Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648)
gp: K = bnfinit(y^16 - 4*y^15 - 29*y^14 + 630*y^13 - 4609*y^12 - 33136*y^11 + 168901*y^10 + 1035038*y^9 - 2476072*y^8 - 17969440*y^7 + 16115584*y^6 + 185656320*y^5 + 24797184*y^4 - 1016004608*y^3 - 749731840*y^2 + 2181038080*y + 2147483648, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648)
\( x^{16} - 4 x^{15} - 29 x^{14} + 630 x^{13} - 4609 x^{12} - 33136 x^{11} + 168901 x^{10} + \cdots + 2147483648 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(123315986626517312705762123587383695257\)
\(\medspace = 7^{12}\cdot 73^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(240.26\) | 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: | $7^{3/4}73^{15/16}\approx 240.26400458491216$ | ||
| Ramified primes: |
\(7\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{73}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{64}a^{7}-\frac{1}{32}a^{6}-\frac{1}{32}a^{5}-\frac{1}{16}a^{4}+\frac{9}{64}a^{3}+\frac{3}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{128}a^{8}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}-\frac{7}{128}a^{4}+\frac{3}{16}a^{3}+\frac{1}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{256}a^{9}-\frac{1}{256}a^{8}-\frac{3}{128}a^{6}-\frac{11}{256}a^{5}+\frac{7}{256}a^{4}+\frac{21}{128}a^{3}-\frac{1}{8}a$, $\frac{1}{2048}a^{10}+\frac{1}{2048}a^{9}+\frac{1}{512}a^{8}+\frac{3}{1024}a^{7}-\frac{19}{2048}a^{6}+\frac{89}{2048}a^{5}-\frac{41}{1024}a^{4}+\frac{9}{64}a^{3}-\frac{9}{64}a^{2}$, $\frac{1}{4096}a^{11}+\frac{3}{4096}a^{9}+\frac{1}{2048}a^{8}+\frac{7}{4096}a^{7}+\frac{11}{1024}a^{6}-\frac{235}{4096}a^{5}-\frac{7}{2048}a^{4}-\frac{25}{128}a^{3}+\frac{23}{128}a^{2}+\frac{1}{16}a$, $\frac{1}{32768}a^{12}+\frac{3}{32768}a^{10}+\frac{17}{16384}a^{9}-\frac{121}{32768}a^{8}+\frac{27}{8192}a^{7}-\frac{1003}{32768}a^{6}-\frac{503}{16384}a^{5}+\frac{51}{1024}a^{4}+\frac{51}{1024}a^{3}+\frac{27}{128}a^{2}-\frac{1}{4}a$, $\frac{1}{262144}a^{13}-\frac{29}{262144}a^{11}+\frac{1}{131072}a^{10}-\frac{249}{262144}a^{9}-\frac{149}{65536}a^{8}-\frac{1419}{262144}a^{7}-\frac{391}{131072}a^{6}-\frac{123}{8192}a^{5}-\frac{445}{8192}a^{4}-\frac{157}{1024}a^{3}-\frac{1}{64}a^{2}+\frac{1}{4}a$, $\frac{1}{4194304}a^{14}+\frac{1}{1048576}a^{13}-\frac{61}{4194304}a^{12}-\frac{185}{2097152}a^{11}+\frac{431}{4194304}a^{10}+\frac{945}{524288}a^{9}-\frac{9659}{4194304}a^{8}-\frac{9693}{2097152}a^{7}-\frac{11367}{524288}a^{6}+\frac{309}{131072}a^{5}-\frac{447}{32768}a^{4}+\frac{33}{1024}a^{3}+\frac{19}{512}a^{2}-\frac{3}{32}a$, $\frac{1}{19\!\cdots\!96}a^{15}-\frac{42\!\cdots\!77}{49\!\cdots\!24}a^{14}+\frac{23\!\cdots\!71}{19\!\cdots\!96}a^{13}-\frac{12\!\cdots\!61}{99\!\cdots\!48}a^{12}-\frac{22\!\cdots\!73}{19\!\cdots\!96}a^{11}-\frac{25\!\cdots\!47}{12\!\cdots\!56}a^{10}+\frac{25\!\cdots\!41}{19\!\cdots\!96}a^{9}+\frac{28\!\cdots\!07}{99\!\cdots\!48}a^{8}+\frac{17\!\cdots\!47}{24\!\cdots\!12}a^{7}+\frac{13\!\cdots\!47}{62\!\cdots\!28}a^{6}-\frac{84\!\cdots\!41}{15\!\cdots\!32}a^{5}-\frac{64\!\cdots\!47}{19\!\cdots\!04}a^{4}-\frac{53\!\cdots\!69}{24\!\cdots\!88}a^{3}-\frac{64\!\cdots\!91}{30\!\cdots\!36}a^{2}+\frac{65\!\cdots\!77}{19\!\cdots\!96}a+\frac{27\!\cdots\!10}{59\!\cdots\!03}$
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
Trivial group, which has order $1$ (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: | $11$ | 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: |
$\frac{50\!\cdots\!97}{24\!\cdots\!12}a^{15}-\frac{17\!\cdots\!79}{12\!\cdots\!56}a^{14}-\frac{40\!\cdots\!57}{24\!\cdots\!12}a^{13}+\frac{20\!\cdots\!43}{15\!\cdots\!32}a^{12}-\frac{33\!\cdots\!85}{24\!\cdots\!12}a^{11}-\frac{34\!\cdots\!51}{12\!\cdots\!56}a^{10}+\frac{10\!\cdots\!85}{24\!\cdots\!12}a^{9}+\frac{48\!\cdots\!17}{62\!\cdots\!28}a^{8}-\frac{47\!\cdots\!97}{62\!\cdots\!28}a^{7}-\frac{20\!\cdots\!67}{15\!\cdots\!32}a^{6}+\frac{31\!\cdots\!49}{38\!\cdots\!08}a^{5}+\frac{14\!\cdots\!67}{97\!\cdots\!52}a^{4}-\frac{71\!\cdots\!93}{15\!\cdots\!68}a^{3}-\frac{13\!\cdots\!83}{15\!\cdots\!68}a^{2}+\frac{11\!\cdots\!47}{95\!\cdots\!48}a+\frac{10\!\cdots\!21}{59\!\cdots\!03}$, $\frac{37\!\cdots\!19}{99\!\cdots\!48}a^{15}-\frac{66\!\cdots\!05}{24\!\cdots\!12}a^{14}-\frac{30\!\cdots\!11}{99\!\cdots\!48}a^{13}+\frac{12\!\cdots\!45}{49\!\cdots\!24}a^{12}-\frac{24\!\cdots\!03}{99\!\cdots\!48}a^{11}-\frac{61\!\cdots\!51}{12\!\cdots\!56}a^{10}+\frac{81\!\cdots\!43}{99\!\cdots\!48}a^{9}+\frac{72\!\cdots\!81}{49\!\cdots\!24}a^{8}-\frac{18\!\cdots\!01}{12\!\cdots\!56}a^{7}-\frac{79\!\cdots\!73}{31\!\cdots\!64}a^{6}+\frac{11\!\cdots\!39}{77\!\cdots\!16}a^{5}+\frac{69\!\cdots\!77}{24\!\cdots\!88}a^{4}-\frac{10\!\cdots\!47}{12\!\cdots\!44}a^{3}-\frac{12\!\cdots\!63}{76\!\cdots\!84}a^{2}+\frac{10\!\cdots\!21}{47\!\cdots\!24}a+\frac{17\!\cdots\!33}{59\!\cdots\!03}$, $\frac{67\!\cdots\!73}{49\!\cdots\!24}a^{15}-\frac{30\!\cdots\!91}{31\!\cdots\!64}a^{14}-\frac{52\!\cdots\!33}{49\!\cdots\!24}a^{13}+\frac{22\!\cdots\!29}{24\!\cdots\!12}a^{12}-\frac{46\!\cdots\!05}{49\!\cdots\!24}a^{11}-\frac{21\!\cdots\!97}{12\!\cdots\!56}a^{10}+\frac{15\!\cdots\!57}{49\!\cdots\!24}a^{9}+\frac{99\!\cdots\!77}{24\!\cdots\!12}a^{8}-\frac{16\!\cdots\!35}{31\!\cdots\!64}a^{7}-\frac{13\!\cdots\!67}{19\!\cdots\!04}a^{6}+\frac{27\!\cdots\!69}{48\!\cdots\!76}a^{5}+\frac{61\!\cdots\!75}{97\!\cdots\!52}a^{4}-\frac{18\!\cdots\!25}{60\!\cdots\!72}a^{3}-\frac{47\!\cdots\!67}{15\!\cdots\!68}a^{2}+\frac{57\!\cdots\!83}{95\!\cdots\!48}a+\frac{37\!\cdots\!27}{59\!\cdots\!03}$, $\frac{41\!\cdots\!77}{99\!\cdots\!48}a^{15}-\frac{81\!\cdots\!39}{24\!\cdots\!12}a^{14}+\frac{89\!\cdots\!47}{99\!\cdots\!48}a^{13}+\frac{13\!\cdots\!11}{49\!\cdots\!24}a^{12}-\frac{30\!\cdots\!29}{99\!\cdots\!48}a^{11}-\frac{23\!\cdots\!11}{12\!\cdots\!56}a^{10}+\frac{83\!\cdots\!61}{99\!\cdots\!48}a^{9}+\frac{14\!\cdots\!19}{49\!\cdots\!24}a^{8}-\frac{16\!\cdots\!59}{12\!\cdots\!56}a^{7}-\frac{18\!\cdots\!99}{31\!\cdots\!64}a^{6}+\frac{96\!\cdots\!09}{77\!\cdots\!16}a^{5}+\frac{54\!\cdots\!03}{48\!\cdots\!76}a^{4}-\frac{79\!\cdots\!53}{12\!\cdots\!44}a^{3}-\frac{10\!\cdots\!29}{11\!\cdots\!06}a^{2}+\frac{74\!\cdots\!57}{47\!\cdots\!24}a+\frac{11\!\cdots\!33}{59\!\cdots\!03}$, $\frac{24\!\cdots\!09}{12\!\cdots\!56}a^{15}-\frac{20\!\cdots\!21}{15\!\cdots\!32}a^{14}-\frac{39\!\cdots\!01}{12\!\cdots\!56}a^{13}+\frac{70\!\cdots\!09}{62\!\cdots\!28}a^{12}-\frac{14\!\cdots\!13}{12\!\cdots\!56}a^{11}-\frac{47\!\cdots\!99}{31\!\cdots\!64}a^{10}+\frac{45\!\cdots\!45}{12\!\cdots\!56}a^{9}+\frac{26\!\cdots\!17}{62\!\cdots\!28}a^{8}-\frac{60\!\cdots\!07}{97\!\cdots\!52}a^{7}-\frac{15\!\cdots\!33}{19\!\cdots\!04}a^{6}+\frac{29\!\cdots\!27}{48\!\cdots\!76}a^{5}+\frac{24\!\cdots\!33}{24\!\cdots\!88}a^{4}-\frac{12\!\cdots\!85}{38\!\cdots\!92}a^{3}-\frac{23\!\cdots\!47}{38\!\cdots\!92}a^{2}+\frac{18\!\cdots\!67}{23\!\cdots\!12}a+\frac{65\!\cdots\!87}{59\!\cdots\!03}$, $\frac{65\!\cdots\!33}{24\!\cdots\!12}a^{15}-\frac{15\!\cdots\!29}{77\!\cdots\!16}a^{14}-\frac{30\!\cdots\!85}{24\!\cdots\!12}a^{13}+\frac{19\!\cdots\!37}{12\!\cdots\!56}a^{12}-\frac{16\!\cdots\!25}{24\!\cdots\!12}a^{11}-\frac{85\!\cdots\!13}{62\!\cdots\!28}a^{10}+\frac{69\!\cdots\!61}{24\!\cdots\!12}a^{9}+\frac{51\!\cdots\!13}{12\!\cdots\!56}a^{8}-\frac{49\!\cdots\!49}{15\!\cdots\!32}a^{7}-\frac{60\!\cdots\!27}{97\!\cdots\!52}a^{6}+\frac{15\!\cdots\!93}{30\!\cdots\!36}a^{5}+\frac{25\!\cdots\!91}{48\!\cdots\!76}a^{4}+\frac{10\!\cdots\!45}{38\!\cdots\!92}a^{3}-\frac{15\!\cdots\!43}{76\!\cdots\!84}a^{2}-\frac{31\!\cdots\!65}{23\!\cdots\!12}a+\frac{14\!\cdots\!57}{59\!\cdots\!03}$, $\frac{31\!\cdots\!21}{99\!\cdots\!48}a^{15}-\frac{57\!\cdots\!99}{24\!\cdots\!12}a^{14}-\frac{25\!\cdots\!05}{99\!\cdots\!48}a^{13}+\frac{10\!\cdots\!99}{49\!\cdots\!24}a^{12}-\frac{21\!\cdots\!65}{99\!\cdots\!48}a^{11}-\frac{52\!\cdots\!89}{12\!\cdots\!56}a^{10}+\frac{69\!\cdots\!09}{99\!\cdots\!48}a^{9}+\frac{62\!\cdots\!43}{49\!\cdots\!24}a^{8}-\frac{15\!\cdots\!67}{12\!\cdots\!56}a^{7}-\frac{67\!\cdots\!07}{31\!\cdots\!64}a^{6}+\frac{10\!\cdots\!65}{77\!\cdots\!16}a^{5}+\frac{59\!\cdots\!43}{24\!\cdots\!88}a^{4}-\frac{90\!\cdots\!09}{12\!\cdots\!44}a^{3}-\frac{10\!\cdots\!11}{76\!\cdots\!84}a^{2}+\frac{11\!\cdots\!55}{59\!\cdots\!03}a+\frac{16\!\cdots\!63}{59\!\cdots\!03}$, $\frac{66\!\cdots\!27}{49\!\cdots\!24}a^{15}-\frac{84\!\cdots\!11}{12\!\cdots\!56}a^{14}-\frac{15\!\cdots\!03}{49\!\cdots\!24}a^{13}+\frac{21\!\cdots\!81}{24\!\cdots\!12}a^{12}-\frac{35\!\cdots\!43}{49\!\cdots\!24}a^{11}-\frac{11\!\cdots\!47}{31\!\cdots\!64}a^{10}+\frac{13\!\cdots\!55}{49\!\cdots\!24}a^{9}+\frac{27\!\cdots\!37}{24\!\cdots\!12}a^{8}-\frac{28\!\cdots\!67}{62\!\cdots\!28}a^{7}-\frac{29\!\cdots\!43}{15\!\cdots\!32}a^{6}+\frac{16\!\cdots\!25}{38\!\cdots\!08}a^{5}+\frac{98\!\cdots\!07}{48\!\cdots\!76}a^{4}-\frac{11\!\cdots\!47}{60\!\cdots\!72}a^{3}-\frac{87\!\cdots\!99}{76\!\cdots\!84}a^{2}+\frac{30\!\cdots\!63}{11\!\cdots\!06}a+\frac{15\!\cdots\!23}{59\!\cdots\!03}$, $\frac{36\!\cdots\!61}{24\!\cdots\!12}a^{15}-\frac{15\!\cdots\!97}{62\!\cdots\!28}a^{14}-\frac{12\!\cdots\!45}{24\!\cdots\!12}a^{13}+\frac{10\!\cdots\!99}{12\!\cdots\!56}a^{12}-\frac{12\!\cdots\!49}{24\!\cdots\!12}a^{11}-\frac{93\!\cdots\!65}{15\!\cdots\!32}a^{10}+\frac{27\!\cdots\!89}{24\!\cdots\!12}a^{9}+\frac{22\!\cdots\!11}{12\!\cdots\!56}a^{8}+\frac{12\!\cdots\!55}{31\!\cdots\!64}a^{7}-\frac{19\!\cdots\!27}{77\!\cdots\!16}a^{6}-\frac{66\!\cdots\!05}{19\!\cdots\!04}a^{5}+\frac{59\!\cdots\!79}{30\!\cdots\!36}a^{4}+\frac{14\!\cdots\!17}{30\!\cdots\!36}a^{3}-\frac{18\!\cdots\!15}{47\!\cdots\!24}a^{2}-\frac{47\!\cdots\!45}{23\!\cdots\!12}a-\frac{82\!\cdots\!39}{59\!\cdots\!03}$, $\frac{30\!\cdots\!87}{99\!\cdots\!48}a^{15}-\frac{13\!\cdots\!85}{24\!\cdots\!12}a^{14}-\frac{99\!\cdots\!27}{99\!\cdots\!48}a^{13}+\frac{84\!\cdots\!09}{49\!\cdots\!24}a^{12}-\frac{10\!\cdots\!27}{99\!\cdots\!48}a^{11}-\frac{15\!\cdots\!27}{12\!\cdots\!56}a^{10}+\frac{23\!\cdots\!99}{99\!\cdots\!48}a^{9}+\frac{18\!\cdots\!89}{49\!\cdots\!24}a^{8}+\frac{10\!\cdots\!59}{12\!\cdots\!56}a^{7}-\frac{16\!\cdots\!69}{31\!\cdots\!64}a^{6}-\frac{55\!\cdots\!09}{77\!\cdots\!16}a^{5}+\frac{97\!\cdots\!41}{24\!\cdots\!88}a^{4}+\frac{12\!\cdots\!37}{12\!\cdots\!44}a^{3}-\frac{62\!\cdots\!61}{76\!\cdots\!84}a^{2}-\frac{49\!\cdots\!17}{11\!\cdots\!06}a-\frac{17\!\cdots\!17}{59\!\cdots\!03}$, $\frac{15\!\cdots\!47}{24\!\cdots\!12}a^{15}-\frac{97\!\cdots\!69}{12\!\cdots\!56}a^{14}+\frac{11\!\cdots\!97}{24\!\cdots\!12}a^{13}-\frac{13\!\cdots\!11}{15\!\cdots\!32}a^{12}-\frac{69\!\cdots\!71}{24\!\cdots\!12}a^{11}+\frac{37\!\cdots\!83}{12\!\cdots\!56}a^{10}+\frac{19\!\cdots\!11}{24\!\cdots\!12}a^{9}-\frac{18\!\cdots\!81}{62\!\cdots\!28}a^{8}-\frac{80\!\cdots\!95}{62\!\cdots\!28}a^{7}-\frac{46\!\cdots\!85}{15\!\cdots\!32}a^{6}+\frac{49\!\cdots\!83}{38\!\cdots\!08}a^{5}+\frac{94\!\cdots\!17}{97\!\cdots\!52}a^{4}-\frac{20\!\cdots\!15}{30\!\cdots\!36}a^{3}-\frac{11\!\cdots\!29}{15\!\cdots\!68}a^{2}+\frac{13\!\cdots\!11}{95\!\cdots\!48}a+\frac{94\!\cdots\!97}{59\!\cdots\!03}$
| 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: | \( 514588263316000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 514588263316000000 \cdot 1}{2\cdot\sqrt{123315986626517312705762123587383695257}}\cr\approx \mathstrut & 9244.41915394571 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648)
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^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648, 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^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648);
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^16 - 4*x^15 - 29*x^14 + 630*x^13 - 4609*x^12 - 33136*x^11 + 168901*x^10 + 1035038*x^9 - 2476072*x^8 - 17969440*x^7 + 16115584*x^6 + 185656320*x^5 + 24797184*x^4 - 1016004608*x^3 - 749731840*x^2 + 2181038080*x + 2147483648);
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
$\OD_{32}$ (as 16T22):
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 32 |
| The 20 conjugacy class representatives for $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.26524803844351897.1 |
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
| Galois closure: | deg 32 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.1.0.1}{1} }^{16}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | $16$ | R | $16$ | $16$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | $16$ | $16$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.16.12.3 | $x^{16} - 84 x^{12} + 6272 x^{8} + 39788 x^{4} + 64827$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |
|
\(73\)
| 73.16.15.1 | $x^{16} + 657$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |