Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979)
gp: K = bnfinit(y^16 - 5*y^15 - 1520*y^14 + 16523*y^13 + 655405*y^12 - 10799728*y^11 - 58356778*y^10 + 1864600258*y^9 - 4621996848*y^8 - 98875841022*y^7 + 638807569630*y^6 + 400115166704*y^5 - 14975139902151*y^4 + 55925689327219*y^3 - 94914904681696*y^2 + 78330056885859*y - 25448748825979, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979)
\( x^{16} - 5 x^{15} - 1520 x^{14} + 16523 x^{13} + 655405 x^{12} - 10799728 x^{11} - 58356778 x^{10} + \cdots - 25448748825979 \)
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: |
\(103591320678630890586725126366901805607295040854289\)
\(\medspace = 37^{14}\cdot 101^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1336.47\) | 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: | $37^{7/8}101^{7/8}\approx 1336.4653720287358$ | ||
| Ramified primes: |
\(37\), \(101\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}+\frac{3}{16}a^{3}-\frac{3}{8}a^{2}-\frac{3}{8}a+\frac{3}{16}$, $\frac{1}{48}a^{10}+\frac{1}{48}a^{9}+\frac{1}{12}a^{8}+\frac{1}{16}a^{7}-\frac{1}{48}a^{6}+\frac{1}{16}a^{4}-\frac{5}{48}a^{3}-\frac{5}{12}a^{2}+\frac{3}{16}a-\frac{11}{48}$, $\frac{1}{48}a^{11}+\frac{5}{48}a^{8}+\frac{1}{24}a^{7}-\frac{1}{24}a^{6}+\frac{1}{16}a^{5}-\frac{1}{6}a^{4}-\frac{1}{48}a^{2}-\frac{1}{24}a-\frac{11}{24}$, $\frac{1}{288}a^{12}+\frac{1}{144}a^{10}-\frac{1}{36}a^{9}+\frac{7}{72}a^{8}+\frac{5}{144}a^{7}+\frac{11}{144}a^{6}-\frac{1}{9}a^{5}+\frac{3}{16}a^{4}+\frac{5}{36}a^{3}-\frac{1}{24}a^{2}-\frac{41}{144}a-\frac{7}{288}$, $\frac{1}{288}a^{13}+\frac{1}{144}a^{11}-\frac{1}{144}a^{10}-\frac{1}{144}a^{9}+\frac{17}{144}a^{8}-\frac{1}{9}a^{7}-\frac{1}{144}a^{6}+\frac{3}{16}a^{5}+\frac{29}{144}a^{4}-\frac{1}{48}a^{3}+\frac{43}{144}a^{2}+\frac{119}{288}a-\frac{17}{48}$, $\frac{1}{864}a^{14}-\frac{1}{864}a^{13}+\frac{1}{864}a^{12}+\frac{1}{108}a^{11}-\frac{1}{432}a^{10}+\frac{13}{432}a^{9}-\frac{35}{432}a^{8}-\frac{2}{27}a^{7}-\frac{5}{54}a^{6}-\frac{1}{4}a^{5}-\frac{107}{432}a^{4}-\frac{73}{432}a^{3}-\frac{25}{288}a^{2}-\frac{199}{864}a+\frac{295}{864}$, $\frac{1}{10\!\cdots\!36}a^{15}+\frac{19\!\cdots\!93}{51\!\cdots\!68}a^{14}-\frac{25\!\cdots\!89}{51\!\cdots\!68}a^{13}-\frac{31\!\cdots\!71}{10\!\cdots\!36}a^{12}-\frac{36\!\cdots\!93}{63\!\cdots\!96}a^{11}+\frac{18\!\cdots\!59}{31\!\cdots\!48}a^{10}-\frac{76\!\cdots\!15}{18\!\cdots\!84}a^{9}-\frac{18\!\cdots\!83}{85\!\cdots\!28}a^{8}-\frac{12\!\cdots\!87}{25\!\cdots\!84}a^{7}-\frac{43\!\cdots\!73}{51\!\cdots\!68}a^{6}+\frac{15\!\cdots\!29}{63\!\cdots\!96}a^{5}+\frac{50\!\cdots\!37}{31\!\cdots\!48}a^{4}-\frac{12\!\cdots\!67}{10\!\cdots\!36}a^{3}+\frac{10\!\cdots\!77}{51\!\cdots\!68}a^{2}+\frac{17\!\cdots\!27}{51\!\cdots\!68}a+\frac{45\!\cdots\!77}{10\!\cdots\!36}$
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
$C_{20}$, which has order $20$ (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: |
$\frac{82\!\cdots\!25}{89\!\cdots\!52}a^{15}-\frac{18\!\cdots\!33}{44\!\cdots\!76}a^{14}-\frac{62\!\cdots\!19}{44\!\cdots\!76}a^{13}+\frac{12\!\cdots\!01}{89\!\cdots\!52}a^{12}+\frac{68\!\cdots\!37}{11\!\cdots\!44}a^{11}-\frac{53\!\cdots\!25}{55\!\cdots\!72}a^{10}-\frac{26\!\cdots\!85}{44\!\cdots\!76}a^{9}+\frac{37\!\cdots\!31}{22\!\cdots\!88}a^{8}-\frac{72\!\cdots\!53}{22\!\cdots\!88}a^{7}-\frac{42\!\cdots\!73}{44\!\cdots\!76}a^{6}+\frac{60\!\cdots\!39}{11\!\cdots\!44}a^{5}+\frac{43\!\cdots\!37}{55\!\cdots\!72}a^{4}-\frac{12\!\cdots\!87}{89\!\cdots\!52}a^{3}+\frac{18\!\cdots\!11}{44\!\cdots\!76}a^{2}-\frac{23\!\cdots\!79}{44\!\cdots\!76}a+\frac{21\!\cdots\!09}{89\!\cdots\!52}$, $\frac{10\!\cdots\!47}{10\!\cdots\!36}a^{15}-\frac{60\!\cdots\!85}{17\!\cdots\!56}a^{14}-\frac{28\!\cdots\!55}{18\!\cdots\!84}a^{13}+\frac{48\!\cdots\!41}{34\!\cdots\!12}a^{12}+\frac{14\!\cdots\!01}{21\!\cdots\!32}a^{11}-\frac{41\!\cdots\!13}{42\!\cdots\!64}a^{10}-\frac{36\!\cdots\!83}{51\!\cdots\!68}a^{9}+\frac{44\!\cdots\!27}{25\!\cdots\!84}a^{8}-\frac{18\!\cdots\!75}{85\!\cdots\!28}a^{7}-\frac{51\!\cdots\!71}{51\!\cdots\!68}a^{6}+\frac{31\!\cdots\!33}{63\!\cdots\!96}a^{5}+\frac{15\!\cdots\!39}{14\!\cdots\!88}a^{4}-\frac{13\!\cdots\!57}{10\!\cdots\!36}a^{3}+\frac{18\!\cdots\!93}{51\!\cdots\!68}a^{2}-\frac{74\!\cdots\!35}{17\!\cdots\!56}a+\frac{19\!\cdots\!35}{10\!\cdots\!36}$, $\frac{14\!\cdots\!45}{10\!\cdots\!36}a^{15}-\frac{27\!\cdots\!55}{51\!\cdots\!68}a^{14}-\frac{10\!\cdots\!73}{51\!\cdots\!68}a^{13}+\frac{20\!\cdots\!25}{10\!\cdots\!36}a^{12}+\frac{60\!\cdots\!83}{63\!\cdots\!96}a^{11}-\frac{11\!\cdots\!21}{79\!\cdots\!37}a^{10}-\frac{16\!\cdots\!27}{17\!\cdots\!56}a^{9}+\frac{21\!\cdots\!77}{85\!\cdots\!28}a^{8}-\frac{87\!\cdots\!51}{25\!\cdots\!84}a^{7}-\frac{72\!\cdots\!77}{51\!\cdots\!68}a^{6}+\frac{46\!\cdots\!33}{63\!\cdots\!96}a^{5}+\frac{11\!\cdots\!03}{79\!\cdots\!37}a^{4}-\frac{19\!\cdots\!07}{10\!\cdots\!36}a^{3}+\frac{28\!\cdots\!05}{51\!\cdots\!68}a^{2}-\frac{33\!\cdots\!05}{51\!\cdots\!68}a+\frac{29\!\cdots\!57}{10\!\cdots\!36}$, $\frac{62\!\cdots\!51}{10\!\cdots\!36}a^{15}-\frac{27\!\cdots\!87}{17\!\cdots\!56}a^{14}-\frac{17\!\cdots\!69}{18\!\cdots\!84}a^{13}+\frac{26\!\cdots\!01}{34\!\cdots\!12}a^{12}+\frac{44\!\cdots\!87}{10\!\cdots\!16}a^{11}-\frac{11\!\cdots\!37}{21\!\cdots\!32}a^{10}-\frac{24\!\cdots\!75}{51\!\cdots\!68}a^{9}+\frac{25\!\cdots\!81}{25\!\cdots\!84}a^{8}-\frac{34\!\cdots\!07}{85\!\cdots\!28}a^{7}-\frac{31\!\cdots\!27}{51\!\cdots\!68}a^{6}+\frac{77\!\cdots\!47}{31\!\cdots\!48}a^{5}+\frac{57\!\cdots\!27}{71\!\cdots\!44}a^{4}-\frac{73\!\cdots\!01}{10\!\cdots\!36}a^{3}+\frac{87\!\cdots\!95}{51\!\cdots\!68}a^{2}-\frac{30\!\cdots\!81}{17\!\cdots\!56}a+\frac{70\!\cdots\!55}{10\!\cdots\!36}$, $\frac{95\!\cdots\!29}{48\!\cdots\!04}a^{15}-\frac{17\!\cdots\!27}{24\!\cdots\!52}a^{14}-\frac{72\!\cdots\!33}{24\!\cdots\!52}a^{13}+\frac{13\!\cdots\!01}{48\!\cdots\!04}a^{12}+\frac{80\!\cdots\!89}{60\!\cdots\!88}a^{11}-\frac{11\!\cdots\!07}{60\!\cdots\!88}a^{10}-\frac{33\!\cdots\!01}{24\!\cdots\!52}a^{9}+\frac{42\!\cdots\!13}{12\!\cdots\!76}a^{8}-\frac{57\!\cdots\!21}{12\!\cdots\!76}a^{7}-\frac{48\!\cdots\!65}{24\!\cdots\!52}a^{6}+\frac{61\!\cdots\!15}{60\!\cdots\!88}a^{5}+\frac{12\!\cdots\!43}{60\!\cdots\!88}a^{4}-\frac{13\!\cdots\!71}{48\!\cdots\!04}a^{3}+\frac{18\!\cdots\!85}{24\!\cdots\!52}a^{2}-\frac{21\!\cdots\!37}{24\!\cdots\!52}a+\frac{18\!\cdots\!85}{48\!\cdots\!04}$, $\frac{29\!\cdots\!63}{10\!\cdots\!36}a^{15}-\frac{74\!\cdots\!73}{51\!\cdots\!68}a^{14}-\frac{22\!\cdots\!15}{51\!\cdots\!68}a^{13}+\frac{49\!\cdots\!11}{10\!\cdots\!36}a^{12}+\frac{24\!\cdots\!49}{12\!\cdots\!92}a^{11}-\frac{20\!\cdots\!89}{63\!\cdots\!96}a^{10}-\frac{91\!\cdots\!11}{51\!\cdots\!68}a^{9}+\frac{14\!\cdots\!59}{25\!\cdots\!84}a^{8}-\frac{31\!\cdots\!85}{25\!\cdots\!84}a^{7}-\frac{51\!\cdots\!89}{17\!\cdots\!56}a^{6}+\frac{23\!\cdots\!47}{12\!\cdots\!92}a^{5}+\frac{13\!\cdots\!05}{63\!\cdots\!96}a^{4}-\frac{52\!\cdots\!81}{11\!\cdots\!04}a^{3}+\frac{74\!\cdots\!83}{51\!\cdots\!68}a^{2}-\frac{95\!\cdots\!83}{51\!\cdots\!68}a+\frac{29\!\cdots\!13}{34\!\cdots\!12}$, $\frac{49\!\cdots\!21}{31\!\cdots\!48}a^{15}-\frac{29\!\cdots\!91}{85\!\cdots\!28}a^{14}-\frac{66\!\cdots\!77}{28\!\cdots\!76}a^{13}+\frac{16\!\cdots\!61}{85\!\cdots\!28}a^{12}+\frac{45\!\cdots\!55}{42\!\cdots\!64}a^{11}-\frac{64\!\cdots\!91}{47\!\cdots\!96}a^{10}-\frac{81\!\cdots\!47}{63\!\cdots\!96}a^{9}+\frac{16\!\cdots\!75}{63\!\cdots\!96}a^{8}-\frac{79\!\cdots\!07}{35\!\cdots\!72}a^{7}-\frac{19\!\cdots\!59}{12\!\cdots\!92}a^{6}+\frac{72\!\cdots\!15}{12\!\cdots\!92}a^{5}+\frac{30\!\cdots\!09}{14\!\cdots\!88}a^{4}-\frac{10\!\cdots\!55}{63\!\cdots\!96}a^{3}+\frac{10\!\cdots\!09}{25\!\cdots\!84}a^{2}-\frac{33\!\cdots\!01}{85\!\cdots\!28}a+\frac{36\!\cdots\!95}{25\!\cdots\!84}$, $\frac{21\!\cdots\!83}{10\!\cdots\!36}a^{15}-\frac{13\!\cdots\!01}{51\!\cdots\!68}a^{14}-\frac{16\!\cdots\!61}{51\!\cdots\!68}a^{13}+\frac{58\!\cdots\!39}{10\!\cdots\!36}a^{12}+\frac{19\!\cdots\!01}{15\!\cdots\!74}a^{11}-\frac{21\!\cdots\!85}{63\!\cdots\!96}a^{10}-\frac{12\!\cdots\!55}{51\!\cdots\!68}a^{9}+\frac{13\!\cdots\!01}{25\!\cdots\!84}a^{8}-\frac{73\!\cdots\!39}{25\!\cdots\!84}a^{7}-\frac{44\!\cdots\!21}{17\!\cdots\!56}a^{6}+\frac{41\!\cdots\!51}{15\!\cdots\!74}a^{5}-\frac{66\!\cdots\!35}{63\!\cdots\!96}a^{4}-\frac{64\!\cdots\!77}{11\!\cdots\!04}a^{3}+\frac{10\!\cdots\!59}{51\!\cdots\!68}a^{2}-\frac{15\!\cdots\!17}{51\!\cdots\!68}a+\frac{52\!\cdots\!09}{37\!\cdots\!68}$, $\frac{13\!\cdots\!91}{10\!\cdots\!36}a^{15}-\frac{25\!\cdots\!35}{51\!\cdots\!68}a^{14}-\frac{10\!\cdots\!17}{51\!\cdots\!68}a^{13}+\frac{20\!\cdots\!23}{10\!\cdots\!36}a^{12}+\frac{11\!\cdots\!87}{12\!\cdots\!92}a^{11}-\frac{17\!\cdots\!17}{12\!\cdots\!92}a^{10}-\frac{49\!\cdots\!63}{51\!\cdots\!68}a^{9}+\frac{61\!\cdots\!97}{25\!\cdots\!84}a^{8}-\frac{77\!\cdots\!17}{25\!\cdots\!84}a^{7}-\frac{78\!\cdots\!63}{56\!\cdots\!52}a^{6}+\frac{86\!\cdots\!97}{12\!\cdots\!92}a^{5}+\frac{18\!\cdots\!85}{12\!\cdots\!92}a^{4}-\frac{62\!\cdots\!31}{34\!\cdots\!12}a^{3}+\frac{26\!\cdots\!57}{51\!\cdots\!68}a^{2}-\frac{30\!\cdots\!77}{51\!\cdots\!68}a+\frac{85\!\cdots\!81}{34\!\cdots\!12}$, $\frac{12\!\cdots\!39}{25\!\cdots\!84}a^{15}-\frac{46\!\cdots\!81}{25\!\cdots\!84}a^{14}-\frac{23\!\cdots\!15}{31\!\cdots\!48}a^{13}+\frac{18\!\cdots\!75}{25\!\cdots\!84}a^{12}+\frac{41\!\cdots\!45}{12\!\cdots\!92}a^{11}-\frac{30\!\cdots\!47}{63\!\cdots\!96}a^{10}-\frac{11\!\cdots\!07}{31\!\cdots\!48}a^{9}+\frac{55\!\cdots\!31}{63\!\cdots\!96}a^{8}-\frac{72\!\cdots\!49}{63\!\cdots\!96}a^{7}-\frac{10\!\cdots\!47}{21\!\cdots\!32}a^{6}+\frac{31\!\cdots\!75}{12\!\cdots\!92}a^{5}+\frac{32\!\cdots\!15}{63\!\cdots\!96}a^{4}-\frac{62\!\cdots\!93}{94\!\cdots\!92}a^{3}+\frac{47\!\cdots\!97}{25\!\cdots\!84}a^{2}-\frac{14\!\cdots\!61}{63\!\cdots\!96}a+\frac{82\!\cdots\!43}{85\!\cdots\!28}$, $\frac{34\!\cdots\!13}{10\!\cdots\!36}a^{15}-\frac{58\!\cdots\!69}{51\!\cdots\!68}a^{14}-\frac{26\!\cdots\!87}{51\!\cdots\!68}a^{13}+\frac{48\!\cdots\!21}{10\!\cdots\!36}a^{12}+\frac{36\!\cdots\!43}{15\!\cdots\!74}a^{11}-\frac{41\!\cdots\!47}{12\!\cdots\!92}a^{10}-\frac{12\!\cdots\!41}{51\!\cdots\!68}a^{9}+\frac{15\!\cdots\!09}{25\!\cdots\!84}a^{8}-\frac{16\!\cdots\!51}{25\!\cdots\!84}a^{7}-\frac{19\!\cdots\!37}{56\!\cdots\!52}a^{6}+\frac{25\!\cdots\!05}{15\!\cdots\!74}a^{5}+\frac{49\!\cdots\!71}{12\!\cdots\!92}a^{4}-\frac{50\!\cdots\!15}{11\!\cdots\!04}a^{3}+\frac{61\!\cdots\!51}{51\!\cdots\!68}a^{2}-\frac{69\!\cdots\!23}{51\!\cdots\!68}a+\frac{19\!\cdots\!15}{34\!\cdots\!12}$, $\frac{59\!\cdots\!21}{11\!\cdots\!04}a^{15}-\frac{83\!\cdots\!45}{51\!\cdots\!68}a^{14}-\frac{40\!\cdots\!49}{51\!\cdots\!68}a^{13}+\frac{73\!\cdots\!33}{10\!\cdots\!36}a^{12}+\frac{28\!\cdots\!81}{79\!\cdots\!37}a^{11}-\frac{31\!\cdots\!95}{63\!\cdots\!96}a^{10}-\frac{20\!\cdots\!65}{51\!\cdots\!68}a^{9}+\frac{22\!\cdots\!61}{25\!\cdots\!84}a^{8}-\frac{18\!\cdots\!79}{25\!\cdots\!84}a^{7}-\frac{27\!\cdots\!09}{51\!\cdots\!68}a^{6}+\frac{12\!\cdots\!43}{53\!\cdots\!58}a^{5}+\frac{40\!\cdots\!83}{63\!\cdots\!96}a^{4}-\frac{67\!\cdots\!71}{10\!\cdots\!36}a^{3}+\frac{96\!\cdots\!83}{56\!\cdots\!52}a^{2}-\frac{95\!\cdots\!73}{51\!\cdots\!68}a+\frac{76\!\cdots\!01}{10\!\cdots\!36}$, $\frac{11\!\cdots\!55}{34\!\cdots\!12}a^{15}-\frac{15\!\cdots\!09}{17\!\cdots\!56}a^{14}-\frac{83\!\cdots\!53}{17\!\cdots\!56}a^{13}+\frac{56\!\cdots\!03}{34\!\cdots\!12}a^{12}+\frac{49\!\cdots\!13}{35\!\cdots\!72}a^{11}-\frac{36\!\cdots\!55}{42\!\cdots\!64}a^{10}+\frac{21\!\cdots\!39}{56\!\cdots\!52}a^{9}+\frac{12\!\cdots\!97}{94\!\cdots\!92}a^{8}-\frac{10\!\cdots\!25}{85\!\cdots\!28}a^{7}-\frac{80\!\cdots\!51}{17\!\cdots\!56}a^{6}+\frac{32\!\cdots\!69}{35\!\cdots\!72}a^{5}-\frac{64\!\cdots\!85}{42\!\cdots\!64}a^{4}-\frac{67\!\cdots\!09}{37\!\cdots\!68}a^{3}+\frac{46\!\cdots\!41}{56\!\cdots\!52}a^{2}-\frac{20\!\cdots\!73}{17\!\cdots\!56}a+\frac{19\!\cdots\!99}{34\!\cdots\!12}$, $\frac{14\!\cdots\!47}{10\!\cdots\!36}a^{15}-\frac{27\!\cdots\!99}{51\!\cdots\!68}a^{14}-\frac{11\!\cdots\!05}{51\!\cdots\!68}a^{13}+\frac{21\!\cdots\!51}{10\!\cdots\!36}a^{12}+\frac{12\!\cdots\!11}{12\!\cdots\!92}a^{11}-\frac{18\!\cdots\!57}{12\!\cdots\!92}a^{10}-\frac{17\!\cdots\!05}{17\!\cdots\!56}a^{9}+\frac{21\!\cdots\!67}{85\!\cdots\!28}a^{8}-\frac{87\!\cdots\!57}{25\!\cdots\!84}a^{7}-\frac{73\!\cdots\!87}{51\!\cdots\!68}a^{6}+\frac{92\!\cdots\!41}{12\!\cdots\!92}a^{5}+\frac{18\!\cdots\!77}{12\!\cdots\!92}a^{4}-\frac{19\!\cdots\!13}{10\!\cdots\!36}a^{3}+\frac{28\!\cdots\!17}{51\!\cdots\!68}a^{2}-\frac{33\!\cdots\!77}{51\!\cdots\!68}a+\frac{29\!\cdots\!31}{10\!\cdots\!36}$, $\frac{92\!\cdots\!09}{37\!\cdots\!68}a^{15}-\frac{45\!\cdots\!97}{51\!\cdots\!68}a^{14}-\frac{19\!\cdots\!01}{51\!\cdots\!68}a^{13}+\frac{36\!\cdots\!43}{10\!\cdots\!36}a^{12}+\frac{21\!\cdots\!43}{12\!\cdots\!92}a^{11}-\frac{77\!\cdots\!99}{31\!\cdots\!48}a^{10}-\frac{90\!\cdots\!39}{51\!\cdots\!68}a^{9}+\frac{11\!\cdots\!43}{25\!\cdots\!84}a^{8}-\frac{13\!\cdots\!63}{25\!\cdots\!84}a^{7}-\frac{12\!\cdots\!15}{51\!\cdots\!68}a^{6}+\frac{17\!\cdots\!89}{14\!\cdots\!88}a^{5}+\frac{85\!\cdots\!63}{31\!\cdots\!48}a^{4}-\frac{33\!\cdots\!09}{10\!\cdots\!36}a^{3}+\frac{15\!\cdots\!37}{17\!\cdots\!56}a^{2}-\frac{54\!\cdots\!01}{51\!\cdots\!68}a+\frac{46\!\cdots\!15}{10\!\cdots\!36}$
| 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: | \( 19385888029400000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 19385888029400000000 \cdot 20}{2\cdot\sqrt{103591320678630890586725126366901805607295040854289}}\cr\approx \mathstrut & 1.24825681453267 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979)
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 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979, 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 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979);
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 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979);
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
$\OD_{16}:C_2$ (as 16T36):
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 11 conjugacy class representatives for $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
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]
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(37\)
| 37.8.7.2 | $x^{8} + 37$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 37.8.7.2 | $x^{8} + 37$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(101\)
| 101.8.7.1 | $x^{8} + 404$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 101.8.7.1 | $x^{8} + 404$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |