Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*x^4 + 256)
gp: K = bnfinit(y^32 - 29*y^28 + 707*y^24 - 9485*y^20 + 58946*y^16 - 110740*y^12 + 61457*y^8 + 6896*y^4 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*x^4 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*x^4 + 256)
\( x^{32} - 29x^{28} + 707x^{24} - 9485x^{20} + 58946x^{16} - 110740x^{12} + 61457x^{8} + 6896x^{4} + 256 \)
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: |
\(11892526956757459821809170921231545794560000000000000000\)
\(\medspace = 2^{64}\cdot 5^{16}\cdot 89^{8}\cdot 181^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.61\) | 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: | $2^{2}5^{1/2}89^{1/2}181^{1/2}\approx 1135.218040730502$ | ||
| Ramified primes: |
\(2\), \(5\), \(89\), \(181\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{225}a^{24}-\frac{71}{225}a^{20}+\frac{88}{225}a^{16}-\frac{4}{15}a^{12}-\frac{47}{225}a^{8}-\frac{31}{225}a^{4}+\frac{31}{225}$, $\frac{1}{450}a^{25}-\frac{71}{450}a^{21}-\frac{137}{450}a^{17}+\frac{11}{30}a^{13}+\frac{89}{225}a^{9}+\frac{97}{225}a^{5}+\frac{31}{450}a$, $\frac{1}{900}a^{26}+\frac{379}{900}a^{22}-\frac{137}{900}a^{18}-\frac{19}{60}a^{14}+\frac{89}{450}a^{10}-\frac{64}{225}a^{6}-\frac{419}{900}a^{2}$, $\frac{1}{1800}a^{27}+\frac{379}{1800}a^{23}+\frac{763}{1800}a^{19}-\frac{19}{120}a^{15}+\frac{89}{900}a^{11}+\frac{161}{450}a^{7}+\frac{481}{1800}a^{3}$, $\frac{1}{14\!\cdots\!00}a^{28}-\frac{574295656225871}{47\!\cdots\!00}a^{24}-\frac{14\!\cdots\!69}{31\!\cdots\!80}a^{20}+\frac{53\!\cdots\!19}{14\!\cdots\!00}a^{16}-\frac{13\!\cdots\!51}{71\!\cdots\!00}a^{12}-\frac{97\!\cdots\!33}{35\!\cdots\!00}a^{8}+\frac{50\!\cdots\!33}{14\!\cdots\!00}a^{4}+\frac{33\!\cdots\!03}{88\!\cdots\!75}$, $\frac{1}{28\!\cdots\!00}a^{29}-\frac{574295656225871}{94\!\cdots\!00}a^{25}+\frac{17\!\cdots\!11}{63\!\cdots\!60}a^{21}+\frac{53\!\cdots\!19}{28\!\cdots\!00}a^{17}+\frac{58\!\cdots\!49}{14\!\cdots\!00}a^{13}-\frac{97\!\cdots\!33}{71\!\cdots\!00}a^{9}+\frac{50\!\cdots\!33}{28\!\cdots\!00}a^{5}-\frac{27\!\cdots\!36}{88\!\cdots\!75}a$, $\frac{1}{56\!\cdots\!00}a^{30}-\frac{574295656225871}{18\!\cdots\!00}a^{26}-\frac{45\!\cdots\!49}{12\!\cdots\!20}a^{22}-\frac{23\!\cdots\!81}{56\!\cdots\!00}a^{18}+\frac{58\!\cdots\!49}{28\!\cdots\!00}a^{14}-\frac{97\!\cdots\!33}{14\!\cdots\!00}a^{10}+\frac{50\!\cdots\!33}{56\!\cdots\!00}a^{6}+\frac{61\!\cdots\!39}{17\!\cdots\!50}a^{2}$, $\frac{1}{11\!\cdots\!00}a^{31}-\frac{574295656225871}{37\!\cdots\!00}a^{27}-\frac{45\!\cdots\!49}{25\!\cdots\!40}a^{23}+\frac{33\!\cdots\!19}{11\!\cdots\!00}a^{19}-\frac{22\!\cdots\!51}{56\!\cdots\!00}a^{15}-\frac{97\!\cdots\!33}{28\!\cdots\!00}a^{11}+\frac{50\!\cdots\!33}{11\!\cdots\!00}a^{7}-\frac{11\!\cdots\!11}{35\!\cdots\!00}a^{3}$
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: |
\( -\frac{95670519029435}{113881633452849888} a^{29} + \frac{2773423963376591}{113881633452849888} a^{25} - \frac{67616374761852145}{113881633452849888} a^{21} + \frac{906906858493250815}{113881633452849888} a^{17} - \frac{2817220008577322395}{56940816726424944} a^{13} + \frac{882973489127041885}{9490136121070824} a^{9} - \frac{680302984930996979}{12653514828094432} a^{5} - \frac{18127430249067545}{7117602090803118} a \)
(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: |
$\frac{85667441749809}{31\!\cdots\!00}a^{30}-\frac{501805128286321}{63\!\cdots\!60}a^{26}+\frac{61\!\cdots\!03}{31\!\cdots\!00}a^{22}-\frac{82\!\cdots\!73}{31\!\cdots\!00}a^{18}+\frac{26\!\cdots\!81}{15\!\cdots\!00}a^{14}-\frac{26\!\cdots\!13}{79\!\cdots\!00}a^{10}+\frac{23\!\cdots\!77}{12\!\cdots\!32}a^{6}+\frac{13\!\cdots\!13}{39\!\cdots\!00}a^{2}$, $\frac{37455946692299}{12\!\cdots\!32}a^{31}-\frac{5434308885}{9517498930496}a^{30}-\frac{98\!\cdots\!63}{11\!\cdots\!88}a^{27}+\frac{156840495697}{9517498930496}a^{26}+\frac{23\!\cdots\!45}{11\!\cdots\!88}a^{23}-\frac{3820912934063}{9517498930496}a^{22}-\frac{32\!\cdots\!79}{11\!\cdots\!88}a^{19}+\frac{51031445232481}{9517498930496}a^{18}+\frac{33\!\cdots\!67}{18\!\cdots\!48}a^{15}-\frac{156833764491053}{4758749465248}a^{14}-\frac{96\!\cdots\!77}{28\!\cdots\!72}a^{11}+\frac{140914469452833}{2379374732624}a^{10}+\frac{23\!\cdots\!31}{11\!\cdots\!88}a^{7}-\frac{285036394765029}{9517498930496}a^{6}+\frac{20\!\cdots\!89}{28\!\cdots\!72}a^{3}-\frac{823061982998}{148710920789}a^{2}-1$, $\frac{21\!\cdots\!09}{11\!\cdots\!00}a^{31}+\frac{5434308885}{9517498930496}a^{30}-\frac{12\!\cdots\!53}{22\!\cdots\!60}a^{27}-\frac{156840495697}{9517498930496}a^{26}+\frac{14\!\cdots\!63}{11\!\cdots\!00}a^{23}+\frac{3820912934063}{9517498930496}a^{22}-\frac{20\!\cdots\!53}{11\!\cdots\!00}a^{19}-\frac{51031445232481}{9517498930496}a^{18}+\frac{62\!\cdots\!81}{56\!\cdots\!00}a^{15}+\frac{156833764491053}{4758749465248}a^{14}-\frac{66\!\cdots\!37}{31\!\cdots\!00}a^{11}-\frac{140914469452833}{2379374732624}a^{10}+\frac{96\!\cdots\!59}{75\!\cdots\!20}a^{7}+\frac{285036394765029}{9517498930496}a^{6}+\frac{32\!\cdots\!09}{71\!\cdots\!00}a^{3}+\frac{823061982998}{148710920789}a^{2}+1$, $\frac{31620569227867}{14\!\cdots\!00}a^{28}-\frac{298039798956133}{47\!\cdots\!00}a^{24}+\frac{72\!\cdots\!51}{47\!\cdots\!00}a^{20}-\frac{28\!\cdots\!91}{14\!\cdots\!00}a^{16}+\frac{83\!\cdots\!23}{71\!\cdots\!00}a^{12}-\frac{58\!\cdots\!07}{35\!\cdots\!00}a^{8}-\frac{24\!\cdots\!21}{14\!\cdots\!00}a^{4}+\frac{19\!\cdots\!53}{88\!\cdots\!75}$, $\frac{11975084690971}{47\!\cdots\!00}a^{28}-\frac{10\!\cdots\!77}{14\!\cdots\!00}a^{24}+\frac{24\!\cdots\!03}{14\!\cdots\!00}a^{20}-\frac{32\!\cdots\!57}{14\!\cdots\!00}a^{16}+\frac{10\!\cdots\!53}{79\!\cdots\!00}a^{12}-\frac{13\!\cdots\!97}{71\!\cdots\!80}a^{8}-\frac{28\!\cdots\!23}{14\!\cdots\!00}a^{4}-\frac{19\!\cdots\!64}{88\!\cdots\!75}$, $\frac{40\!\cdots\!97}{56\!\cdots\!00}a^{30}+\frac{6265173901031}{28\!\cdots\!72}a^{28}-\frac{11\!\cdots\!17}{56\!\cdots\!00}a^{26}-\frac{180542123506027}{28\!\cdots\!72}a^{24}+\frac{28\!\cdots\!91}{56\!\cdots\!00}a^{22}+\frac{43\!\cdots\!61}{28\!\cdots\!72}a^{20}-\frac{25\!\cdots\!19}{37\!\cdots\!60}a^{18}-\frac{19\!\cdots\!93}{94\!\cdots\!24}a^{16}+\frac{12\!\cdots\!33}{28\!\cdots\!00}a^{14}+\frac{18\!\cdots\!71}{14\!\cdots\!36}a^{12}-\frac{11\!\cdots\!93}{14\!\cdots\!00}a^{10}-\frac{16\!\cdots\!59}{71\!\cdots\!18}a^{8}+\frac{28\!\cdots\!37}{56\!\cdots\!00}a^{6}+\frac{37\!\cdots\!95}{28\!\cdots\!72}a^{4}+\frac{42\!\cdots\!83}{23\!\cdots\!60}a^{2}+\frac{299647573538392}{395422338377951}$, $\frac{50\!\cdots\!07}{11\!\cdots\!00}a^{31}-\frac{149883517665851}{22\!\cdots\!76}a^{30}-\frac{22\!\cdots\!21}{28\!\cdots\!00}a^{29}+\frac{277700548658003}{47\!\cdots\!00}a^{28}-\frac{14\!\cdots\!83}{11\!\cdots\!00}a^{27}+\frac{10\!\cdots\!59}{56\!\cdots\!00}a^{26}+\frac{13\!\cdots\!73}{56\!\cdots\!40}a^{25}-\frac{24\!\cdots\!01}{14\!\cdots\!00}a^{24}+\frac{36\!\cdots\!97}{11\!\cdots\!00}a^{23}-\frac{26\!\cdots\!89}{56\!\cdots\!00}a^{22}-\frac{16\!\cdots\!27}{28\!\cdots\!00}a^{21}+\frac{59\!\cdots\!19}{14\!\cdots\!00}a^{20}-\frac{16\!\cdots\!21}{37\!\cdots\!00}a^{19}+\frac{39\!\cdots\!63}{63\!\cdots\!00}a^{18}+\frac{21\!\cdots\!97}{28\!\cdots\!00}a^{17}-\frac{80\!\cdots\!21}{14\!\cdots\!00}a^{16}+\frac{15\!\cdots\!03}{56\!\cdots\!00}a^{15}-\frac{21\!\cdots\!19}{56\!\cdots\!40}a^{14}-\frac{67\!\cdots\!89}{14\!\cdots\!00}a^{13}+\frac{83\!\cdots\!87}{23\!\cdots\!00}a^{12}-\frac{57\!\cdots\!35}{11\!\cdots\!88}a^{11}+\frac{10\!\cdots\!63}{14\!\cdots\!00}a^{10}+\frac{21\!\cdots\!79}{23\!\cdots\!00}a^{9}-\frac{49\!\cdots\!77}{71\!\cdots\!80}a^{8}+\frac{34\!\cdots\!83}{11\!\cdots\!00}a^{7}-\frac{20\!\cdots\!79}{56\!\cdots\!00}a^{6}-\frac{34\!\cdots\!81}{63\!\cdots\!60}a^{5}+\frac{60\!\cdots\!01}{14\!\cdots\!00}a^{4}+\frac{21\!\cdots\!29}{11\!\cdots\!00}a^{3}-\frac{80\!\cdots\!27}{11\!\cdots\!00}a^{2}-\frac{16\!\cdots\!41}{17\!\cdots\!50}a+\frac{19\!\cdots\!83}{88\!\cdots\!75}$, $\frac{532580566629993}{63\!\cdots\!00}a^{31}-\frac{5434308885}{9517498930496}a^{30}+\frac{523068579011831}{28\!\cdots\!00}a^{29}-\frac{30\!\cdots\!57}{12\!\cdots\!20}a^{27}+\frac{156840495697}{9517498930496}a^{26}-\frac{15\!\cdots\!03}{28\!\cdots\!00}a^{25}+\frac{37\!\cdots\!31}{63\!\cdots\!00}a^{23}-\frac{3820912934063}{9517498930496}a^{22}+\frac{74\!\cdots\!49}{56\!\cdots\!40}a^{21}-\frac{50\!\cdots\!21}{63\!\cdots\!00}a^{19}+\frac{51031445232481}{9517498930496}a^{18}-\frac{55\!\cdots\!79}{31\!\cdots\!00}a^{17}+\frac{15\!\cdots\!37}{31\!\cdots\!00}a^{15}-\frac{156833764491053}{4758749465248}a^{14}+\frac{15\!\cdots\!19}{14\!\cdots\!00}a^{13}-\frac{14\!\cdots\!01}{15\!\cdots\!00}a^{11}+\frac{140914469452833}{2379374732624}a^{10}-\frac{15\!\cdots\!23}{71\!\cdots\!00}a^{9}+\frac{12\!\cdots\!65}{25\!\cdots\!64}a^{7}-\frac{285036394765029}{9517498930496}a^{6}+\frac{34\!\cdots\!23}{28\!\cdots\!00}a^{5}+\frac{35\!\cdots\!13}{39\!\cdots\!00}a^{3}-\frac{823061982998}{148710920789}a^{2}+\frac{64\!\cdots\!78}{29\!\cdots\!25}a$, $\frac{21\!\cdots\!09}{11\!\cdots\!00}a^{31}-\frac{380600133343607}{47\!\cdots\!00}a^{30}-\frac{11\!\cdots\!19}{28\!\cdots\!00}a^{29}-\frac{12\!\cdots\!53}{22\!\cdots\!60}a^{27}+\frac{33\!\cdots\!29}{14\!\cdots\!00}a^{26}+\frac{70\!\cdots\!79}{56\!\cdots\!40}a^{25}+\frac{14\!\cdots\!63}{11\!\cdots\!00}a^{23}-\frac{81\!\cdots\!71}{14\!\cdots\!00}a^{22}-\frac{85\!\cdots\!13}{28\!\cdots\!00}a^{21}-\frac{20\!\cdots\!53}{11\!\cdots\!00}a^{19}+\frac{10\!\cdots\!29}{14\!\cdots\!00}a^{18}+\frac{38\!\cdots\!21}{94\!\cdots\!00}a^{17}+\frac{62\!\cdots\!81}{56\!\cdots\!00}a^{15}-\frac{11\!\cdots\!53}{23\!\cdots\!00}a^{14}-\frac{36\!\cdots\!71}{14\!\cdots\!00}a^{13}-\frac{66\!\cdots\!37}{31\!\cdots\!00}a^{11}+\frac{67\!\cdots\!27}{71\!\cdots\!80}a^{10}+\frac{37\!\cdots\!63}{71\!\cdots\!00}a^{9}+\frac{96\!\cdots\!59}{75\!\cdots\!20}a^{7}-\frac{73\!\cdots\!29}{14\!\cdots\!00}a^{6}-\frac{19\!\cdots\!23}{56\!\cdots\!40}a^{5}+\frac{32\!\cdots\!09}{71\!\cdots\!00}a^{3}-\frac{33\!\cdots\!93}{35\!\cdots\!00}a^{2}-\frac{47\!\cdots\!19}{29\!\cdots\!25}a$, $\frac{22\!\cdots\!83}{12\!\cdots\!00}a^{31}+\frac{626669964860191}{56\!\cdots\!40}a^{30}+\frac{95670519029435}{11\!\cdots\!88}a^{29}+\frac{349183949124463}{14\!\cdots\!00}a^{28}-\frac{59\!\cdots\!51}{11\!\cdots\!00}a^{27}-\frac{90\!\cdots\!43}{28\!\cdots\!00}a^{26}-\frac{27\!\cdots\!91}{11\!\cdots\!88}a^{25}-\frac{11\!\cdots\!03}{15\!\cdots\!00}a^{24}+\frac{29\!\cdots\!61}{22\!\cdots\!60}a^{23}+\frac{22\!\cdots\!13}{28\!\cdots\!00}a^{22}+\frac{67\!\cdots\!45}{11\!\cdots\!88}a^{21}+\frac{81\!\cdots\!51}{47\!\cdots\!00}a^{20}-\frac{19\!\cdots\!27}{11\!\cdots\!00}a^{19}-\frac{98\!\cdots\!73}{94\!\cdots\!00}a^{18}-\frac{90\!\cdots\!15}{11\!\cdots\!88}a^{17}-\frac{32\!\cdots\!07}{14\!\cdots\!00}a^{16}+\frac{20\!\cdots\!01}{18\!\cdots\!00}a^{15}+\frac{18\!\cdots\!27}{28\!\cdots\!20}a^{14}+\frac{28\!\cdots\!95}{56\!\cdots\!44}a^{13}+\frac{99\!\cdots\!27}{71\!\cdots\!00}a^{12}-\frac{56\!\cdots\!31}{28\!\cdots\!00}a^{11}-\frac{85\!\cdots\!11}{71\!\cdots\!00}a^{10}-\frac{88\!\cdots\!85}{94\!\cdots\!24}a^{9}-\frac{17\!\cdots\!87}{71\!\cdots\!80}a^{8}+\frac{12\!\cdots\!91}{11\!\cdots\!00}a^{7}+\frac{19\!\cdots\!83}{28\!\cdots\!00}a^{6}+\frac{68\!\cdots\!79}{12\!\cdots\!32}a^{5}+\frac{18\!\cdots\!27}{14\!\cdots\!00}a^{4}+\frac{32\!\cdots\!63}{35\!\cdots\!00}a^{3}+\frac{16\!\cdots\!01}{39\!\cdots\!00}a^{2}+\frac{11\!\cdots\!27}{71\!\cdots\!18}a-\frac{34\!\cdots\!89}{88\!\cdots\!75}$, $\frac{489485542427291}{31\!\cdots\!00}a^{31}+\frac{23670000045669}{19\!\cdots\!50}a^{29}-\frac{25\!\cdots\!91}{56\!\cdots\!40}a^{27}-\frac{20\!\cdots\!73}{59\!\cdots\!50}a^{25}+\frac{31\!\cdots\!73}{28\!\cdots\!00}a^{23}+\frac{50\!\cdots\!87}{59\!\cdots\!50}a^{21}-\frac{42\!\cdots\!43}{28\!\cdots\!00}a^{19}-\frac{67\!\cdots\!33}{59\!\cdots\!50}a^{17}+\frac{44\!\cdots\!07}{47\!\cdots\!00}a^{15}+\frac{69\!\cdots\!51}{98\!\cdots\!75}a^{13}-\frac{13\!\cdots\!33}{71\!\cdots\!00}a^{11}-\frac{77\!\cdots\!12}{59\!\cdots\!65}a^{9}+\frac{14\!\cdots\!39}{11\!\cdots\!88}a^{7}+\frac{43\!\cdots\!73}{59\!\cdots\!50}a^{5}-\frac{50\!\cdots\!09}{71\!\cdots\!00}a^{3}+\frac{18\!\cdots\!97}{29\!\cdots\!25}a$, $\frac{31\!\cdots\!83}{11\!\cdots\!00}a^{31}+\frac{117869163402881}{37\!\cdots\!60}a^{30}-\frac{703869832185419}{28\!\cdots\!00}a^{29}-\frac{662442644176013}{14\!\cdots\!00}a^{28}-\frac{10\!\cdots\!07}{12\!\cdots\!00}a^{27}-\frac{50\!\cdots\!39}{56\!\cdots\!00}a^{26}+\frac{20\!\cdots\!11}{28\!\cdots\!00}a^{25}+\frac{19\!\cdots\!77}{14\!\cdots\!00}a^{24}+\frac{74\!\cdots\!83}{37\!\cdots\!00}a^{23}+\frac{12\!\cdots\!49}{56\!\cdots\!00}a^{22}-\frac{49\!\cdots\!49}{28\!\cdots\!00}a^{21}-\frac{46\!\cdots\!03}{14\!\cdots\!00}a^{20}-\frac{60\!\cdots\!83}{22\!\cdots\!60}a^{19}-\frac{16\!\cdots\!87}{56\!\cdots\!00}a^{18}+\frac{22\!\cdots\!57}{94\!\cdots\!00}a^{17}+\frac{61\!\cdots\!57}{14\!\cdots\!00}a^{16}+\frac{93\!\cdots\!87}{56\!\cdots\!00}a^{15}+\frac{33\!\cdots\!37}{18\!\cdots\!80}a^{14}-\frac{21\!\cdots\!51}{14\!\cdots\!00}a^{13}-\frac{18\!\cdots\!77}{71\!\cdots\!00}a^{12}-\frac{89\!\cdots\!27}{28\!\cdots\!00}a^{11}-\frac{45\!\cdots\!03}{14\!\cdots\!00}a^{10}+\frac{83\!\cdots\!87}{28\!\cdots\!72}a^{9}+\frac{11\!\cdots\!59}{23\!\cdots\!60}a^{8}+\frac{20\!\cdots\!43}{11\!\cdots\!00}a^{7}+\frac{90\!\cdots\!59}{56\!\cdots\!00}a^{6}-\frac{56\!\cdots\!11}{28\!\cdots\!00}a^{5}-\frac{41\!\cdots\!53}{15\!\cdots\!00}a^{4}+\frac{21\!\cdots\!01}{14\!\cdots\!60}a^{3}+\frac{10\!\cdots\!41}{35\!\cdots\!00}a^{2}+\frac{17\!\cdots\!63}{19\!\cdots\!50}a-\frac{12\!\cdots\!86}{88\!\cdots\!75}$, $\frac{29\!\cdots\!03}{11\!\cdots\!00}a^{31}-\frac{18\!\cdots\!39}{11\!\cdots\!80}a^{30}+\frac{129574321822297}{56\!\cdots\!40}a^{29}+\frac{402570233757223}{14\!\cdots\!00}a^{28}-\frac{28\!\cdots\!33}{37\!\cdots\!00}a^{27}+\frac{26\!\cdots\!43}{56\!\cdots\!00}a^{26}-\frac{18\!\cdots\!13}{28\!\cdots\!00}a^{25}-\frac{11\!\cdots\!71}{14\!\cdots\!00}a^{24}+\frac{45\!\cdots\!41}{25\!\cdots\!40}a^{23}-\frac{65\!\cdots\!93}{56\!\cdots\!00}a^{22}+\frac{45\!\cdots\!43}{28\!\cdots\!00}a^{21}+\frac{28\!\cdots\!97}{14\!\cdots\!00}a^{20}-\frac{27\!\cdots\!23}{11\!\cdots\!00}a^{19}+\frac{87\!\cdots\!99}{56\!\cdots\!00}a^{18}-\frac{68\!\cdots\!21}{31\!\cdots\!00}a^{17}-\frac{13\!\cdots\!33}{47\!\cdots\!00}a^{16}+\frac{86\!\cdots\!47}{56\!\cdots\!00}a^{15}-\frac{54\!\cdots\!59}{56\!\cdots\!40}a^{14}+\frac{38\!\cdots\!41}{28\!\cdots\!20}a^{13}+\frac{12\!\cdots\!87}{71\!\cdots\!00}a^{12}-\frac{83\!\cdots\!19}{28\!\cdots\!00}a^{11}+\frac{88\!\cdots\!97}{47\!\cdots\!00}a^{10}-\frac{17\!\cdots\!61}{71\!\cdots\!00}a^{9}-\frac{13\!\cdots\!63}{35\!\cdots\!00}a^{8}+\frac{20\!\cdots\!59}{11\!\cdots\!00}a^{7}-\frac{21\!\cdots\!61}{18\!\cdots\!00}a^{6}+\frac{33\!\cdots\!53}{28\!\cdots\!00}a^{5}+\frac{35\!\cdots\!91}{14\!\cdots\!00}a^{4}+\frac{45\!\cdots\!99}{71\!\cdots\!00}a^{3}-\frac{59\!\cdots\!33}{88\!\cdots\!75}a^{2}+\frac{11\!\cdots\!92}{29\!\cdots\!25}a+\frac{78\!\cdots\!13}{98\!\cdots\!75}$, $\frac{17\!\cdots\!29}{56\!\cdots\!40}a^{31}+\frac{29\!\cdots\!01}{18\!\cdots\!00}a^{30}+\frac{375660210101377}{14\!\cdots\!00}a^{29}-\frac{129682195025081}{35\!\cdots\!00}a^{28}-\frac{85\!\cdots\!83}{94\!\cdots\!00}a^{27}-\frac{25\!\cdots\!79}{56\!\cdots\!00}a^{26}-\frac{21\!\cdots\!61}{28\!\cdots\!20}a^{25}+\frac{417001480613977}{39\!\cdots\!00}a^{24}+\frac{20\!\cdots\!73}{94\!\cdots\!00}a^{23}+\frac{25\!\cdots\!81}{22\!\cdots\!76}a^{22}+\frac{26\!\cdots\!99}{14\!\cdots\!00}a^{21}-\frac{60\!\cdots\!09}{23\!\cdots\!60}a^{20}-\frac{83\!\cdots\!77}{28\!\cdots\!00}a^{19}-\frac{85\!\cdots\!63}{56\!\cdots\!00}a^{18}-\frac{11\!\cdots\!63}{47\!\cdots\!00}a^{17}+\frac{12\!\cdots\!81}{35\!\cdots\!00}a^{16}+\frac{10\!\cdots\!99}{56\!\cdots\!44}a^{15}+\frac{88\!\cdots\!49}{94\!\cdots\!00}a^{14}+\frac{10\!\cdots\!93}{71\!\cdots\!00}a^{13}-\frac{37\!\cdots\!69}{17\!\cdots\!50}a^{12}-\frac{24\!\cdots\!83}{71\!\cdots\!00}a^{11}-\frac{25\!\cdots\!79}{14\!\cdots\!00}a^{10}-\frac{92\!\cdots\!69}{35\!\cdots\!00}a^{9}+\frac{35\!\cdots\!28}{88\!\cdots\!75}a^{8}+\frac{57\!\cdots\!69}{28\!\cdots\!00}a^{7}+\frac{58\!\cdots\!39}{56\!\cdots\!00}a^{6}+\frac{36\!\cdots\!13}{28\!\cdots\!20}a^{5}-\frac{81\!\cdots\!13}{35\!\cdots\!00}a^{4}+\frac{10\!\cdots\!69}{71\!\cdots\!00}a^{3}+\frac{25\!\cdots\!19}{35\!\cdots\!00}a^{2}+\frac{23\!\cdots\!38}{98\!\cdots\!75}a-\frac{11\!\cdots\!12}{88\!\cdots\!75}$, $\frac{100055454734174}{17\!\cdots\!95}a^{31}-\frac{18\!\cdots\!07}{14\!\cdots\!00}a^{30}-\frac{60\!\cdots\!63}{28\!\cdots\!00}a^{29}-\frac{173697474333817}{14\!\cdots\!60}a^{28}-\frac{11\!\cdots\!27}{71\!\cdots\!00}a^{27}+\frac{17\!\cdots\!17}{47\!\cdots\!00}a^{26}+\frac{17\!\cdots\!79}{28\!\cdots\!00}a^{25}+\frac{50\!\cdots\!53}{14\!\cdots\!60}a^{24}+\frac{28\!\cdots\!87}{71\!\cdots\!00}a^{23}-\frac{17\!\cdots\!87}{18\!\cdots\!48}a^{22}-\frac{85\!\cdots\!09}{56\!\cdots\!40}a^{21}-\frac{12\!\cdots\!87}{14\!\cdots\!60}a^{20}-\frac{38\!\cdots\!21}{71\!\cdots\!00}a^{19}+\frac{17\!\cdots\!47}{14\!\cdots\!00}a^{18}+\frac{19\!\cdots\!61}{94\!\cdots\!00}a^{17}+\frac{18\!\cdots\!37}{15\!\cdots\!40}a^{16}+\frac{95\!\cdots\!19}{28\!\cdots\!72}a^{15}-\frac{54\!\cdots\!93}{71\!\cdots\!00}a^{14}-\frac{18\!\cdots\!87}{14\!\cdots\!00}a^{13}-\frac{51\!\cdots\!33}{71\!\cdots\!80}a^{12}-\frac{25\!\cdots\!27}{39\!\cdots\!00}a^{11}+\frac{51\!\cdots\!01}{35\!\cdots\!00}a^{10}+\frac{17\!\cdots\!99}{71\!\cdots\!00}a^{9}+\frac{99\!\cdots\!31}{71\!\cdots\!18}a^{8}+\frac{23\!\cdots\!01}{59\!\cdots\!50}a^{7}-\frac{11\!\cdots\!91}{14\!\cdots\!00}a^{6}-\frac{41\!\cdots\!39}{28\!\cdots\!00}a^{5}-\frac{11\!\cdots\!53}{14\!\cdots\!60}a^{4}+\frac{11\!\cdots\!73}{71\!\cdots\!00}a^{3}-\frac{39\!\cdots\!69}{35\!\cdots\!00}a^{2}-\frac{25\!\cdots\!81}{19\!\cdots\!50}a-\frac{38\!\cdots\!21}{59\!\cdots\!65}$
| 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: | \( 11519211186268.066 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 11519211186268.066 \cdot 208}{8\cdot\sqrt{11892526956757459821809170921231545794560000000000000000}}\cr\approx \mathstrut & 0.512432787535290 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*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 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*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 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*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 - 29*x^28 + 707*x^24 - 9485*x^20 + 58946*x^16 - 110740*x^12 + 61457*x^8 + 6896*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 | ${\href{/padicField/7.8.0.1}{8} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 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 $16$ | $4$ | $4$ | $32$ | ||||
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(89\)
| 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 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(181\)
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |