Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 - 36*y^14 + 184*y^13 + 276*y^12 - 2532*y^11 + 1576*y^10 + 10204*y^9 - 12939*y^8 - 15196*y^7 + 25960*y^6 + 6876*y^5 - 16638*y^4 + 32*y^3 + 1620*y^2 + 196*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1)
\( x^{16} - 4 x^{15} - 36 x^{14} + 184 x^{13} + 276 x^{12} - 2532 x^{11} + 1576 x^{10} + 10204 x^{9} + \cdots + 1 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2282521714753536000000000000\)
\(\medspace = 2^{44}\cdot 3^{12}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(51.27\) | 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: | $2^{3}3^{3/4}5^{3/4}\approx 60.97592977855377$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{38\!\cdots\!49}a^{15}-\frac{2613668157701}{66149120091131}a^{14}+\frac{61\!\cdots\!20}{38\!\cdots\!49}a^{13}+\frac{46\!\cdots\!23}{38\!\cdots\!49}a^{12}-\frac{48\!\cdots\!15}{35\!\cdots\!59}a^{11}-\frac{20\!\cdots\!76}{38\!\cdots\!49}a^{10}+\frac{60\!\cdots\!63}{22\!\cdots\!97}a^{9}+\frac{78\!\cdots\!00}{38\!\cdots\!49}a^{8}+\frac{51\!\cdots\!50}{38\!\cdots\!49}a^{7}+\frac{38\!\cdots\!95}{55\!\cdots\!07}a^{6}-\frac{45\!\cdots\!57}{38\!\cdots\!49}a^{5}-\frac{91\!\cdots\!93}{55\!\cdots\!07}a^{4}+\frac{70\!\cdots\!39}{38\!\cdots\!49}a^{3}+\frac{18\!\cdots\!26}{55\!\cdots\!07}a^{2}+\frac{18\!\cdots\!04}{38\!\cdots\!49}a+\frac{72\!\cdots\!76}{38\!\cdots\!49}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (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: |
$\frac{254327836804}{4675796575919}a^{15}-\frac{184747788}{795338761}a^{14}-\frac{8922297839472}{4675796575919}a^{13}+\frac{49314203503120}{4675796575919}a^{12}+\frac{59225677758540}{4675796575919}a^{11}-\frac{665612511166272}{4675796575919}a^{10}+\frac{32515776620772}{275046857407}a^{9}+\frac{25\!\cdots\!99}{4675796575919}a^{8}-\frac{38\!\cdots\!80}{4675796575919}a^{7}-\frac{455476584648160}{667970939417}a^{6}+\frac{74\!\cdots\!60}{4675796575919}a^{5}+\frac{53290945629450}{667970939417}a^{4}-\frac{44\!\cdots\!80}{4675796575919}a^{3}+\frac{121706188780500}{667970939417}a^{2}+\frac{264896741731668}{4675796575919}a-\frac{6176681687430}{4675796575919}$, $\frac{67\!\cdots\!96}{38\!\cdots\!49}a^{15}-\frac{5085483802878}{66149120091131}a^{14}-\frac{23\!\cdots\!67}{38\!\cdots\!49}a^{13}+\frac{13\!\cdots\!69}{38\!\cdots\!49}a^{12}+\frac{12\!\cdots\!15}{35\!\cdots\!59}a^{11}-\frac{18\!\cdots\!23}{38\!\cdots\!49}a^{10}+\frac{10\!\cdots\!08}{22\!\cdots\!97}a^{9}+\frac{67\!\cdots\!29}{38\!\cdots\!49}a^{8}-\frac{11\!\cdots\!57}{38\!\cdots\!49}a^{7}-\frac{10\!\cdots\!69}{55\!\cdots\!07}a^{6}+\frac{22\!\cdots\!72}{38\!\cdots\!49}a^{5}-\frac{30\!\cdots\!08}{55\!\cdots\!07}a^{4}-\frac{13\!\cdots\!67}{38\!\cdots\!49}a^{3}+\frac{63\!\cdots\!04}{55\!\cdots\!07}a^{2}+\frac{61\!\cdots\!19}{38\!\cdots\!49}a-\frac{11\!\cdots\!58}{38\!\cdots\!49}$, $\frac{71\!\cdots\!52}{38\!\cdots\!49}a^{15}-\frac{50971991314856}{66149120091131}a^{14}-\frac{25\!\cdots\!19}{38\!\cdots\!49}a^{13}+\frac{13\!\cdots\!54}{38\!\cdots\!49}a^{12}+\frac{16\!\cdots\!07}{35\!\cdots\!59}a^{11}-\frac{18\!\cdots\!48}{38\!\cdots\!49}a^{10}+\frac{84\!\cdots\!44}{22\!\cdots\!97}a^{9}+\frac{71\!\cdots\!16}{38\!\cdots\!49}a^{8}-\frac{10\!\cdots\!63}{38\!\cdots\!49}a^{7}-\frac{13\!\cdots\!92}{55\!\cdots\!07}a^{6}+\frac{20\!\cdots\!88}{38\!\cdots\!49}a^{5}+\frac{24\!\cdots\!55}{55\!\cdots\!07}a^{4}-\frac{12\!\cdots\!23}{38\!\cdots\!49}a^{3}+\frac{28\!\cdots\!37}{55\!\cdots\!07}a^{2}+\frac{82\!\cdots\!53}{38\!\cdots\!49}a+\frac{65\!\cdots\!16}{38\!\cdots\!49}$, $\frac{260140726908892}{32\!\cdots\!71}a^{15}-\frac{184754369054}{555874958749}a^{14}-\frac{92\!\cdots\!71}{32\!\cdots\!71}a^{13}+\frac{49\!\cdots\!01}{32\!\cdots\!71}a^{12}+\frac{58\!\cdots\!63}{297089898407761}a^{11}-\frac{67\!\cdots\!78}{32\!\cdots\!71}a^{10}+\frac{51\!\cdots\!12}{32\!\cdots\!71}a^{9}+\frac{25\!\cdots\!89}{32\!\cdots\!71}a^{8}-\frac{37\!\cdots\!87}{32\!\cdots\!71}a^{7}-\frac{34\!\cdots\!66}{32\!\cdots\!71}a^{6}+\frac{73\!\cdots\!96}{32\!\cdots\!71}a^{5}+\frac{67\!\cdots\!30}{32\!\cdots\!71}a^{4}-\frac{44\!\cdots\!27}{32\!\cdots\!71}a^{3}+\frac{71\!\cdots\!61}{32\!\cdots\!71}a^{2}+\frac{30\!\cdots\!17}{32\!\cdots\!71}a+\frac{10\!\cdots\!56}{32\!\cdots\!71}$, $\frac{710065945994778}{35\!\cdots\!59}a^{15}-\frac{563275833090}{6013556371921}a^{14}-\frac{24\!\cdots\!80}{35\!\cdots\!59}a^{13}+\frac{14\!\cdots\!15}{35\!\cdots\!59}a^{12}+\frac{14\!\cdots\!18}{35\!\cdots\!59}a^{11}-\frac{19\!\cdots\!76}{35\!\cdots\!59}a^{10}+\frac{11\!\cdots\!26}{20\!\cdots\!27}a^{9}+\frac{75\!\cdots\!92}{35\!\cdots\!59}a^{8}-\frac{12\!\cdots\!66}{35\!\cdots\!59}a^{7}-\frac{13\!\cdots\!60}{50\!\cdots\!37}a^{6}+\frac{23\!\cdots\!98}{35\!\cdots\!59}a^{5}+\frac{28\!\cdots\!92}{50\!\cdots\!37}a^{4}-\frac{14\!\cdots\!28}{35\!\cdots\!59}a^{3}+\frac{26\!\cdots\!38}{50\!\cdots\!37}a^{2}+\frac{11\!\cdots\!10}{35\!\cdots\!59}a+\frac{57\!\cdots\!60}{35\!\cdots\!59}$, $\frac{625593585316944}{35\!\cdots\!59}a^{15}-\frac{494585854540}{6013556371921}a^{14}-\frac{21\!\cdots\!35}{35\!\cdots\!59}a^{13}+\frac{13\!\cdots\!40}{35\!\cdots\!59}a^{12}+\frac{12\!\cdots\!72}{35\!\cdots\!59}a^{11}-\frac{17\!\cdots\!24}{35\!\cdots\!59}a^{10}+\frac{96\!\cdots\!82}{20\!\cdots\!27}a^{9}+\frac{67\!\cdots\!20}{35\!\cdots\!59}a^{8}-\frac{11\!\cdots\!96}{35\!\cdots\!59}a^{7}-\frac{12\!\cdots\!76}{50\!\cdots\!37}a^{6}+\frac{21\!\cdots\!82}{35\!\cdots\!59}a^{5}+\frac{15\!\cdots\!01}{50\!\cdots\!37}a^{4}-\frac{13\!\cdots\!32}{35\!\cdots\!59}a^{3}+\frac{31\!\cdots\!12}{50\!\cdots\!37}a^{2}+\frac{99\!\cdots\!44}{35\!\cdots\!59}a+\frac{13\!\cdots\!65}{35\!\cdots\!59}$, $\frac{84472360677834}{35\!\cdots\!59}a^{15}-\frac{68689978550}{6013556371921}a^{14}-\frac{27\!\cdots\!45}{35\!\cdots\!59}a^{13}+\frac{17\!\cdots\!75}{35\!\cdots\!59}a^{12}+\frac{12\!\cdots\!46}{35\!\cdots\!59}a^{11}-\frac{22\!\cdots\!52}{35\!\cdots\!59}a^{10}+\frac{14\!\cdots\!44}{20\!\cdots\!27}a^{9}+\frac{76\!\cdots\!72}{35\!\cdots\!59}a^{8}-\frac{12\!\cdots\!70}{35\!\cdots\!59}a^{7}-\frac{15\!\cdots\!84}{50\!\cdots\!37}a^{6}+\frac{18\!\cdots\!16}{35\!\cdots\!59}a^{5}+\frac{13\!\cdots\!91}{50\!\cdots\!37}a^{4}-\frac{98\!\cdots\!96}{35\!\cdots\!59}a^{3}-\frac{52\!\cdots\!74}{50\!\cdots\!37}a^{2}+\frac{17\!\cdots\!66}{35\!\cdots\!59}a+\frac{20\!\cdots\!36}{35\!\cdots\!59}$, $\frac{558069356}{5289683161}a^{15}-\frac{394270}{899759}a^{14}-\frac{19702870830}{5289683161}a^{13}+\frac{105610714215}{5289683161}a^{12}+\frac{136630885690}{5289683161}a^{11}-\frac{1429506962474}{5289683161}a^{10}+\frac{65303921040}{311157833}a^{9}+\frac{5464836668325}{5289683161}a^{8}-\frac{8055616848290}{5289683161}a^{7}-\frac{1006367178830}{755669023}a^{6}+\frac{15469186120296}{5289683161}a^{5}+\frac{173849017205}{755669023}a^{4}-\frac{9361483590790}{5289683161}a^{3}+\frac{223015580500}{755669023}a^{2}+\frac{640815458431}{5289683161}a+\frac{7218480328}{5289683161}$, $\frac{23\!\cdots\!61}{38\!\cdots\!49}a^{15}-\frac{2371926156594}{66149120091131}a^{14}-\frac{74\!\cdots\!31}{38\!\cdots\!49}a^{13}+\frac{59\!\cdots\!74}{38\!\cdots\!49}a^{12}+\frac{11\!\cdots\!54}{35\!\cdots\!59}a^{11}-\frac{75\!\cdots\!29}{38\!\cdots\!49}a^{10}+\frac{64\!\cdots\!55}{22\!\cdots\!97}a^{9}+\frac{23\!\cdots\!13}{38\!\cdots\!49}a^{8}-\frac{55\!\cdots\!24}{38\!\cdots\!49}a^{7}-\frac{23\!\cdots\!10}{55\!\cdots\!07}a^{6}+\frac{90\!\cdots\!07}{38\!\cdots\!49}a^{5}-\frac{25\!\cdots\!92}{55\!\cdots\!07}a^{4}-\frac{49\!\cdots\!59}{38\!\cdots\!49}a^{3}+\frac{26\!\cdots\!00}{55\!\cdots\!07}a^{2}+\frac{15\!\cdots\!02}{38\!\cdots\!49}a-\frac{61\!\cdots\!76}{38\!\cdots\!49}$, $\frac{17\!\cdots\!07}{55\!\cdots\!07}a^{15}-\frac{1168870326905}{9449874298733}a^{14}-\frac{62\!\cdots\!75}{55\!\cdots\!07}a^{13}+\frac{31\!\cdots\!06}{55\!\cdots\!07}a^{12}+\frac{43\!\cdots\!06}{50\!\cdots\!37}a^{11}-\frac{43\!\cdots\!33}{55\!\cdots\!07}a^{10}+\frac{15\!\cdots\!81}{32\!\cdots\!71}a^{9}+\frac{16\!\cdots\!49}{55\!\cdots\!07}a^{8}-\frac{21\!\cdots\!36}{55\!\cdots\!07}a^{7}-\frac{25\!\cdots\!60}{55\!\cdots\!07}a^{6}+\frac{41\!\cdots\!63}{55\!\cdots\!07}a^{5}+\frac{12\!\cdots\!94}{55\!\cdots\!07}a^{4}-\frac{26\!\cdots\!31}{55\!\cdots\!07}a^{3}-\frac{14\!\cdots\!35}{55\!\cdots\!07}a^{2}+\frac{28\!\cdots\!82}{55\!\cdots\!07}a+\frac{47\!\cdots\!08}{55\!\cdots\!07}$, $\frac{55\!\cdots\!75}{38\!\cdots\!49}a^{15}-\frac{4380724419405}{66149120091131}a^{14}-\frac{18\!\cdots\!40}{38\!\cdots\!49}a^{13}+\frac{11\!\cdots\!68}{38\!\cdots\!49}a^{12}+\frac{85\!\cdots\!01}{35\!\cdots\!59}a^{11}-\frac{14\!\cdots\!13}{38\!\cdots\!49}a^{10}+\frac{92\!\cdots\!00}{22\!\cdots\!97}a^{9}+\frac{48\!\cdots\!21}{38\!\cdots\!49}a^{8}-\frac{87\!\cdots\!44}{38\!\cdots\!49}a^{7}-\frac{71\!\cdots\!33}{55\!\cdots\!07}a^{6}+\frac{14\!\cdots\!58}{38\!\cdots\!49}a^{5}+\frac{28\!\cdots\!64}{55\!\cdots\!07}a^{4}-\frac{83\!\cdots\!97}{38\!\cdots\!49}a^{3}+\frac{16\!\cdots\!04}{55\!\cdots\!07}a^{2}+\frac{70\!\cdots\!44}{38\!\cdots\!49}a+\frac{40\!\cdots\!87}{38\!\cdots\!49}$, $\frac{25\!\cdots\!08}{38\!\cdots\!49}a^{15}-\frac{17292813078397}{66149120091131}a^{14}-\frac{89\!\cdots\!60}{38\!\cdots\!49}a^{13}+\frac{46\!\cdots\!57}{38\!\cdots\!49}a^{12}+\frac{59\!\cdots\!17}{35\!\cdots\!59}a^{11}-\frac{63\!\cdots\!23}{38\!\cdots\!49}a^{10}+\frac{26\!\cdots\!18}{22\!\cdots\!97}a^{9}+\frac{25\!\cdots\!54}{38\!\cdots\!49}a^{8}-\frac{35\!\cdots\!11}{38\!\cdots\!49}a^{7}-\frac{49\!\cdots\!57}{55\!\cdots\!07}a^{6}+\frac{69\!\cdots\!85}{38\!\cdots\!49}a^{5}+\frac{13\!\cdots\!35}{55\!\cdots\!07}a^{4}-\frac{43\!\cdots\!23}{38\!\cdots\!49}a^{3}+\frac{78\!\cdots\!73}{55\!\cdots\!07}a^{2}+\frac{31\!\cdots\!89}{38\!\cdots\!49}a+\frac{28\!\cdots\!06}{38\!\cdots\!49}$, $\frac{66\!\cdots\!12}{38\!\cdots\!49}a^{15}-\frac{46279034475827}{66149120091131}a^{14}-\frac{23\!\cdots\!30}{38\!\cdots\!49}a^{13}+\frac{12\!\cdots\!92}{38\!\cdots\!49}a^{12}+\frac{15\!\cdots\!27}{35\!\cdots\!59}a^{11}-\frac{16\!\cdots\!89}{38\!\cdots\!49}a^{10}+\frac{74\!\cdots\!78}{22\!\cdots\!97}a^{9}+\frac{65\!\cdots\!79}{38\!\cdots\!49}a^{8}-\frac{94\!\cdots\!21}{38\!\cdots\!49}a^{7}-\frac{12\!\cdots\!27}{55\!\cdots\!07}a^{6}+\frac{18\!\cdots\!49}{38\!\cdots\!49}a^{5}+\frac{26\!\cdots\!80}{55\!\cdots\!07}a^{4}-\frac{11\!\cdots\!33}{38\!\cdots\!49}a^{3}+\frac{24\!\cdots\!73}{55\!\cdots\!07}a^{2}+\frac{78\!\cdots\!19}{38\!\cdots\!49}a+\frac{15\!\cdots\!56}{38\!\cdots\!49}$, $\frac{45\!\cdots\!28}{38\!\cdots\!49}a^{15}-\frac{32646284228622}{66149120091131}a^{14}-\frac{16\!\cdots\!78}{38\!\cdots\!49}a^{13}+\frac{87\!\cdots\!87}{38\!\cdots\!49}a^{12}+\frac{98\!\cdots\!78}{35\!\cdots\!59}a^{11}-\frac{11\!\cdots\!85}{38\!\cdots\!49}a^{10}+\frac{55\!\cdots\!92}{22\!\cdots\!97}a^{9}+\frac{44\!\cdots\!63}{38\!\cdots\!49}a^{8}-\frac{66\!\cdots\!60}{38\!\cdots\!49}a^{7}-\frac{81\!\cdots\!99}{55\!\cdots\!07}a^{6}+\frac{12\!\cdots\!88}{38\!\cdots\!49}a^{5}+\frac{13\!\cdots\!43}{55\!\cdots\!07}a^{4}-\frac{76\!\cdots\!90}{38\!\cdots\!49}a^{3}+\frac{18\!\cdots\!59}{55\!\cdots\!07}a^{2}+\frac{48\!\cdots\!93}{38\!\cdots\!49}a+\frac{13\!\cdots\!84}{38\!\cdots\!49}$, $\frac{53\!\cdots\!45}{38\!\cdots\!49}a^{15}-\frac{2517013541293}{66149120091131}a^{14}-\frac{20\!\cdots\!76}{38\!\cdots\!49}a^{13}+\frac{72\!\cdots\!86}{38\!\cdots\!49}a^{12}+\frac{19\!\cdots\!00}{35\!\cdots\!59}a^{11}-\frac{10\!\cdots\!37}{38\!\cdots\!49}a^{10}-\frac{13\!\cdots\!76}{22\!\cdots\!97}a^{9}+\frac{48\!\cdots\!13}{38\!\cdots\!49}a^{8}-\frac{21\!\cdots\!42}{38\!\cdots\!49}a^{7}-\frac{13\!\cdots\!98}{55\!\cdots\!07}a^{6}+\frac{61\!\cdots\!94}{38\!\cdots\!49}a^{5}+\frac{11\!\cdots\!85}{55\!\cdots\!07}a^{4}-\frac{49\!\cdots\!47}{38\!\cdots\!49}a^{3}-\frac{41\!\cdots\!95}{55\!\cdots\!07}a^{2}+\frac{99\!\cdots\!38}{38\!\cdots\!49}a+\frac{25\!\cdots\!49}{38\!\cdots\!49}$
| 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: | \( 279059448.286 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 279059448.286 \cdot 2}{2\cdot\sqrt{2282521714753536000000000000}}\cr\approx \mathstrut & 0.382797597100 \end{aligned}\] (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 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1)
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 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1, 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 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1);
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 - 36*x^14 + 184*x^13 + 276*x^12 - 2532*x^11 + 1576*x^10 + 10204*x^9 - 12939*x^8 - 15196*x^7 + 25960*x^6 + 6876*x^5 - 16638*x^4 + 32*x^3 + 1620*x^2 + 196*x + 1);
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
$C_4^2:C_2$ (as 16T30):
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 14 conjugacy class representatives for $C_4^2:C_2$ |
| Character table for $C_4^2:C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), 4.4.576000.2, 4.4.576000.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.8.331776000000.1, 8.8.29859840000.1, 8.8.2985984000000.1 |
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 |
| Degree 16 sibling: | deg 16 |
| 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 | R | R | R | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 $16$ | $8$ | $2$ | $44$ | |||
|
\(3\)
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |