Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 44*x^30 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*x^2 + 256)
gp: K = bnfinit(y^32 - 44*y^30 + 874*y^28 - 10054*y^26 + 71163*y^24 - 296956*y^22 + 554793*y^20 + 537460*y^18 - 3539799*y^16 - 66968*y^14 + 9372831*y^12 + 10803870*y^10 + 6888129*y^8 + 2624028*y^6 + 464976*y^4 + 14784*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 44*x^30 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*x^2 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 44*x^30 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*x^2 + 256)
\( x^{32} - 44 x^{30} + 874 x^{28} - 10054 x^{26} + 71163 x^{24} - 296956 x^{22} + 554793 x^{20} + \cdots + 256 \)
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: |
\(13192724911186950673358584730276205272104960000000000000000\)
\(\medspace = 2^{32}\cdot 5^{16}\cdot 7^{16}\cdot 29^{8}\cdot 1049^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.50\) | 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: | $2\cdot 5^{1/2}7^{1/2}29^{1/2}1049^{1/2}\approx 2063.7199422402255$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(29\), \(1049\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{14}+\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{16}+\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}+\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{10}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{11}-\frac{1}{2}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{21}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{22}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{23}-\frac{1}{8}a^{19}-\frac{1}{8}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{24}-\frac{1}{8}a^{22}+\frac{1}{16}a^{20}-\frac{1}{8}a^{18}-\frac{1}{8}a^{16}-\frac{1}{8}a^{14}-\frac{1}{16}a^{12}-\frac{3}{8}a^{8}-\frac{1}{4}a^{6}-\frac{3}{16}a^{4}$, $\frac{1}{16}a^{25}+\frac{1}{16}a^{21}-\frac{1}{8}a^{17}-\frac{1}{8}a^{15}-\frac{1}{16}a^{13}+\frac{1}{8}a^{11}+\frac{3}{8}a^{9}-\frac{1}{2}a^{7}+\frac{5}{16}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{26}+\frac{1}{16}a^{22}-\frac{1}{8}a^{18}-\frac{1}{8}a^{16}-\frac{1}{16}a^{14}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{7}{16}a^{6}-\frac{3}{8}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{27}-\frac{1}{32}a^{25}-\frac{1}{32}a^{23}-\frac{1}{32}a^{21}-\frac{1}{8}a^{17}-\frac{3}{32}a^{15}+\frac{3}{32}a^{13}+\frac{1}{16}a^{11}-\frac{1}{16}a^{9}+\frac{9}{32}a^{7}+\frac{1}{32}a^{5}-\frac{3}{8}a^{3}$, $\frac{1}{1935808}a^{28}+\frac{20075}{1935808}a^{26}+\frac{8795}{1935808}a^{24}-\frac{219293}{1935808}a^{22}+\frac{6723}{483952}a^{20}+\frac{2241}{483952}a^{18}+\frac{1345}{12992}a^{16}+\frac{135255}{1935808}a^{14}-\frac{4183}{138272}a^{12}-\frac{244157}{967904}a^{10}+\frac{711569}{1935808}a^{8}-\frac{87525}{276544}a^{6}+\frac{668}{4321}a^{4}-\frac{6291}{60494}a^{2}-\frac{5892}{30247}$, $\frac{1}{1935808}a^{29}+\frac{20075}{1935808}a^{27}+\frac{8795}{1935808}a^{25}+\frac{22683}{1935808}a^{23}+\frac{6723}{483952}a^{21}-\frac{58253}{483952}a^{19}+\frac{1345}{12992}a^{17}+\frac{135255}{1935808}a^{15}-\frac{4183}{138272}a^{13}+\frac{118807}{967904}a^{11}-\frac{740287}{1935808}a^{9}+\frac{50747}{276544}a^{7}+\frac{6993}{17284}a^{5}+\frac{5083}{241976}a^{3}+\frac{18463}{60494}a$, $\frac{1}{39\!\cdots\!72}a^{30}+\frac{37\!\cdots\!63}{39\!\cdots\!72}a^{28}-\frac{42\!\cdots\!91}{56\!\cdots\!96}a^{26}-\frac{10\!\cdots\!41}{39\!\cdots\!72}a^{24}+\frac{80\!\cdots\!22}{88\!\cdots\!39}a^{22}-\frac{39\!\cdots\!85}{49\!\cdots\!84}a^{20}+\frac{27\!\cdots\!79}{56\!\cdots\!96}a^{18}-\frac{53\!\cdots\!01}{81\!\cdots\!28}a^{16}+\frac{66\!\cdots\!53}{68\!\cdots\!84}a^{14}-\frac{74\!\cdots\!47}{19\!\cdots\!36}a^{12}+\frac{46\!\cdots\!45}{39\!\cdots\!72}a^{10}+\frac{18\!\cdots\!09}{39\!\cdots\!72}a^{8}-\frac{31\!\cdots\!93}{14\!\cdots\!24}a^{6}-\frac{16\!\cdots\!11}{99\!\cdots\!68}a^{4}-\frac{15\!\cdots\!32}{62\!\cdots\!73}a^{2}-\frac{65\!\cdots\!27}{62\!\cdots\!73}$, $\frac{1}{79\!\cdots\!44}a^{31}-\frac{41\!\cdots\!99}{19\!\cdots\!36}a^{29}-\frac{50\!\cdots\!83}{56\!\cdots\!96}a^{27}+\frac{60\!\cdots\!23}{39\!\cdots\!72}a^{25}+\frac{44\!\cdots\!37}{11\!\cdots\!92}a^{23}+\frac{10\!\cdots\!19}{99\!\cdots\!68}a^{21}-\frac{46\!\cdots\!01}{11\!\cdots\!92}a^{19}-\frac{63\!\cdots\!45}{28\!\cdots\!48}a^{17}+\frac{60\!\cdots\!13}{79\!\cdots\!44}a^{15}+\frac{17\!\cdots\!35}{19\!\cdots\!36}a^{13}-\frac{15\!\cdots\!33}{79\!\cdots\!44}a^{11}-\frac{18\!\cdots\!41}{39\!\cdots\!72}a^{9}+\frac{19\!\cdots\!27}{11\!\cdots\!92}a^{7}+\frac{24\!\cdots\!47}{99\!\cdots\!68}a^{5}+\frac{22\!\cdots\!39}{12\!\cdots\!46}a^{3}-\frac{28\!\cdots\!86}{62\!\cdots\!73}a$
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
$C_{2}\times C_{2}\times C_{114}$, which has order $456$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $456$ (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: |
\( \frac{1212198436306899677985}{5020023985943106325355968} a^{31} - \frac{13304064985537798123215}{1255005996485776581338992} a^{29} + \frac{5378279244032814706597}{25612367275219930231408} a^{27} - \frac{755085071138638901781819}{313751499121444145334748} a^{25} + \frac{12148111413127701250618927}{717146283706158046479424} a^{23} - \frac{175608202718792884749281523}{2510011992971553162677984} a^{21} + \frac{90781906578874341227807831}{717146283706158046479424} a^{19} + \frac{891152206664961809111701}{6182295549191017642064} a^{17} - \frac{4235793906432875880761439733}{5020023985943106325355968} a^{15} - \frac{259252036904033812748951733}{2510011992971553162677984} a^{13} + \frac{11418936268628627747753300155}{5020023985943106325355968} a^{11} + \frac{245340555604445323655962241}{86552137688674246988896} a^{9} + \frac{46656671613700758856017153}{24729182196764070568256} a^{7} + \frac{1907703902834059885134020843}{2510011992971553162677984} a^{5} + \frac{12206814291498928056218167}{78437874780361036333687} a^{3} + \frac{659169053797579020919973}{78437874780361036333687} a \)
(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: |
$\frac{22\!\cdots\!53}{15\!\cdots\!44}a^{31}-\frac{24\!\cdots\!93}{38\!\cdots\!36}a^{29}+\frac{20\!\cdots\!41}{15\!\cdots\!28}a^{27}-\frac{11\!\cdots\!15}{77\!\cdots\!72}a^{25}+\frac{22\!\cdots\!73}{22\!\cdots\!92}a^{23}-\frac{16\!\cdots\!03}{38\!\cdots\!36}a^{21}+\frac{17\!\cdots\!55}{22\!\cdots\!92}a^{19}+\frac{16\!\cdots\!61}{19\!\cdots\!12}a^{17}-\frac{79\!\cdots\!47}{15\!\cdots\!44}a^{15}-\frac{95\!\cdots\!19}{19\!\cdots\!68}a^{13}+\frac{21\!\cdots\!63}{15\!\cdots\!44}a^{11}+\frac{45\!\cdots\!23}{26\!\cdots\!68}a^{9}+\frac{86\!\cdots\!43}{76\!\cdots\!48}a^{7}+\frac{17\!\cdots\!51}{38\!\cdots\!36}a^{5}+\frac{93\!\cdots\!41}{97\!\cdots\!84}a^{3}+\frac{63\!\cdots\!53}{12\!\cdots\!23}a$, $\frac{14\!\cdots\!43}{29\!\cdots\!44}a^{30}-\frac{12\!\cdots\!55}{58\!\cdots\!88}a^{28}+\frac{35\!\cdots\!57}{83\!\cdots\!84}a^{26}-\frac{28\!\cdots\!97}{58\!\cdots\!88}a^{24}+\frac{29\!\cdots\!87}{83\!\cdots\!84}a^{22}-\frac{10\!\cdots\!67}{73\!\cdots\!36}a^{20}+\frac{12\!\cdots\!75}{41\!\cdots\!92}a^{18}+\frac{17\!\cdots\!91}{83\!\cdots\!84}a^{16}-\frac{35\!\cdots\!19}{20\!\cdots\!72}a^{14}+\frac{88\!\cdots\!97}{29\!\cdots\!44}a^{12}+\frac{16\!\cdots\!57}{36\!\cdots\!68}a^{10}+\frac{24\!\cdots\!57}{58\!\cdots\!88}a^{8}+\frac{19\!\cdots\!01}{83\!\cdots\!84}a^{6}+\frac{10\!\cdots\!37}{14\!\cdots\!72}a^{4}+\frac{84\!\cdots\!05}{36\!\cdots\!68}a^{2}-\frac{39\!\cdots\!29}{91\!\cdots\!67}$, $\frac{27\!\cdots\!89}{79\!\cdots\!44}a^{31}-\frac{60\!\cdots\!17}{39\!\cdots\!72}a^{29}+\frac{61\!\cdots\!63}{20\!\cdots\!32}a^{27}-\frac{17\!\cdots\!79}{49\!\cdots\!84}a^{25}+\frac{28\!\cdots\!43}{11\!\cdots\!92}a^{23}-\frac{20\!\cdots\!65}{19\!\cdots\!36}a^{21}+\frac{22\!\cdots\!47}{11\!\cdots\!92}a^{19}+\frac{10\!\cdots\!93}{56\!\cdots\!96}a^{17}-\frac{98\!\cdots\!13}{79\!\cdots\!44}a^{15}-\frac{16\!\cdots\!39}{19\!\cdots\!36}a^{13}+\frac{25\!\cdots\!67}{79\!\cdots\!44}a^{11}+\frac{92\!\cdots\!01}{24\!\cdots\!92}a^{9}+\frac{27\!\cdots\!05}{11\!\cdots\!92}a^{7}+\frac{18\!\cdots\!83}{19\!\cdots\!36}a^{5}+\frac{43\!\cdots\!29}{24\!\cdots\!92}a^{3}+\frac{57\!\cdots\!14}{62\!\cdots\!73}a$, $\frac{67\!\cdots\!49}{76\!\cdots\!08}a^{31}-\frac{35\!\cdots\!51}{14\!\cdots\!12}a^{30}-\frac{36\!\cdots\!53}{95\!\cdots\!76}a^{29}+\frac{78\!\cdots\!91}{72\!\cdots\!56}a^{28}+\frac{41\!\cdots\!39}{54\!\cdots\!72}a^{27}-\frac{11\!\cdots\!47}{51\!\cdots\!04}a^{26}-\frac{33\!\cdots\!55}{38\!\cdots\!04}a^{25}+\frac{11\!\cdots\!63}{45\!\cdots\!66}a^{24}+\frac{68\!\cdots\!61}{10\!\cdots\!44}a^{23}-\frac{36\!\cdots\!65}{20\!\cdots\!16}a^{22}-\frac{12\!\cdots\!77}{47\!\cdots\!88}a^{21}+\frac{27\!\cdots\!79}{36\!\cdots\!28}a^{20}+\frac{53\!\cdots\!79}{10\!\cdots\!44}a^{19}-\frac{29\!\cdots\!25}{20\!\cdots\!16}a^{18}+\frac{21\!\cdots\!55}{46\!\cdots\!92}a^{17}-\frac{11\!\cdots\!11}{10\!\cdots\!08}a^{16}-\frac{23\!\cdots\!79}{76\!\cdots\!08}a^{15}+\frac{12\!\cdots\!55}{14\!\cdots\!12}a^{14}+\frac{50\!\cdots\!07}{19\!\cdots\!52}a^{13}-\frac{27\!\cdots\!17}{36\!\cdots\!28}a^{12}+\frac{63\!\cdots\!51}{76\!\cdots\!08}a^{11}-\frac{32\!\cdots\!29}{14\!\cdots\!12}a^{10}+\frac{12\!\cdots\!33}{13\!\cdots\!76}a^{9}-\frac{21\!\cdots\!07}{90\!\cdots\!32}a^{8}+\frac{21\!\cdots\!39}{37\!\cdots\!36}a^{7}-\frac{30\!\cdots\!71}{20\!\cdots\!16}a^{6}+\frac{98\!\cdots\!07}{47\!\cdots\!88}a^{5}-\frac{19\!\cdots\!59}{36\!\cdots\!28}a^{4}+\frac{14\!\cdots\!07}{47\!\cdots\!88}a^{3}-\frac{19\!\cdots\!11}{22\!\cdots\!83}a^{2}-\frac{24\!\cdots\!20}{59\!\cdots\!11}a-\frac{59\!\cdots\!94}{22\!\cdots\!83}$, $\frac{26\!\cdots\!65}{39\!\cdots\!72}a^{31}-\frac{62\!\cdots\!47}{12\!\cdots\!68}a^{30}-\frac{11\!\cdots\!05}{39\!\cdots\!72}a^{29}+\frac{27\!\cdots\!97}{12\!\cdots\!68}a^{28}+\frac{47\!\cdots\!67}{81\!\cdots\!28}a^{27}-\frac{77\!\cdots\!53}{17\!\cdots\!24}a^{26}-\frac{26\!\cdots\!09}{39\!\cdots\!72}a^{25}+\frac{61\!\cdots\!41}{12\!\cdots\!68}a^{24}+\frac{68\!\cdots\!51}{14\!\cdots\!24}a^{23}-\frac{30\!\cdots\!61}{88\!\cdots\!12}a^{22}-\frac{10\!\cdots\!71}{49\!\cdots\!84}a^{21}+\frac{43\!\cdots\!29}{30\!\cdots\!92}a^{20}+\frac{22\!\cdots\!31}{56\!\cdots\!96}a^{19}-\frac{40\!\cdots\!17}{17\!\cdots\!24}a^{18}+\frac{18\!\cdots\!01}{56\!\cdots\!96}a^{17}-\frac{68\!\cdots\!01}{17\!\cdots\!24}a^{16}-\frac{47\!\cdots\!57}{19\!\cdots\!36}a^{15}+\frac{55\!\cdots\!57}{30\!\cdots\!92}a^{14}+\frac{32\!\cdots\!81}{19\!\cdots\!36}a^{13}+\frac{34\!\cdots\!13}{61\!\cdots\!84}a^{12}+\frac{24\!\cdots\!57}{39\!\cdots\!72}a^{11}-\frac{64\!\cdots\!27}{12\!\cdots\!68}a^{10}+\frac{26\!\cdots\!21}{39\!\cdots\!72}a^{9}-\frac{81\!\cdots\!65}{12\!\cdots\!68}a^{8}+\frac{74\!\cdots\!77}{17\!\cdots\!78}a^{7}-\frac{35\!\cdots\!09}{88\!\cdots\!12}a^{6}+\frac{15\!\cdots\!63}{99\!\cdots\!68}a^{5}-\frac{12\!\cdots\!11}{77\!\cdots\!48}a^{4}+\frac{12\!\cdots\!87}{49\!\cdots\!84}a^{3}-\frac{18\!\cdots\!99}{77\!\cdots\!48}a^{2}+\frac{44\!\cdots\!70}{62\!\cdots\!73}a+\frac{44\!\cdots\!24}{19\!\cdots\!37}$, $\frac{27\!\cdots\!03}{99\!\cdots\!68}a^{31}-\frac{92\!\cdots\!91}{17\!\cdots\!24}a^{30}-\frac{29\!\cdots\!13}{24\!\cdots\!92}a^{29}+\frac{40\!\cdots\!39}{17\!\cdots\!24}a^{28}+\frac{97\!\cdots\!39}{40\!\cdots\!64}a^{27}-\frac{11\!\cdots\!79}{25\!\cdots\!32}a^{26}-\frac{54\!\cdots\!13}{19\!\cdots\!36}a^{25}+\frac{93\!\cdots\!31}{17\!\cdots\!24}a^{24}+\frac{55\!\cdots\!73}{28\!\cdots\!48}a^{23}-\frac{23\!\cdots\!35}{62\!\cdots\!08}a^{22}-\frac{16\!\cdots\!17}{19\!\cdots\!36}a^{21}+\frac{68\!\cdots\!15}{44\!\cdots\!56}a^{20}+\frac{22\!\cdots\!71}{14\!\cdots\!24}a^{19}-\frac{71\!\cdots\!37}{25\!\cdots\!32}a^{18}+\frac{48\!\cdots\!25}{35\!\cdots\!56}a^{17}-\frac{74\!\cdots\!59}{25\!\cdots\!32}a^{16}-\frac{19\!\cdots\!55}{19\!\cdots\!36}a^{15}+\frac{16\!\cdots\!23}{88\!\cdots\!12}a^{14}+\frac{81\!\cdots\!51}{19\!\cdots\!36}a^{13}+\frac{97\!\cdots\!63}{88\!\cdots\!12}a^{12}+\frac{63\!\cdots\!81}{24\!\cdots\!92}a^{11}-\frac{87\!\cdots\!35}{17\!\cdots\!24}a^{10}+\frac{27\!\cdots\!73}{99\!\cdots\!68}a^{9}-\frac{10\!\cdots\!63}{17\!\cdots\!24}a^{8}+\frac{48\!\cdots\!27}{28\!\cdots\!48}a^{7}-\frac{11\!\cdots\!83}{31\!\cdots\!04}a^{6}+\frac{12\!\cdots\!85}{19\!\cdots\!36}a^{5}-\frac{16\!\cdots\!05}{11\!\cdots\!64}a^{4}+\frac{44\!\cdots\!71}{49\!\cdots\!84}a^{3}-\frac{29\!\cdots\!27}{11\!\cdots\!64}a^{2}-\frac{23\!\cdots\!39}{12\!\cdots\!46}a-\frac{24\!\cdots\!12}{27\!\cdots\!91}$, $\frac{83\!\cdots\!71}{40\!\cdots\!64}a^{31}+\frac{18\!\cdots\!93}{61\!\cdots\!84}a^{30}-\frac{36\!\cdots\!53}{40\!\cdots\!64}a^{29}-\frac{39\!\cdots\!23}{30\!\cdots\!92}a^{28}+\frac{36\!\cdots\!27}{20\!\cdots\!32}a^{27}+\frac{28\!\cdots\!25}{11\!\cdots\!64}a^{26}-\frac{21\!\cdots\!35}{10\!\cdots\!16}a^{25}-\frac{57\!\cdots\!28}{19\!\cdots\!37}a^{24}+\frac{59\!\cdots\!67}{40\!\cdots\!64}a^{23}+\frac{18\!\cdots\!67}{88\!\cdots\!12}a^{22}-\frac{24\!\cdots\!91}{40\!\cdots\!64}a^{21}-\frac{67\!\cdots\!11}{77\!\cdots\!48}a^{20}+\frac{46\!\cdots\!87}{40\!\cdots\!64}a^{19}+\frac{14\!\cdots\!35}{88\!\cdots\!12}a^{18}+\frac{45\!\cdots\!35}{40\!\cdots\!64}a^{17}+\frac{67\!\cdots\!87}{44\!\cdots\!56}a^{16}-\frac{29\!\cdots\!01}{40\!\cdots\!64}a^{15}-\frac{64\!\cdots\!85}{61\!\cdots\!84}a^{14}-\frac{71\!\cdots\!97}{40\!\cdots\!64}a^{13}+\frac{23\!\cdots\!99}{77\!\cdots\!48}a^{12}+\frac{78\!\cdots\!45}{40\!\cdots\!64}a^{11}+\frac{16\!\cdots\!63}{61\!\cdots\!84}a^{10}+\frac{90\!\cdots\!55}{40\!\cdots\!64}a^{9}+\frac{48\!\cdots\!17}{15\!\cdots\!96}a^{8}+\frac{58\!\cdots\!37}{40\!\cdots\!64}a^{7}+\frac{17\!\cdots\!05}{88\!\cdots\!12}a^{6}+\frac{22\!\cdots\!11}{40\!\cdots\!64}a^{5}+\frac{28\!\cdots\!79}{38\!\cdots\!74}a^{4}+\frac{12\!\cdots\!06}{12\!\cdots\!77}a^{3}+\frac{44\!\cdots\!03}{38\!\cdots\!74}a^{2}+\frac{29\!\cdots\!93}{12\!\cdots\!77}a+\frac{27\!\cdots\!28}{19\!\cdots\!37}$, $\frac{13\!\cdots\!39}{79\!\cdots\!44}a^{31}+\frac{11\!\cdots\!57}{12\!\cdots\!68}a^{30}-\frac{83\!\cdots\!09}{99\!\cdots\!68}a^{29}-\frac{30\!\cdots\!41}{61\!\cdots\!84}a^{28}+\frac{14\!\cdots\!79}{81\!\cdots\!28}a^{27}+\frac{52\!\cdots\!31}{44\!\cdots\!56}a^{26}-\frac{92\!\cdots\!53}{39\!\cdots\!72}a^{25}-\frac{26\!\cdots\!77}{15\!\cdots\!96}a^{24}+\frac{21\!\cdots\!51}{11\!\cdots\!92}a^{23}+\frac{27\!\cdots\!11}{17\!\cdots\!24}a^{22}-\frac{98\!\cdots\!77}{99\!\cdots\!68}a^{21}-\frac{29\!\cdots\!53}{30\!\cdots\!92}a^{20}+\frac{33\!\cdots\!81}{11\!\cdots\!92}a^{19}+\frac{60\!\cdots\!15}{17\!\cdots\!24}a^{18}-\frac{40\!\cdots\!73}{14\!\cdots\!24}a^{17}-\frac{51\!\cdots\!43}{88\!\cdots\!12}a^{16}-\frac{74\!\cdots\!21}{79\!\cdots\!44}a^{15}-\frac{62\!\cdots\!49}{12\!\cdots\!68}a^{14}+\frac{46\!\cdots\!43}{19\!\cdots\!36}a^{13}+\frac{98\!\cdots\!75}{30\!\cdots\!92}a^{12}+\frac{10\!\cdots\!05}{79\!\cdots\!44}a^{11}-\frac{67\!\cdots\!97}{12\!\cdots\!68}a^{10}-\frac{16\!\cdots\!81}{39\!\cdots\!72}a^{9}-\frac{10\!\cdots\!87}{15\!\cdots\!96}a^{8}-\frac{53\!\cdots\!27}{11\!\cdots\!92}a^{7}-\frac{95\!\cdots\!87}{17\!\cdots\!24}a^{6}-\frac{27\!\cdots\!67}{99\!\cdots\!68}a^{5}-\frac{92\!\cdots\!99}{30\!\cdots\!92}a^{4}-\frac{16\!\cdots\!69}{17\!\cdots\!96}a^{3}-\frac{13\!\cdots\!36}{19\!\cdots\!37}a^{2}-\frac{33\!\cdots\!48}{62\!\cdots\!73}a-\frac{24\!\cdots\!76}{19\!\cdots\!37}$, $\frac{12\!\cdots\!35}{79\!\cdots\!44}a^{31}+\frac{57\!\cdots\!27}{19\!\cdots\!36}a^{30}-\frac{28\!\cdots\!77}{39\!\cdots\!72}a^{29}-\frac{63\!\cdots\!15}{49\!\cdots\!84}a^{28}+\frac{99\!\cdots\!31}{71\!\cdots\!12}a^{27}+\frac{36\!\cdots\!99}{14\!\cdots\!24}a^{26}-\frac{80\!\cdots\!83}{49\!\cdots\!84}a^{25}-\frac{29\!\cdots\!41}{99\!\cdots\!68}a^{24}+\frac{13\!\cdots\!53}{11\!\cdots\!92}a^{23}+\frac{58\!\cdots\!63}{28\!\cdots\!48}a^{22}-\frac{23\!\cdots\!49}{49\!\cdots\!84}a^{21}-\frac{43\!\cdots\!89}{49\!\cdots\!84}a^{20}+\frac{10\!\cdots\!17}{11\!\cdots\!92}a^{19}+\frac{46\!\cdots\!05}{28\!\cdots\!48}a^{18}+\frac{47\!\cdots\!01}{56\!\cdots\!96}a^{17}+\frac{11\!\cdots\!37}{71\!\cdots\!12}a^{16}-\frac{45\!\cdots\!27}{79\!\cdots\!44}a^{15}-\frac{20\!\cdots\!05}{19\!\cdots\!36}a^{14}+\frac{43\!\cdots\!21}{62\!\cdots\!73}a^{13}-\frac{29\!\cdots\!57}{24\!\cdots\!92}a^{12}+\frac{11\!\cdots\!73}{79\!\cdots\!44}a^{11}+\frac{54\!\cdots\!25}{19\!\cdots\!36}a^{10}+\frac{33\!\cdots\!17}{19\!\cdots\!36}a^{9}+\frac{31\!\cdots\!57}{99\!\cdots\!68}a^{8}+\frac{11\!\cdots\!55}{11\!\cdots\!92}a^{7}+\frac{56\!\cdots\!81}{28\!\cdots\!48}a^{6}+\frac{38\!\cdots\!95}{99\!\cdots\!68}a^{5}+\frac{37\!\cdots\!93}{49\!\cdots\!84}a^{4}+\frac{30\!\cdots\!73}{49\!\cdots\!84}a^{3}+\frac{31\!\cdots\!81}{24\!\cdots\!92}a^{2}+\frac{10\!\cdots\!11}{62\!\cdots\!73}a+\frac{18\!\cdots\!89}{62\!\cdots\!73}$, $\frac{41\!\cdots\!97}{11\!\cdots\!92}a^{31}+\frac{28\!\cdots\!71}{58\!\cdots\!88}a^{30}-\frac{45\!\cdots\!47}{28\!\cdots\!48}a^{29}-\frac{12\!\cdots\!09}{58\!\cdots\!88}a^{28}+\frac{18\!\cdots\!79}{56\!\cdots\!96}a^{27}+\frac{35\!\cdots\!13}{83\!\cdots\!84}a^{26}-\frac{20\!\cdots\!85}{56\!\cdots\!96}a^{25}-\frac{28\!\cdots\!25}{58\!\cdots\!88}a^{24}+\frac{29\!\cdots\!99}{11\!\cdots\!92}a^{23}+\frac{14\!\cdots\!53}{41\!\cdots\!92}a^{22}-\frac{15\!\cdots\!61}{14\!\cdots\!24}a^{21}-\frac{21\!\cdots\!11}{14\!\cdots\!72}a^{20}+\frac{23\!\cdots\!81}{11\!\cdots\!92}a^{19}+\frac{23\!\cdots\!73}{83\!\cdots\!84}a^{18}+\frac{54\!\cdots\!97}{28\!\cdots\!48}a^{17}+\frac{18\!\cdots\!13}{83\!\cdots\!84}a^{16}-\frac{14\!\cdots\!91}{11\!\cdots\!92}a^{15}-\frac{88\!\cdots\!41}{50\!\cdots\!68}a^{14}+\frac{56\!\cdots\!11}{28\!\cdots\!48}a^{13}+\frac{59\!\cdots\!75}{29\!\cdots\!44}a^{12}+\frac{39\!\cdots\!03}{11\!\cdots\!92}a^{11}+\frac{26\!\cdots\!75}{58\!\cdots\!88}a^{10}+\frac{22\!\cdots\!43}{56\!\cdots\!96}a^{9}+\frac{26\!\cdots\!65}{58\!\cdots\!88}a^{8}+\frac{39\!\cdots\!03}{16\!\cdots\!56}a^{7}+\frac{10\!\cdots\!53}{41\!\cdots\!92}a^{6}+\frac{12\!\cdots\!53}{14\!\cdots\!24}a^{5}+\frac{54\!\cdots\!73}{73\!\cdots\!36}a^{4}+\frac{10\!\cdots\!41}{71\!\cdots\!12}a^{3}+\frac{90\!\cdots\!85}{36\!\cdots\!68}a^{2}+\frac{14\!\cdots\!55}{88\!\cdots\!39}a-\frac{27\!\cdots\!75}{91\!\cdots\!67}$, $\frac{49\!\cdots\!15}{99\!\cdots\!68}a^{31}-\frac{17\!\cdots\!67}{58\!\cdots\!88}a^{30}-\frac{21\!\cdots\!37}{99\!\cdots\!68}a^{29}+\frac{99\!\cdots\!99}{73\!\cdots\!36}a^{28}+\frac{62\!\cdots\!75}{14\!\cdots\!24}a^{27}-\frac{11\!\cdots\!13}{41\!\cdots\!92}a^{26}-\frac{62\!\cdots\!41}{12\!\cdots\!46}a^{25}+\frac{93\!\cdots\!01}{29\!\cdots\!44}a^{24}+\frac{88\!\cdots\!75}{24\!\cdots\!28}a^{23}-\frac{19\!\cdots\!55}{83\!\cdots\!84}a^{22}-\frac{15\!\cdots\!39}{99\!\cdots\!68}a^{21}+\frac{72\!\cdots\!05}{73\!\cdots\!36}a^{20}+\frac{41\!\cdots\!11}{14\!\cdots\!24}a^{19}-\frac{17\!\cdots\!13}{83\!\cdots\!84}a^{18}+\frac{47\!\cdots\!17}{20\!\cdots\!32}a^{17}-\frac{10\!\cdots\!31}{10\!\cdots\!48}a^{16}-\frac{44\!\cdots\!07}{24\!\cdots\!92}a^{15}+\frac{23\!\cdots\!61}{20\!\cdots\!72}a^{14}+\frac{15\!\cdots\!57}{99\!\cdots\!68}a^{13}-\frac{60\!\cdots\!75}{14\!\cdots\!72}a^{12}+\frac{45\!\cdots\!95}{99\!\cdots\!68}a^{11}-\frac{16\!\cdots\!37}{58\!\cdots\!88}a^{10}+\frac{48\!\cdots\!19}{99\!\cdots\!68}a^{9}-\frac{63\!\cdots\!79}{29\!\cdots\!44}a^{8}+\frac{53\!\cdots\!89}{17\!\cdots\!78}a^{7}-\frac{90\!\cdots\!97}{83\!\cdots\!84}a^{6}+\frac{37\!\cdots\!13}{34\!\cdots\!92}a^{5}-\frac{25\!\cdots\!93}{73\!\cdots\!36}a^{4}+\frac{42\!\cdots\!69}{24\!\cdots\!92}a^{3}-\frac{42\!\cdots\!31}{36\!\cdots\!68}a^{2}+\frac{11\!\cdots\!32}{62\!\cdots\!73}a+\frac{59\!\cdots\!21}{91\!\cdots\!67}$, $\frac{25\!\cdots\!33}{11\!\cdots\!92}a^{31}-\frac{98\!\cdots\!31}{19\!\cdots\!36}a^{30}-\frac{35\!\cdots\!07}{35\!\cdots\!56}a^{29}+\frac{43\!\cdots\!47}{19\!\cdots\!36}a^{28}+\frac{15\!\cdots\!65}{81\!\cdots\!28}a^{27}-\frac{17\!\cdots\!99}{40\!\cdots\!64}a^{26}-\frac{12\!\cdots\!69}{56\!\cdots\!96}a^{25}+\frac{99\!\cdots\!37}{19\!\cdots\!36}a^{24}+\frac{25\!\cdots\!65}{16\!\cdots\!56}a^{23}-\frac{50\!\cdots\!83}{14\!\cdots\!24}a^{22}-\frac{18\!\cdots\!17}{28\!\cdots\!48}a^{21}+\frac{14\!\cdots\!77}{99\!\cdots\!68}a^{20}+\frac{20\!\cdots\!67}{16\!\cdots\!56}a^{19}-\frac{79\!\cdots\!73}{28\!\cdots\!48}a^{18}+\frac{12\!\cdots\!93}{10\!\cdots\!16}a^{17}-\frac{71\!\cdots\!23}{28\!\cdots\!48}a^{16}-\frac{90\!\cdots\!55}{11\!\cdots\!92}a^{15}+\frac{87\!\cdots\!55}{49\!\cdots\!84}a^{14}-\frac{48\!\cdots\!13}{14\!\cdots\!24}a^{13}-\frac{25\!\cdots\!99}{49\!\cdots\!84}a^{12}+\frac{24\!\cdots\!83}{11\!\cdots\!92}a^{11}-\frac{91\!\cdots\!35}{19\!\cdots\!36}a^{10}+\frac{14\!\cdots\!89}{56\!\cdots\!96}a^{9}-\frac{10\!\cdots\!47}{19\!\cdots\!36}a^{8}+\frac{25\!\cdots\!63}{16\!\cdots\!56}a^{7}-\frac{45\!\cdots\!99}{14\!\cdots\!24}a^{6}+\frac{16\!\cdots\!57}{28\!\cdots\!48}a^{5}-\frac{11\!\cdots\!11}{99\!\cdots\!68}a^{4}+\frac{19\!\cdots\!89}{17\!\cdots\!78}a^{3}-\frac{47\!\cdots\!63}{24\!\cdots\!92}a^{2}+\frac{10\!\cdots\!93}{17\!\cdots\!78}a-\frac{21\!\cdots\!97}{62\!\cdots\!73}$, $\frac{36\!\cdots\!21}{39\!\cdots\!72}a^{31}+\frac{25\!\cdots\!91}{39\!\cdots\!72}a^{30}-\frac{15\!\cdots\!79}{39\!\cdots\!72}a^{29}-\frac{11\!\cdots\!05}{39\!\cdots\!72}a^{28}+\frac{45\!\cdots\!03}{56\!\cdots\!96}a^{27}+\frac{33\!\cdots\!61}{56\!\cdots\!96}a^{26}-\frac{36\!\cdots\!59}{39\!\cdots\!72}a^{25}-\frac{27\!\cdots\!77}{39\!\cdots\!72}a^{24}+\frac{26\!\cdots\!31}{40\!\cdots\!64}a^{23}+\frac{21\!\cdots\!67}{40\!\cdots\!64}a^{22}-\frac{27\!\cdots\!15}{99\!\cdots\!68}a^{21}-\frac{23\!\cdots\!73}{99\!\cdots\!68}a^{20}+\frac{29\!\cdots\!03}{56\!\cdots\!96}a^{19}+\frac{32\!\cdots\!41}{56\!\cdots\!96}a^{18}+\frac{25\!\cdots\!19}{56\!\cdots\!96}a^{17}-\frac{73\!\cdots\!59}{56\!\cdots\!96}a^{16}-\frac{16\!\cdots\!59}{49\!\cdots\!84}a^{15}-\frac{24\!\cdots\!65}{99\!\cdots\!68}a^{14}+\frac{26\!\cdots\!25}{19\!\cdots\!36}a^{13}+\frac{49\!\cdots\!91}{19\!\cdots\!36}a^{12}+\frac{33\!\cdots\!93}{39\!\cdots\!72}a^{11}+\frac{19\!\cdots\!03}{39\!\cdots\!72}a^{10}+\frac{37\!\cdots\!23}{39\!\cdots\!72}a^{9}+\frac{22\!\cdots\!01}{39\!\cdots\!72}a^{8}+\frac{16\!\cdots\!21}{28\!\cdots\!48}a^{7}-\frac{23\!\cdots\!99}{28\!\cdots\!48}a^{6}+\frac{10\!\cdots\!67}{49\!\cdots\!84}a^{5}-\frac{13\!\cdots\!47}{12\!\cdots\!46}a^{4}+\frac{82\!\cdots\!75}{21\!\cdots\!37}a^{3}-\frac{10\!\cdots\!43}{24\!\cdots\!92}a^{2}+\frac{10\!\cdots\!27}{12\!\cdots\!46}a-\frac{55\!\cdots\!96}{62\!\cdots\!73}$, $\frac{97\!\cdots\!23}{56\!\cdots\!96}a^{31}-\frac{17\!\cdots\!17}{17\!\cdots\!78}a^{30}-\frac{43\!\cdots\!03}{56\!\cdots\!96}a^{29}+\frac{24\!\cdots\!21}{56\!\cdots\!96}a^{28}+\frac{86\!\cdots\!27}{56\!\cdots\!96}a^{27}-\frac{46\!\cdots\!69}{56\!\cdots\!96}a^{26}-\frac{14\!\cdots\!23}{81\!\cdots\!28}a^{25}+\frac{50\!\cdots\!31}{56\!\cdots\!96}a^{24}+\frac{35\!\cdots\!91}{28\!\cdots\!48}a^{23}-\frac{32\!\cdots\!57}{56\!\cdots\!96}a^{22}-\frac{15\!\cdots\!31}{28\!\cdots\!48}a^{21}+\frac{35\!\cdots\!61}{17\!\cdots\!78}a^{20}+\frac{59\!\cdots\!31}{56\!\cdots\!96}a^{19}-\frac{20\!\cdots\!67}{14\!\cdots\!24}a^{18}+\frac{42\!\cdots\!41}{56\!\cdots\!96}a^{17}-\frac{78\!\cdots\!75}{56\!\cdots\!96}a^{16}-\frac{43\!\cdots\!57}{71\!\cdots\!12}a^{15}+\frac{25\!\cdots\!77}{81\!\cdots\!28}a^{14}+\frac{77\!\cdots\!84}{88\!\cdots\!39}a^{13}+\frac{13\!\cdots\!97}{28\!\cdots\!48}a^{12}+\frac{90\!\cdots\!51}{56\!\cdots\!96}a^{11}-\frac{31\!\cdots\!49}{28\!\cdots\!48}a^{10}+\frac{91\!\cdots\!87}{56\!\cdots\!96}a^{9}-\frac{18\!\cdots\!85}{81\!\cdots\!28}a^{8}+\frac{38\!\cdots\!05}{40\!\cdots\!64}a^{7}-\frac{13\!\cdots\!65}{81\!\cdots\!28}a^{6}+\frac{89\!\cdots\!57}{28\!\cdots\!48}a^{5}-\frac{11\!\cdots\!89}{14\!\cdots\!24}a^{4}+\frac{26\!\cdots\!23}{71\!\cdots\!12}a^{3}-\frac{32\!\cdots\!69}{17\!\cdots\!78}a^{2}-\frac{48\!\cdots\!49}{17\!\cdots\!78}a-\frac{10\!\cdots\!30}{12\!\cdots\!77}$, $\frac{96\!\cdots\!83}{99\!\cdots\!68}a^{30}-\frac{84\!\cdots\!61}{19\!\cdots\!36}a^{28}+\frac{24\!\cdots\!71}{28\!\cdots\!48}a^{26}-\frac{19\!\cdots\!77}{19\!\cdots\!36}a^{24}+\frac{19\!\cdots\!97}{28\!\cdots\!48}a^{22}-\frac{29\!\cdots\!25}{99\!\cdots\!68}a^{20}+\frac{80\!\cdots\!01}{14\!\cdots\!24}a^{18}+\frac{13\!\cdots\!61}{28\!\cdots\!48}a^{16}-\frac{68\!\cdots\!09}{19\!\cdots\!36}a^{14}+\frac{68\!\cdots\!37}{24\!\cdots\!92}a^{12}+\frac{44\!\cdots\!95}{49\!\cdots\!84}a^{10}+\frac{19\!\cdots\!99}{19\!\cdots\!36}a^{8}+\frac{16\!\cdots\!79}{28\!\cdots\!48}a^{6}+\frac{21\!\cdots\!49}{99\!\cdots\!68}a^{4}+\frac{83\!\cdots\!29}{24\!\cdots\!92}a^{2}+\frac{37\!\cdots\!64}{62\!\cdots\!73}$
| 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: | \( 37277164025166.29 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 37277164025166.29 \cdot 456}{4\cdot\sqrt{13192724911186950673358584730276205272104960000000000000000}}\cr\approx \mathstrut & 0.218302322576636 \end{aligned}\] (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 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*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 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*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 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*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 + 874*x^28 - 10054*x^26 + 71163*x^24 - 296956*x^22 + 554793*x^20 + 537460*x^18 - 3539799*x^16 - 66968*x^14 + 9372831*x^12 + 10803870*x^10 + 6888129*x^8 + 2624028*x^6 + 464976*x^4 + 14784*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 | ${\href{/padicField/3.8.0.1}{8} }^{4}$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1049\)
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |