92}a^{26}+\frac{25\!\cdots\!09}{39\!\cdots\!44}a^{25}-\frac{54\!\cdots\!49}{19\!\cdots\!72}a^{24}+\frac{50888343416445}{48912841117888}a^{23}-\frac{24\!\cdots\!41}{35\!\cdots\!84}a^{22}+\frac{18\!\cdots\!55}{78\!\cdots\!88}a^{21}-\frac{40\!\cdots\!11}{39\!\cdots\!44}a^{20}+\frac{10\!\cdots\!15}{97825682235776}a^{19}-\frac{49\!\cdots\!15}{71\!\cdots\!68}a^{18}+\frac{38\!\cdots\!49}{15\!\cdots\!76}a^{17}-\frac{82\!\cdots\!05}{78\!\cdots\!88}a^{16}+\frac{51\!\cdots\!15}{97825682235776}a^{15}-\frac{14\!\cdots\!27}{42\!\cdots\!04}a^{14}+\frac{11\!\cdots\!77}{92\!\cdots\!28}a^{13}-\frac{24\!\cdots\!93}{46\!\cdots\!64}a^{12}-\frac{21\!\cdots\!17}{97825682235776}a^{11}+\frac{98\!\cdots\!77}{71\!\cdots\!68}a^{10}-\frac{81\!\cdots\!39}{15\!\cdots\!76}a^{9}+\frac{16\!\cdots\!15}{78\!\cdots\!88}a^{8}+\frac{17\!\cdots\!41}{97825682235776}a^{7}-\frac{81\!\cdots\!33}{71\!\cdots\!68}a^{6}+\frac{70\!\cdots\!31}{15\!\cdots\!76}a^{5}-\frac{13\!\cdots\!99}{78\!\cdots\!88}a^{4}-\frac{52052954765509}{6114105139736}a^{3}+\frac{23\!\cdots\!35}{44\!\cdots\!48}a^{2}-\frac{30\!\cdots\!81}{98\!\cdots\!36}a+\frac{59\!\cdots\!29}{49\!\cdots\!68}$, $\frac{60\!\cdots\!81}{62\!\cdots\!04}a^{31}-\frac{10\!\cdots\!37}{71\!\cdots\!68}a^{30}-\frac{85\!\cdots\!11}{15\!\cdots\!76}a^{29}+\frac{199155243598767}{73\!\cdots\!88}a^{28}+\frac{18\!\cdots\!55}{15\!\cdots\!76}a^{27}-\frac{32\!\cdots\!99}{17\!\cdots\!92}a^{26}-\frac{25\!\cdots\!09}{39\!\cdots\!44}a^{25}+\frac{609844244287857}{18\!\cdots\!72}a^{24}+\frac{13\!\cdots\!45}{31\!\cdots\!52}a^{23}-\frac{24\!\cdots\!41}{35\!\cdots\!84}a^{22}-\frac{18\!\cdots\!55}{78\!\cdots\!88}a^{21}+\frac{44\!\cdots\!15}{36\!\cdots\!44}a^{20}+\frac{27\!\cdots\!03}{62\!\cdots\!04}a^{19}-\frac{49\!\cdots\!15}{71\!\cdots\!68}a^{18}-\frac{38\!\cdots\!49}{15\!\cdots\!76}a^{17}+\frac{91\!\cdots\!13}{73\!\cdots\!88}a^{16}+\frac{80\!\cdots\!67}{36\!\cdots\!12}a^{15}-\frac{14\!\cdots\!27}{42\!\cdots\!04}a^{14}-\frac{11\!\cdots\!77}{92\!\cdots\!28}a^{13}+\frac{27\!\cdots\!85}{42\!\cdots\!64}a^{12}-\frac{56\!\cdots\!49}{62\!\cdots\!04}a^{11}+\frac{98\!\cdots\!77}{71\!\cdots\!68}a^{10}+\frac{81\!\cdots\!39}{15\!\cdots\!76}a^{9}-\frac{17\!\cdots\!67}{73\!\cdots\!88}a^{8}+\frac{49\!\cdots\!29}{62\!\cdots\!04}a^{7}-\frac{81\!\cdots\!33}{71\!\cdots\!68}a^{6}-\frac{70\!\cdots\!31}{15\!\cdots\!76}a^{5}+\frac{13\!\cdots\!11}{73\!\cdots\!88}a^{4}-\frac{30\!\cdots\!59}{39\!\cdots\!44}a^{3}+\frac{23\!\cdots\!35}{44\!\cdots\!48}a^{2}+\frac{30\!\cdots\!81}{98\!\cdots\!36}a-\frac{27\!\cdots\!37}{45\!\cdots\!68}$, $\frac{10\!\cdots\!77}{20\!\cdots\!84}a^{31}-\frac{88\!\cdots\!11}{78\!\cdots\!88}a^{30}+\frac{97\!\cdots\!65}{15\!\cdots\!76}a^{29}+\frac{40\!\cdots\!97}{35\!\cdots\!04}a^{28}+\frac{31\!\cdots\!79}{50\!\cdots\!96}a^{27}-\frac{26\!\cdots\!01}{19\!\cdots\!72}a^{26}+\frac{29\!\cdots\!59}{39\!\cdots\!44}a^{25}+\frac{12\!\cdots\!47}{89\!\cdots\!76}a^{24}+\frac{22\!\cdots\!17}{10\!\cdots\!92}a^{23}-\frac{19\!\cdots\!43}{39\!\cdots\!44}a^{22}+\frac{21\!\cdots\!89}{78\!\cdots\!88}a^{21}+\frac{90\!\cdots\!57}{17\!\cdots\!52}a^{20}+\frac{46\!\cdots\!71}{20\!\cdots\!84}a^{19}-\frac{39\!\cdots\!73}{78\!\cdots\!88}a^{18}+\frac{43\!\cdots\!63}{15\!\cdots\!76}a^{17}+\frac{18\!\cdots\!19}{35\!\cdots\!04}a^{16}+\frac{13\!\cdots\!79}{11\!\cdots\!52}a^{15}-\frac{11\!\cdots\!05}{46\!\cdots\!64}a^{14}+\frac{12\!\cdots\!67}{92\!\cdots\!28}a^{13}+\frac{56\!\cdots\!87}{21\!\cdots\!12}a^{12}-\frac{97\!\cdots\!69}{20\!\cdots\!84}a^{11}+\frac{85\!\cdots\!71}{78\!\cdots\!88}a^{10}-\frac{94\!\cdots\!57}{15\!\cdots\!76}a^{9}-\frac{34\!\cdots\!97}{35\!\cdots\!04}a^{8}+\frac{84\!\cdots\!17}{20\!\cdots\!84}a^{7}-\frac{74\!\cdots\!07}{78\!\cdots\!88}a^{6}+\frac{80\!\cdots\!57}{15\!\cdots\!76}a^{5}+\frac{27\!\cdots\!93}{35\!\cdots\!04}a^{4}-\frac{38\!\cdots\!95}{12\!\cdots\!24}a^{3}+\frac{21\!\cdots\!29}{49\!\cdots\!68}a^{2}+\frac{25\!\cdots\!93}{98\!\cdots\!36}a-\frac{86\!\cdots\!11}{22\!\cdots\!44}$, $\frac{13\!\cdots\!53}{19\!\cdots\!72}a^{31}+\frac{94\!\cdots\!27}{78\!\cdots\!88}a^{30}+\frac{61\!\cdots\!41}{12\!\cdots\!92}a^{29}+\frac{372127308109879}{10\!\cdots\!56}a^{28}+\frac{41\!\cdots\!15}{49\!\cdots\!68}a^{27}+\frac{28\!\cdots\!33}{19\!\cdots\!72}a^{26}+\frac{18\!\cdots\!71}{30\!\cdots\!98}a^{25}+\frac{10\!\cdots\!57}{26\!\cdots\!64}a^{24}+\frac{30\!\cdots\!97}{98\!\cdots\!36}a^{23}+\frac{21\!\cdots\!15}{39\!\cdots\!44}a^{22}+\frac{13\!\cdots\!21}{61\!\cdots\!96}a^{21}+\frac{80\!\cdots\!19}{52\!\cdots\!28}a^{20}+\frac{62\!\cdots\!15}{19\!\cdots\!72}a^{19}+\frac{43\!\cdots\!01}{78\!\cdots\!88}a^{18}+\frac{28\!\cdots\!19}{12\!\cdots\!92}a^{17}+\frac{15\!\cdots\!05}{10\!\cdots\!56}a^{16}+\frac{18\!\cdots\!63}{11\!\cdots\!16}a^{15}+\frac{12\!\cdots\!01}{46\!\cdots\!64}a^{14}+\frac{85\!\cdots\!15}{72\!\cdots\!76}a^{13}+\frac{72\!\cdots\!49}{10\!\cdots\!56}a^{12}-\frac{12\!\cdots\!21}{19\!\cdots\!72}a^{11}-\frac{84\!\cdots\!31}{78\!\cdots\!88}a^{10}-\frac{53\!\cdots\!29}{12\!\cdots\!92}a^{9}-\frac{40\!\cdots\!07}{10\!\cdots\!56}a^{8}+\frac{11\!\cdots\!37}{19\!\cdots\!72}a^{7}+\frac{69\!\cdots\!99}{78\!\cdots\!88}a^{6}+\frac{46\!\cdots\!61}{12\!\cdots\!92}a^{5}+\frac{41\!\cdots\!31}{10\!\cdots\!56}a^{4}-\frac{53\!\cdots\!27}{15\!\cdots\!49}a^{3}-\frac{38\!\cdots\!73}{49\!\cdots\!68}a^{2}-\frac{39\!\cdots\!85}{15\!\cdots\!49}a-\frac{15\!\cdots\!29}{65\!\cdots\!16}$, $\frac{66\!\cdots\!35}{15\!\cdots\!76}a^{31}+\frac{65\!\cdots\!03}{39\!\cdots\!44}a^{29}-\frac{10\!\cdots\!81}{71\!\cdots\!08}a^{28}+\frac{20\!\cdots\!45}{39\!\cdots\!44}a^{27}+\frac{19\!\cdots\!97}{98\!\cdots\!36}a^{25}-\frac{31\!\cdots\!67}{17\!\cdots\!52}a^{24}+\frac{14\!\cdots\!47}{78\!\cdots\!88}a^{23}+\frac{14\!\cdots\!07}{19\!\cdots\!72}a^{21}-\frac{23\!\cdots\!81}{35\!\cdots\!04}a^{20}+\frac{30\!\cdots\!29}{15\!\cdots\!76}a^{19}+\frac{29\!\cdots\!13}{39\!\cdots\!44}a^{17}-\frac{46\!\cdots\!95}{71\!\cdots\!08}a^{16}+\frac{88\!\cdots\!81}{92\!\cdots\!28}a^{15}+\frac{85\!\cdots\!89}{23\!\cdots\!32}a^{13}-\frac{13\!\cdots\!83}{42\!\cdots\!24}a^{12}-\frac{60\!\cdots\!31}{15\!\cdots\!76}a^{11}-\frac{60\!\cdots\!55}{39\!\cdots\!44}a^{9}+\frac{99\!\cdots\!57}{71\!\cdots\!08}a^{8}+\frac{49\!\cdots\!59}{15\!\cdots\!76}a^{7}+\frac{52\!\cdots\!23}{39\!\cdots\!44}a^{5}-\frac{86\!\cdots\!93}{71\!\cdots\!08}a^{4}-\frac{22\!\cdots\!01}{98\!\cdots\!36}a^{3}-\frac{22\!\cdots\!37}{24\!\cdots\!84}a+\frac{27\!\cdots\!63}{44\!\cdots\!88}$, $\frac{12\!\cdots\!77}{78\!\cdots\!88}a^{31}+\frac{32\!\cdots\!71}{50\!\cdots\!96}a^{30}+\frac{45\!\cdots\!47}{15\!\cdots\!76}a^{29}+\frac{10\!\cdots\!41}{98\!\cdots\!36}a^{28}+\frac{38\!\cdots\!87}{19\!\cdots\!72}a^{27}+\frac{97\!\cdots\!37}{12\!\cdots\!24}a^{26}+\frac{13\!\cdots\!57}{39\!\cdots\!44}a^{25}+\frac{30\!\cdots\!99}{24\!\cdots\!84}a^{24}+\frac{28\!\cdots\!45}{39\!\cdots\!44}a^{23}+\frac{71\!\cdots\!59}{25\!\cdots\!48}a^{22}+\frac{10\!\cdots\!27}{78\!\cdots\!88}a^{21}+\frac{22\!\cdots\!53}{49\!\cdots\!68}a^{20}+\frac{57\!\cdots\!47}{78\!\cdots\!88}a^{19}+\frac{14\!\cdots\!21}{50\!\cdots\!96}a^{18}+\frac{20\!\cdots\!85}{15\!\cdots\!76}a^{17}+\frac{45\!\cdots\!15}{98\!\cdots\!36}a^{16}+\frac{16\!\cdots\!63}{46\!\cdots\!64}a^{15}+\frac{42\!\cdots\!17}{29\!\cdots\!88}a^{14}+\frac{59\!\cdots\!85}{92\!\cdots\!28}a^{13}+\frac{13\!\cdots\!07}{57\!\cdots\!08}a^{12}-\frac{11\!\cdots\!37}{78\!\cdots\!88}a^{11}-\frac{30\!\cdots\!23}{50\!\cdots\!96}a^{10}-\frac{42\!\cdots\!07}{15\!\cdots\!76}a^{9}-\frac{97\!\cdots\!21}{98\!\cdots\!36}a^{8}+\frac{10\!\cdots\!41}{78\!\cdots\!88}a^{7}+\frac{26\!\cdots\!35}{50\!\cdots\!96}a^{6}+\frac{38\!\cdots\!55}{15\!\cdots\!76}a^{5}+\frac{87\!\cdots\!49}{98\!\cdots\!36}a^{4}-\frac{48\!\cdots\!39}{49\!\cdots\!68}a^{3}-\frac{13\!\cdots\!33}{31\!\cdots\!56}a^{2}-\frac{21\!\cdots\!29}{98\!\cdots\!36}a-\frac{58\!\cdots\!55}{61\!\cdots\!96}$
| 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: | \( 11215015279939.906 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 11215015279939.906 \cdot 208}{8\cdot\sqrt{30472057165568685620821441442859880284160000000000000000}}\cr\approx \mathstrut & 0.311673748393457 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 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 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 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 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 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 + 12*x^28 + 442*x^24 + 4491*x^20 + 22291*x^16 - 93453*x^12 + 82801*x^8 - 8160*x^4 + 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 | ${\href{/padicField/7.8.0.1}{8} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| Deg $16$ | $4$ | $4$ | $32$ | ||||
|
\(5\)
| 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 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(89\)
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(229\)
| 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 $4$ | $2$ | $2$ | $2$ |