Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536)
gp: K = bnfinit(y^32 + 48*y^30 + 1128*y^28 + 16800*y^26 + 175155*y^24 + 1339752*y^22 + 7694204*y^20 + 33445200*y^18 + 109690049*y^16 + 267441600*y^14 + 471188096*y^12 + 570415104*y^10 + 436417280*y^8 + 180264960*y^6 + 39043072*y^4 + 2359296*y^2 + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536)
\( x^{32} + 48 x^{30} + 1128 x^{28} + 16800 x^{26} + 175155 x^{24} + 1339752 x^{22} + 7694204 x^{20} + \cdots + 65536 \)
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: |
\(156938077449417789520626992646455296000000000000000000000000\)
\(\medspace = 2^{96}\cdot 5^{24}\cdot 7^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.77\) | 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: | $2^{3}5^{3/4}7^{1/2}\approx 70.7728215461241$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(560=2^{4}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(321,·)$, $\chi_{560}(267,·)$, $\chi_{560}(141,·)$, $\chi_{560}(407,·)$, $\chi_{560}(281,·)$, $\chi_{560}(27,·)$, $\chi_{560}(29,·)$, $\chi_{560}(547,·)$, $\chi_{560}(421,·)$, $\chi_{560}(167,·)$, $\chi_{560}(41,·)$, $\chi_{560}(43,·)$, $\chi_{560}(307,·)$, $\chi_{560}(181,·)$, $\chi_{560}(183,·)$, $\chi_{560}(309,·)$, $\chi_{560}(449,·)$, $\chi_{560}(323,·)$, $\chi_{560}(69,·)$, $\chi_{560}(461,·)$, $\chi_{560}(463,·)$, $\chi_{560}(209,·)$, $\chi_{560}(83,·)$, $\chi_{560}(169,·)$, $\chi_{560}(349,·)$, $\chi_{560}(223,·)$, $\chi_{560}(489,·)$, $\chi_{560}(363,·)$, $\chi_{560}(503,·)$, $\chi_{560}(447,·)$, $\chi_{560}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{10}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}+\frac{3}{8}a^{11}-\frac{1}{2}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{2}a^{16}+\frac{3}{16}a^{12}-\frac{1}{2}a^{10}-\frac{1}{4}a^{8}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{21}-\frac{1}{4}a^{17}-\frac{13}{32}a^{13}-\frac{1}{4}a^{11}+\frac{3}{8}a^{9}-\frac{1}{2}a^{7}+\frac{1}{32}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{22}-\frac{1}{8}a^{18}-\frac{1}{2}a^{16}-\frac{13}{64}a^{14}+\frac{3}{8}a^{12}-\frac{5}{16}a^{10}+\frac{1}{4}a^{8}+\frac{1}{64}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{128}a^{23}-\frac{1}{16}a^{19}-\frac{1}{4}a^{17}+\frac{51}{128}a^{15}-\frac{5}{16}a^{13}+\frac{11}{32}a^{11}-\frac{3}{8}a^{9}+\frac{1}{128}a^{7}+\frac{1}{8}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{24}-\frac{1}{32}a^{20}-\frac{1}{8}a^{18}-\frac{77}{256}a^{16}-\frac{5}{32}a^{14}-\frac{21}{64}a^{12}-\frac{3}{16}a^{10}-\frac{127}{256}a^{8}-\frac{7}{16}a^{6}+\frac{1}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{512}a^{25}-\frac{1}{64}a^{21}-\frac{1}{16}a^{19}-\frac{77}{512}a^{17}-\frac{5}{64}a^{15}+\frac{43}{128}a^{13}+\frac{13}{32}a^{11}+\frac{129}{512}a^{9}-\frac{7}{32}a^{7}-\frac{15}{32}a^{5}+\frac{3}{8}a^{3}$, $\frac{1}{1024}a^{26}-\frac{1}{128}a^{22}-\frac{1}{32}a^{20}-\frac{77}{1024}a^{18}-\frac{5}{128}a^{16}+\frac{43}{256}a^{14}+\frac{13}{64}a^{12}-\frac{383}{1024}a^{10}-\frac{7}{64}a^{8}-\frac{15}{64}a^{6}-\frac{5}{16}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2048}a^{27}-\frac{1}{256}a^{23}-\frac{1}{64}a^{21}-\frac{77}{2048}a^{19}-\frac{5}{256}a^{17}-\frac{213}{512}a^{15}+\frac{13}{128}a^{13}+\frac{641}{2048}a^{11}+\frac{57}{128}a^{9}-\frac{15}{128}a^{7}+\frac{11}{32}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{323584}a^{28}+\frac{13}{40448}a^{24}+\frac{53}{10112}a^{22}+\frac{7475}{323584}a^{20}-\frac{3461}{40448}a^{18}-\frac{21825}{80896}a^{16}+\frac{6777}{20224}a^{14}-\frac{128319}{323584}a^{12}-\frac{9735}{20224}a^{10}-\frac{1057}{2528}a^{8}-\frac{623}{2528}a^{6}+\frac{153}{632}a^{4}+\frac{51}{158}a^{2}+\frac{21}{79}$, $\frac{1}{647168}a^{29}+\frac{13}{80896}a^{25}+\frac{53}{20224}a^{23}+\frac{7475}{647168}a^{21}-\frac{3461}{80896}a^{19}-\frac{21825}{161792}a^{17}+\frac{6777}{40448}a^{15}-\frac{128319}{647168}a^{13}+\frac{10489}{40448}a^{11}+\frac{1471}{5056}a^{9}+\frac{1905}{5056}a^{7}+\frac{153}{1264}a^{5}+\frac{51}{316}a^{3}-\frac{29}{79}a$, $\frac{1}{28\!\cdots\!36}a^{30}+\frac{21\!\cdots\!17}{35\!\cdots\!92}a^{28}-\frac{14\!\cdots\!29}{35\!\cdots\!92}a^{26}-\frac{15\!\cdots\!27}{89\!\cdots\!48}a^{24}+\frac{53\!\cdots\!47}{28\!\cdots\!36}a^{22}+\frac{17\!\cdots\!27}{17\!\cdots\!96}a^{20}-\frac{23\!\cdots\!33}{71\!\cdots\!84}a^{18}+\frac{29\!\cdots\!39}{17\!\cdots\!96}a^{16}+\frac{19\!\cdots\!45}{28\!\cdots\!36}a^{14}-\frac{16\!\cdots\!65}{35\!\cdots\!92}a^{12}+\frac{15\!\cdots\!61}{17\!\cdots\!96}a^{10}+\frac{15\!\cdots\!87}{44\!\cdots\!24}a^{8}-\frac{13\!\cdots\!95}{11\!\cdots\!56}a^{6}-\frac{13\!\cdots\!27}{27\!\cdots\!64}a^{4}-\frac{26\!\cdots\!53}{69\!\cdots\!16}a^{2}-\frac{10\!\cdots\!69}{17\!\cdots\!79}$, $\frac{1}{57\!\cdots\!72}a^{31}+\frac{21\!\cdots\!17}{71\!\cdots\!84}a^{29}-\frac{14\!\cdots\!29}{71\!\cdots\!84}a^{27}-\frac{15\!\cdots\!27}{17\!\cdots\!96}a^{25}+\frac{53\!\cdots\!47}{57\!\cdots\!72}a^{23}+\frac{17\!\cdots\!27}{35\!\cdots\!92}a^{21}-\frac{23\!\cdots\!33}{14\!\cdots\!68}a^{19}+\frac{29\!\cdots\!39}{35\!\cdots\!92}a^{17}-\frac{26\!\cdots\!91}{57\!\cdots\!72}a^{15}+\frac{19\!\cdots\!27}{71\!\cdots\!84}a^{13}+\frac{15\!\cdots\!61}{35\!\cdots\!92}a^{11}+\frac{15\!\cdots\!87}{89\!\cdots\!48}a^{9}-\frac{13\!\cdots\!95}{22\!\cdots\!12}a^{7}+\frac{13\!\cdots\!37}{55\!\cdots\!28}a^{5}-\frac{26\!\cdots\!53}{13\!\cdots\!32}a^{3}+\frac{82\!\cdots\!55}{17\!\cdots\!79}a$
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
$C_{2}\times C_{4}\times C_{80}\times C_{1360}$, which has order $870400$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{95\!\cdots\!55}{16\!\cdots\!28}a^{30}+\frac{91\!\cdots\!49}{33\!\cdots\!56}a^{28}+\frac{53\!\cdots\!23}{84\!\cdots\!64}a^{26}+\frac{12\!\cdots\!85}{13\!\cdots\!24}a^{24}+\frac{16\!\cdots\!79}{16\!\cdots\!28}a^{22}+\frac{24\!\cdots\!81}{33\!\cdots\!56}a^{20}+\frac{35\!\cdots\!19}{84\!\cdots\!64}a^{18}+\frac{24\!\cdots\!67}{13\!\cdots\!24}a^{16}+\frac{12\!\cdots\!01}{21\!\cdots\!16}a^{14}+\frac{14\!\cdots\!15}{10\!\cdots\!58}a^{12}+\frac{12\!\cdots\!88}{52\!\cdots\!79}a^{10}+\frac{36\!\cdots\!73}{13\!\cdots\!24}a^{8}+\frac{76\!\cdots\!43}{42\!\cdots\!32}a^{6}+\frac{11\!\cdots\!29}{21\!\cdots\!16}a^{4}+\frac{17\!\cdots\!84}{52\!\cdots\!79}a^{2}-\frac{27\!\cdots\!75}{52\!\cdots\!79}$, $\frac{45\!\cdots\!75}{11\!\cdots\!56}a^{30}+\frac{13\!\cdots\!43}{69\!\cdots\!16}a^{28}+\frac{12\!\cdots\!25}{27\!\cdots\!64}a^{26}+\frac{46\!\cdots\!61}{69\!\cdots\!16}a^{24}+\frac{96\!\cdots\!83}{13\!\cdots\!32}a^{22}+\frac{91\!\cdots\!01}{17\!\cdots\!79}a^{20}+\frac{51\!\cdots\!34}{17\!\cdots\!79}a^{18}+\frac{22\!\cdots\!16}{17\!\cdots\!79}a^{16}+\frac{70\!\cdots\!80}{17\!\cdots\!79}a^{14}+\frac{16\!\cdots\!76}{17\!\cdots\!79}a^{12}+\frac{28\!\cdots\!76}{17\!\cdots\!79}a^{10}+\frac{31\!\cdots\!40}{17\!\cdots\!79}a^{8}+\frac{13\!\cdots\!31}{11\!\cdots\!56}a^{6}+\frac{49\!\cdots\!99}{13\!\cdots\!32}a^{4}+\frac{15\!\cdots\!59}{69\!\cdots\!16}a^{2}-\frac{29\!\cdots\!14}{17\!\cdots\!79}$, $\frac{90\!\cdots\!65}{28\!\cdots\!36}a^{30}+\frac{54\!\cdots\!47}{35\!\cdots\!92}a^{28}+\frac{12\!\cdots\!05}{35\!\cdots\!92}a^{26}+\frac{47\!\cdots\!67}{89\!\cdots\!48}a^{24}+\frac{15\!\cdots\!23}{28\!\cdots\!36}a^{22}+\frac{18\!\cdots\!27}{44\!\cdots\!24}a^{20}+\frac{17\!\cdots\!87}{71\!\cdots\!84}a^{18}+\frac{18\!\cdots\!63}{17\!\cdots\!96}a^{16}+\frac{94\!\cdots\!05}{28\!\cdots\!36}a^{14}+\frac{27\!\cdots\!41}{35\!\cdots\!92}a^{12}+\frac{59\!\cdots\!67}{44\!\cdots\!24}a^{10}+\frac{83\!\cdots\!75}{55\!\cdots\!28}a^{8}+\frac{56\!\cdots\!45}{55\!\cdots\!28}a^{6}+\frac{20\!\cdots\!09}{69\!\cdots\!16}a^{4}+\frac{32\!\cdots\!32}{17\!\cdots\!79}a^{2}-\frac{45\!\cdots\!40}{17\!\cdots\!79}$, $\frac{67\!\cdots\!65}{57\!\cdots\!72}a^{31}+\frac{81\!\cdots\!97}{14\!\cdots\!68}a^{29}+\frac{95\!\cdots\!57}{71\!\cdots\!84}a^{27}+\frac{17\!\cdots\!09}{89\!\cdots\!48}a^{25}+\frac{11\!\cdots\!75}{57\!\cdots\!72}a^{23}+\frac{22\!\cdots\!53}{14\!\cdots\!68}a^{21}+\frac{12\!\cdots\!11}{14\!\cdots\!68}a^{19}+\frac{35\!\cdots\!27}{89\!\cdots\!48}a^{17}+\frac{73\!\cdots\!97}{57\!\cdots\!72}a^{15}+\frac{44\!\cdots\!41}{14\!\cdots\!68}a^{13}+\frac{98\!\cdots\!21}{17\!\cdots\!96}a^{11}+\frac{14\!\cdots\!75}{22\!\cdots\!12}a^{9}+\frac{55\!\cdots\!33}{11\!\cdots\!56}a^{7}+\frac{54\!\cdots\!73}{27\!\cdots\!64}a^{5}+\frac{13\!\cdots\!23}{34\!\cdots\!58}a^{3}+\frac{17\!\cdots\!84}{17\!\cdots\!79}a$, $\frac{18\!\cdots\!71}{22\!\cdots\!12}a^{31}+\frac{88\!\cdots\!51}{22\!\cdots\!12}a^{29}+\frac{32\!\cdots\!73}{34\!\cdots\!58}a^{27}+\frac{12\!\cdots\!23}{89\!\cdots\!48}a^{25}+\frac{15\!\cdots\!29}{11\!\cdots\!56}a^{23}+\frac{24\!\cdots\!59}{22\!\cdots\!12}a^{21}+\frac{34\!\cdots\!09}{55\!\cdots\!28}a^{19}+\frac{24\!\cdots\!73}{89\!\cdots\!48}a^{17}+\frac{12\!\cdots\!15}{13\!\cdots\!32}a^{15}+\frac{14\!\cdots\!65}{69\!\cdots\!16}a^{13}+\frac{65\!\cdots\!00}{17\!\cdots\!79}a^{11}+\frac{39\!\cdots\!47}{89\!\cdots\!48}a^{9}+\frac{73\!\cdots\!69}{22\!\cdots\!12}a^{7}+\frac{71\!\cdots\!45}{55\!\cdots\!28}a^{5}+\frac{17\!\cdots\!79}{69\!\cdots\!16}a^{3}+\frac{23\!\cdots\!35}{34\!\cdots\!58}a$, $\frac{47\!\cdots\!13}{44\!\cdots\!24}a^{31}+\frac{72\!\cdots\!59}{14\!\cdots\!68}a^{29}+\frac{42\!\cdots\!21}{35\!\cdots\!92}a^{27}+\frac{31\!\cdots\!13}{17\!\cdots\!96}a^{25}+\frac{82\!\cdots\!27}{44\!\cdots\!24}a^{23}+\frac{20\!\cdots\!69}{14\!\cdots\!68}a^{21}+\frac{36\!\cdots\!59}{45\!\cdots\!48}a^{19}+\frac{12\!\cdots\!13}{35\!\cdots\!92}a^{17}+\frac{63\!\cdots\!41}{55\!\cdots\!28}a^{15}+\frac{39\!\cdots\!71}{14\!\cdots\!68}a^{13}+\frac{17\!\cdots\!85}{35\!\cdots\!92}a^{11}+\frac{51\!\cdots\!67}{89\!\cdots\!48}a^{9}+\frac{24\!\cdots\!15}{55\!\cdots\!28}a^{7}+\frac{47\!\cdots\!69}{27\!\cdots\!64}a^{5}+\frac{46\!\cdots\!71}{13\!\cdots\!32}a^{3}+\frac{30\!\cdots\!79}{34\!\cdots\!58}a$, $\frac{17\!\cdots\!16}{17\!\cdots\!79}a^{31}+\frac{85\!\cdots\!20}{17\!\cdots\!79}a^{29}+\frac{20\!\cdots\!00}{17\!\cdots\!79}a^{27}+\frac{59\!\cdots\!65}{34\!\cdots\!58}a^{25}+\frac{30\!\cdots\!00}{17\!\cdots\!79}a^{23}+\frac{23\!\cdots\!18}{17\!\cdots\!79}a^{21}+\frac{13\!\cdots\!40}{17\!\cdots\!79}a^{19}+\frac{58\!\cdots\!20}{17\!\cdots\!79}a^{17}+\frac{18\!\cdots\!60}{17\!\cdots\!79}a^{15}+\frac{45\!\cdots\!60}{17\!\cdots\!79}a^{13}+\frac{79\!\cdots\!76}{17\!\cdots\!79}a^{11}+\frac{93\!\cdots\!80}{17\!\cdots\!79}a^{9}+\frac{69\!\cdots\!60}{17\!\cdots\!79}a^{7}+\frac{26\!\cdots\!00}{17\!\cdots\!79}a^{5}+\frac{52\!\cdots\!80}{17\!\cdots\!79}a^{3}+\frac{27\!\cdots\!51}{34\!\cdots\!58}a$, $\frac{43\!\cdots\!91}{38\!\cdots\!08}a^{31}+\frac{52\!\cdots\!31}{96\!\cdots\!52}a^{29}+\frac{61\!\cdots\!15}{48\!\cdots\!76}a^{27}+\frac{57\!\cdots\!65}{30\!\cdots\!36}a^{25}+\frac{76\!\cdots\!17}{38\!\cdots\!08}a^{23}+\frac{14\!\cdots\!83}{96\!\cdots\!52}a^{21}+\frac{83\!\cdots\!89}{96\!\cdots\!52}a^{19}+\frac{11\!\cdots\!89}{30\!\cdots\!36}a^{17}+\frac{47\!\cdots\!19}{38\!\cdots\!08}a^{15}+\frac{28\!\cdots\!31}{96\!\cdots\!52}a^{13}+\frac{63\!\cdots\!31}{12\!\cdots\!44}a^{11}+\frac{37\!\cdots\!95}{60\!\cdots\!72}a^{9}+\frac{35\!\cdots\!79}{75\!\cdots\!84}a^{7}+\frac{35\!\cdots\!49}{18\!\cdots\!96}a^{5}+\frac{85\!\cdots\!73}{23\!\cdots\!62}a^{3}+\frac{22\!\cdots\!07}{23\!\cdots\!62}a+1$, $\frac{58\!\cdots\!56}{11\!\cdots\!81}a^{31}+\frac{89\!\cdots\!53}{37\!\cdots\!92}a^{29}+\frac{52\!\cdots\!85}{94\!\cdots\!48}a^{27}+\frac{77\!\cdots\!43}{94\!\cdots\!48}a^{25}+\frac{10\!\cdots\!67}{11\!\cdots\!81}a^{23}+\frac{30\!\cdots\!93}{47\!\cdots\!24}a^{21}+\frac{44\!\cdots\!71}{11\!\cdots\!81}a^{19}+\frac{19\!\cdots\!32}{11\!\cdots\!81}a^{17}+\frac{62\!\cdots\!40}{11\!\cdots\!81}a^{15}+\frac{15\!\cdots\!04}{11\!\cdots\!81}a^{13}+\frac{26\!\cdots\!12}{11\!\cdots\!81}a^{11}+\frac{30\!\cdots\!80}{11\!\cdots\!81}a^{9}+\frac{22\!\cdots\!68}{11\!\cdots\!81}a^{7}+\frac{28\!\cdots\!45}{37\!\cdots\!92}a^{5}+\frac{13\!\cdots\!71}{94\!\cdots\!48}a^{3}+\frac{92\!\cdots\!61}{23\!\cdots\!62}a+1$, $\frac{18\!\cdots\!71}{22\!\cdots\!12}a^{31}-\frac{95\!\cdots\!55}{16\!\cdots\!28}a^{30}+\frac{88\!\cdots\!51}{22\!\cdots\!12}a^{29}-\frac{91\!\cdots\!49}{33\!\cdots\!56}a^{28}+\frac{32\!\cdots\!73}{34\!\cdots\!58}a^{27}-\frac{53\!\cdots\!23}{84\!\cdots\!64}a^{26}+\frac{12\!\cdots\!23}{89\!\cdots\!48}a^{25}-\frac{12\!\cdots\!85}{13\!\cdots\!24}a^{24}+\frac{15\!\cdots\!29}{11\!\cdots\!56}a^{23}-\frac{16\!\cdots\!79}{16\!\cdots\!28}a^{22}+\frac{24\!\cdots\!59}{22\!\cdots\!12}a^{21}-\frac{24\!\cdots\!81}{33\!\cdots\!56}a^{20}+\frac{34\!\cdots\!09}{55\!\cdots\!28}a^{19}-\frac{35\!\cdots\!19}{84\!\cdots\!64}a^{18}+\frac{24\!\cdots\!73}{89\!\cdots\!48}a^{17}-\frac{24\!\cdots\!67}{13\!\cdots\!24}a^{16}+\frac{12\!\cdots\!15}{13\!\cdots\!32}a^{15}-\frac{12\!\cdots\!01}{21\!\cdots\!16}a^{14}+\frac{14\!\cdots\!65}{69\!\cdots\!16}a^{13}-\frac{14\!\cdots\!15}{10\!\cdots\!58}a^{12}+\frac{65\!\cdots\!00}{17\!\cdots\!79}a^{11}-\frac{12\!\cdots\!88}{52\!\cdots\!79}a^{10}+\frac{39\!\cdots\!47}{89\!\cdots\!48}a^{9}-\frac{36\!\cdots\!73}{13\!\cdots\!24}a^{8}+\frac{73\!\cdots\!69}{22\!\cdots\!12}a^{7}-\frac{76\!\cdots\!43}{42\!\cdots\!32}a^{6}+\frac{71\!\cdots\!45}{55\!\cdots\!28}a^{5}-\frac{11\!\cdots\!29}{21\!\cdots\!16}a^{4}+\frac{17\!\cdots\!79}{69\!\cdots\!16}a^{3}-\frac{17\!\cdots\!84}{52\!\cdots\!79}a^{2}+\frac{23\!\cdots\!35}{34\!\cdots\!58}a+\frac{80\!\cdots\!54}{52\!\cdots\!79}$, $\frac{52\!\cdots\!36}{17\!\cdots\!79}a^{31}-\frac{95\!\cdots\!55}{16\!\cdots\!28}a^{30}+\frac{25\!\cdots\!48}{17\!\cdots\!79}a^{29}-\frac{91\!\cdots\!49}{33\!\cdots\!56}a^{28}+\frac{47\!\cdots\!65}{13\!\cdots\!32}a^{27}-\frac{53\!\cdots\!23}{84\!\cdots\!64}a^{26}+\frac{17\!\cdots\!05}{34\!\cdots\!58}a^{25}-\frac{12\!\cdots\!85}{13\!\cdots\!24}a^{24}+\frac{18\!\cdots\!43}{34\!\cdots\!58}a^{23}-\frac{16\!\cdots\!79}{16\!\cdots\!28}a^{22}+\frac{69\!\cdots\!48}{17\!\cdots\!79}a^{21}-\frac{24\!\cdots\!81}{33\!\cdots\!56}a^{20}+\frac{50\!\cdots\!75}{22\!\cdots\!01}a^{19}-\frac{35\!\cdots\!19}{84\!\cdots\!64}a^{18}+\frac{17\!\cdots\!04}{17\!\cdots\!79}a^{17}-\frac{24\!\cdots\!67}{13\!\cdots\!24}a^{16}+\frac{56\!\cdots\!20}{17\!\cdots\!79}a^{15}-\frac{12\!\cdots\!01}{21\!\cdots\!16}a^{14}+\frac{13\!\cdots\!40}{17\!\cdots\!79}a^{13}-\frac{14\!\cdots\!15}{10\!\cdots\!58}a^{12}+\frac{23\!\cdots\!32}{17\!\cdots\!79}a^{11}-\frac{12\!\cdots\!88}{52\!\cdots\!79}a^{10}+\frac{27\!\cdots\!60}{17\!\cdots\!79}a^{9}-\frac{36\!\cdots\!73}{13\!\cdots\!24}a^{8}+\frac{20\!\cdots\!40}{17\!\cdots\!79}a^{7}-\frac{76\!\cdots\!43}{42\!\cdots\!32}a^{6}+\frac{78\!\cdots\!32}{17\!\cdots\!79}a^{5}-\frac{11\!\cdots\!29}{21\!\cdots\!16}a^{4}+\frac{12\!\cdots\!41}{13\!\cdots\!32}a^{3}-\frac{17\!\cdots\!84}{52\!\cdots\!79}a^{2}+\frac{83\!\cdots\!65}{34\!\cdots\!58}a+\frac{27\!\cdots\!75}{52\!\cdots\!79}$, $\frac{14\!\cdots\!57}{22\!\cdots\!12}a^{31}+\frac{17\!\cdots\!51}{55\!\cdots\!28}a^{29}+\frac{42\!\cdots\!19}{55\!\cdots\!28}a^{27}+\frac{39\!\cdots\!49}{34\!\cdots\!58}a^{25}+\frac{32\!\cdots\!37}{27\!\cdots\!64}a^{23}+\frac{61\!\cdots\!93}{69\!\cdots\!16}a^{21}+\frac{88\!\cdots\!42}{17\!\cdots\!79}a^{19}+\frac{38\!\cdots\!84}{17\!\cdots\!79}a^{17}+\frac{12\!\cdots\!88}{17\!\cdots\!79}a^{15}+\frac{30\!\cdots\!64}{17\!\cdots\!79}a^{13}+\frac{52\!\cdots\!16}{17\!\cdots\!79}a^{11}+\frac{62\!\cdots\!16}{17\!\cdots\!79}a^{9}+\frac{59\!\cdots\!45}{22\!\cdots\!12}a^{7}+\frac{57\!\cdots\!81}{55\!\cdots\!28}a^{5}+\frac{14\!\cdots\!27}{69\!\cdots\!16}a^{3}+\frac{18\!\cdots\!29}{34\!\cdots\!58}a-1$, $\frac{10\!\cdots\!75}{89\!\cdots\!48}a^{31}+\frac{83\!\cdots\!15}{14\!\cdots\!68}a^{29}+\frac{24\!\cdots\!69}{17\!\cdots\!96}a^{27}+\frac{36\!\cdots\!43}{17\!\cdots\!96}a^{25}+\frac{18\!\cdots\!67}{89\!\cdots\!48}a^{23}+\frac{23\!\cdots\!05}{14\!\cdots\!68}a^{21}+\frac{10\!\cdots\!45}{11\!\cdots\!56}a^{19}+\frac{14\!\cdots\!69}{35\!\cdots\!92}a^{17}+\frac{73\!\cdots\!11}{55\!\cdots\!28}a^{15}+\frac{45\!\cdots\!19}{14\!\cdots\!68}a^{13}+\frac{10\!\cdots\!51}{17\!\cdots\!96}a^{11}+\frac{15\!\cdots\!01}{22\!\cdots\!12}a^{9}+\frac{14\!\cdots\!81}{27\!\cdots\!64}a^{7}+\frac{56\!\cdots\!27}{27\!\cdots\!64}a^{5}+\frac{13\!\cdots\!91}{34\!\cdots\!58}a^{3}+\frac{18\!\cdots\!64}{17\!\cdots\!79}a-1$, $\frac{52\!\cdots\!36}{17\!\cdots\!79}a^{31}+\frac{25\!\cdots\!48}{17\!\cdots\!79}a^{29}+\frac{47\!\cdots\!65}{13\!\cdots\!32}a^{27}+\frac{17\!\cdots\!05}{34\!\cdots\!58}a^{25}+\frac{18\!\cdots\!43}{34\!\cdots\!58}a^{23}+\frac{69\!\cdots\!48}{17\!\cdots\!79}a^{21}+\frac{50\!\cdots\!75}{22\!\cdots\!01}a^{19}+\frac{17\!\cdots\!04}{17\!\cdots\!79}a^{17}+\frac{56\!\cdots\!20}{17\!\cdots\!79}a^{15}+\frac{13\!\cdots\!40}{17\!\cdots\!79}a^{13}+\frac{23\!\cdots\!32}{17\!\cdots\!79}a^{11}+\frac{27\!\cdots\!60}{17\!\cdots\!79}a^{9}+\frac{20\!\cdots\!40}{17\!\cdots\!79}a^{7}+\frac{78\!\cdots\!32}{17\!\cdots\!79}a^{5}+\frac{12\!\cdots\!41}{13\!\cdots\!32}a^{3}+\frac{83\!\cdots\!65}{34\!\cdots\!58}a+1$, $\frac{67\!\cdots\!65}{57\!\cdots\!72}a^{31}+\frac{95\!\cdots\!55}{16\!\cdots\!28}a^{30}+\frac{81\!\cdots\!97}{14\!\cdots\!68}a^{29}+\frac{91\!\cdots\!49}{33\!\cdots\!56}a^{28}+\frac{95\!\cdots\!57}{71\!\cdots\!84}a^{27}+\frac{53\!\cdots\!23}{84\!\cdots\!64}a^{26}+\frac{17\!\cdots\!09}{89\!\cdots\!48}a^{25}+\frac{12\!\cdots\!85}{13\!\cdots\!24}a^{24}+\frac{11\!\cdots\!75}{57\!\cdots\!72}a^{23}+\frac{16\!\cdots\!79}{16\!\cdots\!28}a^{22}+\frac{22\!\cdots\!53}{14\!\cdots\!68}a^{21}+\frac{24\!\cdots\!81}{33\!\cdots\!56}a^{20}+\frac{12\!\cdots\!11}{14\!\cdots\!68}a^{19}+\frac{35\!\cdots\!19}{84\!\cdots\!64}a^{18}+\frac{35\!\cdots\!27}{89\!\cdots\!48}a^{17}+\frac{24\!\cdots\!67}{13\!\cdots\!24}a^{16}+\frac{73\!\cdots\!97}{57\!\cdots\!72}a^{15}+\frac{12\!\cdots\!01}{21\!\cdots\!16}a^{14}+\frac{44\!\cdots\!41}{14\!\cdots\!68}a^{13}+\frac{14\!\cdots\!15}{10\!\cdots\!58}a^{12}+\frac{98\!\cdots\!21}{17\!\cdots\!96}a^{11}+\frac{12\!\cdots\!88}{52\!\cdots\!79}a^{10}+\frac{14\!\cdots\!75}{22\!\cdots\!12}a^{9}+\frac{36\!\cdots\!73}{13\!\cdots\!24}a^{8}+\frac{55\!\cdots\!33}{11\!\cdots\!56}a^{7}+\frac{76\!\cdots\!43}{42\!\cdots\!32}a^{6}+\frac{54\!\cdots\!73}{27\!\cdots\!64}a^{5}+\frac{11\!\cdots\!29}{21\!\cdots\!16}a^{4}+\frac{13\!\cdots\!23}{34\!\cdots\!58}a^{3}+\frac{17\!\cdots\!84}{52\!\cdots\!79}a^{2}+\frac{17\!\cdots\!84}{17\!\cdots\!79}a-\frac{27\!\cdots\!75}{52\!\cdots\!79}$
| 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: | \( 48613521256.81357 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 48613521256.81357 \cdot 870400}{2\cdot\sqrt{156938077449417789520626992646455296000000000000000000000000}}\cr\approx \mathstrut & 0.315108193677728 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536)
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 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536, 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 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536);
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 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\times C_4^2$ (as 32T36):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_4^2$ |
| Character table for $C_2\times C_4^2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{8}$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $8$ | $2$ | $48$ | |||
| Deg $16$ | $8$ | $2$ | $48$ | ||||
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ | |
|
\(7\)
| 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |