Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649)
gp: K = bnfinit(y^16 - 6*y^15 - 61*y^14 + 129*y^13 + 1709*y^12 + 11091*y^11 - 31783*y^10 - 454524*y^9 + 25606*y^8 + 6052602*y^7 + 10610353*y^6 - 30220476*y^5 - 125101939*y^4 - 49470471*y^3 + 344285185*y^2 + 698844519*y + 585525649, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649)
\( x^{16} - 6 x^{15} - 61 x^{14} + 129 x^{13} + 1709 x^{12} + 11091 x^{11} - 31783 x^{10} + \cdots + 585525649 \)
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: |
\(92534813992455271438265413802888409\)
\(\medspace = 3^{12}\cdot 89^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(153.25\) | 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: | $3^{3/4}89^{15/16}\approx 153.24773778749656$ | ||
| Ramified primes: |
\(3\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{89}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{44}a^{14}-\frac{1}{22}a^{13}+\frac{1}{11}a^{12}-\frac{1}{44}a^{11}-\frac{5}{44}a^{10}-\frac{1}{22}a^{9}-\frac{7}{22}a^{8}-\frac{2}{11}a^{7}-\frac{1}{2}a^{6}-\frac{3}{22}a^{5}-\frac{7}{44}a^{4}-\frac{1}{11}a^{3}-\frac{4}{11}a^{2}-\frac{21}{44}a+\frac{15}{44}$, $\frac{1}{14\!\cdots\!92}a^{15}-\frac{46\!\cdots\!63}{14\!\cdots\!92}a^{14}+\frac{85\!\cdots\!17}{73\!\cdots\!96}a^{13}+\frac{52\!\cdots\!43}{14\!\cdots\!92}a^{12}+\frac{16\!\cdots\!01}{73\!\cdots\!96}a^{11}-\frac{33\!\cdots\!03}{14\!\cdots\!92}a^{10}+\frac{34\!\cdots\!75}{18\!\cdots\!49}a^{9}+\frac{79\!\cdots\!39}{18\!\cdots\!49}a^{8}+\frac{49\!\cdots\!77}{73\!\cdots\!96}a^{7}-\frac{98\!\cdots\!05}{36\!\cdots\!98}a^{6}+\frac{49\!\cdots\!77}{14\!\cdots\!92}a^{5}-\frac{64\!\cdots\!45}{14\!\cdots\!92}a^{4}-\frac{14\!\cdots\!09}{73\!\cdots\!96}a^{3}+\frac{57\!\cdots\!01}{13\!\cdots\!72}a^{2}+\frac{36\!\cdots\!57}{73\!\cdots\!96}a+\frac{41\!\cdots\!57}{14\!\cdots\!92}$
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: |
$\frac{26\!\cdots\!41}{22\!\cdots\!76}a^{15}-\frac{25\!\cdots\!43}{22\!\cdots\!76}a^{14}-\frac{53\!\cdots\!49}{11\!\cdots\!88}a^{13}+\frac{76\!\cdots\!15}{22\!\cdots\!76}a^{12}+\frac{17\!\cdots\!81}{11\!\cdots\!88}a^{11}+\frac{20\!\cdots\!77}{22\!\cdots\!76}a^{10}-\frac{21\!\cdots\!94}{27\!\cdots\!47}a^{9}-\frac{24\!\cdots\!43}{55\!\cdots\!94}a^{8}+\frac{13\!\cdots\!85}{11\!\cdots\!88}a^{7}+\frac{16\!\cdots\!58}{27\!\cdots\!47}a^{6}+\frac{33\!\cdots\!77}{22\!\cdots\!76}a^{5}-\frac{97\!\cdots\!69}{22\!\cdots\!76}a^{4}-\frac{77\!\cdots\!97}{11\!\cdots\!88}a^{3}+\frac{14\!\cdots\!93}{20\!\cdots\!16}a^{2}+\frac{25\!\cdots\!99}{11\!\cdots\!88}a+\frac{69\!\cdots\!25}{22\!\cdots\!76}$, $\frac{30\!\cdots\!73}{11\!\cdots\!88}a^{15}-\frac{87\!\cdots\!31}{11\!\cdots\!88}a^{14}-\frac{77\!\cdots\!35}{55\!\cdots\!94}a^{13}-\frac{54\!\cdots\!37}{11\!\cdots\!88}a^{12}+\frac{16\!\cdots\!99}{27\!\cdots\!47}a^{11}+\frac{49\!\cdots\!53}{11\!\cdots\!88}a^{10}+\frac{62\!\cdots\!75}{55\!\cdots\!94}a^{9}-\frac{25\!\cdots\!51}{55\!\cdots\!94}a^{8}-\frac{96\!\cdots\!59}{27\!\cdots\!47}a^{7}-\frac{35\!\cdots\!00}{27\!\cdots\!47}a^{6}+\frac{36\!\cdots\!91}{11\!\cdots\!88}a^{5}+\frac{23\!\cdots\!09}{11\!\cdots\!88}a^{4}+\frac{18\!\cdots\!50}{27\!\cdots\!47}a^{3}-\frac{63\!\cdots\!37}{10\!\cdots\!08}a^{2}-\frac{39\!\cdots\!71}{27\!\cdots\!47}a-\frac{19\!\cdots\!85}{11\!\cdots\!88}$, $\frac{83\!\cdots\!33}{22\!\cdots\!76}a^{15}-\frac{82\!\cdots\!07}{22\!\cdots\!76}a^{14}-\frac{20\!\cdots\!63}{11\!\cdots\!88}a^{13}+\frac{33\!\cdots\!99}{22\!\cdots\!76}a^{12}+\frac{78\!\cdots\!09}{11\!\cdots\!88}a^{11}+\frac{47\!\cdots\!61}{22\!\cdots\!76}a^{10}-\frac{17\!\cdots\!97}{55\!\cdots\!94}a^{9}-\frac{49\!\cdots\!57}{27\!\cdots\!47}a^{8}+\frac{64\!\cdots\!21}{11\!\cdots\!88}a^{7}+\frac{92\!\cdots\!60}{27\!\cdots\!47}a^{6}-\frac{33\!\cdots\!11}{22\!\cdots\!76}a^{5}-\frac{65\!\cdots\!77}{22\!\cdots\!76}a^{4}-\frac{41\!\cdots\!29}{11\!\cdots\!88}a^{3}+\frac{19\!\cdots\!17}{20\!\cdots\!16}a^{2}+\frac{24\!\cdots\!15}{11\!\cdots\!88}a+\frac{13\!\cdots\!73}{22\!\cdots\!76}$, $\frac{97\!\cdots\!51}{11\!\cdots\!88}a^{15}-\frac{27\!\cdots\!87}{11\!\cdots\!88}a^{14}-\frac{17\!\cdots\!11}{27\!\cdots\!47}a^{13}-\frac{94\!\cdots\!57}{11\!\cdots\!88}a^{12}+\frac{34\!\cdots\!45}{27\!\cdots\!47}a^{11}+\frac{15\!\cdots\!31}{11\!\cdots\!88}a^{10}+\frac{44\!\cdots\!78}{27\!\cdots\!47}a^{9}-\frac{98\!\cdots\!27}{27\!\cdots\!47}a^{8}-\frac{31\!\cdots\!14}{27\!\cdots\!47}a^{7}+\frac{49\!\cdots\!07}{27\!\cdots\!47}a^{6}+\frac{17\!\cdots\!69}{11\!\cdots\!88}a^{5}+\frac{27\!\cdots\!23}{11\!\cdots\!88}a^{4}-\frac{12\!\cdots\!00}{27\!\cdots\!47}a^{3}-\frac{19\!\cdots\!33}{10\!\cdots\!08}a^{2}-\frac{79\!\cdots\!41}{27\!\cdots\!47}a-\frac{21\!\cdots\!15}{11\!\cdots\!88}$, $\frac{16\!\cdots\!81}{11\!\cdots\!88}a^{15}-\frac{57\!\cdots\!97}{11\!\cdots\!88}a^{14}-\frac{27\!\cdots\!48}{27\!\cdots\!47}a^{13}-\frac{10\!\cdots\!43}{11\!\cdots\!88}a^{12}+\frac{57\!\cdots\!82}{27\!\cdots\!47}a^{11}+\frac{24\!\cdots\!85}{11\!\cdots\!88}a^{10}+\frac{43\!\cdots\!46}{27\!\cdots\!47}a^{9}-\frac{16\!\cdots\!26}{27\!\cdots\!47}a^{8}-\frac{42\!\cdots\!09}{27\!\cdots\!47}a^{7}+\frac{94\!\cdots\!79}{27\!\cdots\!47}a^{6}+\frac{26\!\cdots\!99}{11\!\cdots\!88}a^{5}+\frac{33\!\cdots\!69}{11\!\cdots\!88}a^{4}-\frac{19\!\cdots\!43}{27\!\cdots\!47}a^{3}-\frac{27\!\cdots\!87}{10\!\cdots\!08}a^{2}-\frac{10\!\cdots\!22}{27\!\cdots\!47}a-\frac{25\!\cdots\!37}{11\!\cdots\!88}$, $\frac{67\!\cdots\!81}{11\!\cdots\!88}a^{15}-\frac{39\!\cdots\!25}{11\!\cdots\!88}a^{14}-\frac{92\!\cdots\!98}{27\!\cdots\!47}a^{13}+\frac{57\!\cdots\!57}{11\!\cdots\!88}a^{12}+\frac{21\!\cdots\!83}{27\!\cdots\!47}a^{11}+\frac{77\!\cdots\!81}{11\!\cdots\!88}a^{10}-\frac{34\!\cdots\!84}{27\!\cdots\!47}a^{9}-\frac{61\!\cdots\!31}{27\!\cdots\!47}a^{8}-\frac{14\!\cdots\!21}{27\!\cdots\!47}a^{7}+\frac{56\!\cdots\!15}{27\!\cdots\!47}a^{6}+\frac{52\!\cdots\!31}{11\!\cdots\!88}a^{5}-\frac{55\!\cdots\!43}{11\!\cdots\!88}a^{4}-\frac{82\!\cdots\!27}{27\!\cdots\!47}a^{3}-\frac{22\!\cdots\!03}{10\!\cdots\!08}a^{2}+\frac{99\!\cdots\!49}{27\!\cdots\!47}a+\frac{58\!\cdots\!27}{11\!\cdots\!88}$, $\frac{64\!\cdots\!59}{22\!\cdots\!76}a^{15}-\frac{16\!\cdots\!89}{22\!\cdots\!76}a^{14}-\frac{31\!\cdots\!21}{11\!\cdots\!88}a^{13}+\frac{39\!\cdots\!93}{22\!\cdots\!76}a^{12}+\frac{92\!\cdots\!13}{11\!\cdots\!88}a^{11}+\frac{75\!\cdots\!95}{22\!\cdots\!76}a^{10}-\frac{11\!\cdots\!51}{27\!\cdots\!47}a^{9}-\frac{10\!\cdots\!45}{55\!\cdots\!94}a^{8}-\frac{22\!\cdots\!07}{11\!\cdots\!88}a^{7}+\frac{96\!\cdots\!35}{27\!\cdots\!47}a^{6}+\frac{88\!\cdots\!63}{22\!\cdots\!76}a^{5}-\frac{52\!\cdots\!75}{22\!\cdots\!76}a^{4}-\frac{41\!\cdots\!85}{11\!\cdots\!88}a^{3}+\frac{61\!\cdots\!75}{20\!\cdots\!16}a^{2}+\frac{15\!\cdots\!95}{11\!\cdots\!88}a+\frac{35\!\cdots\!91}{22\!\cdots\!76}$, $\frac{10\!\cdots\!43}{36\!\cdots\!98}a^{15}+\frac{52\!\cdots\!67}{18\!\cdots\!49}a^{14}-\frac{75\!\cdots\!05}{36\!\cdots\!98}a^{13}-\frac{20\!\cdots\!16}{18\!\cdots\!49}a^{12}+\frac{43\!\cdots\!77}{36\!\cdots\!98}a^{11}+\frac{99\!\cdots\!13}{18\!\cdots\!49}a^{10}+\frac{42\!\cdots\!87}{18\!\cdots\!49}a^{9}-\frac{96\!\cdots\!93}{18\!\cdots\!49}a^{8}-\frac{11\!\cdots\!19}{18\!\cdots\!49}a^{7}-\frac{40\!\cdots\!63}{36\!\cdots\!98}a^{6}+\frac{15\!\cdots\!49}{36\!\cdots\!98}a^{5}+\frac{30\!\cdots\!64}{18\!\cdots\!49}a^{4}+\frac{26\!\cdots\!77}{36\!\cdots\!98}a^{3}-\frac{62\!\cdots\!15}{15\!\cdots\!69}a^{2}-\frac{15\!\cdots\!56}{18\!\cdots\!49}a-\frac{13\!\cdots\!65}{18\!\cdots\!49}$, $\frac{37\!\cdots\!49}{14\!\cdots\!92}a^{15}+\frac{87\!\cdots\!01}{14\!\cdots\!92}a^{14}-\frac{50\!\cdots\!59}{73\!\cdots\!96}a^{13}-\frac{36\!\cdots\!09}{14\!\cdots\!92}a^{12}+\frac{62\!\cdots\!45}{73\!\cdots\!96}a^{11}+\frac{16\!\cdots\!93}{14\!\cdots\!92}a^{10}+\frac{27\!\cdots\!21}{36\!\cdots\!98}a^{9}-\frac{58\!\cdots\!11}{18\!\cdots\!49}a^{8}-\frac{14\!\cdots\!83}{73\!\cdots\!96}a^{7}+\frac{49\!\cdots\!48}{18\!\cdots\!49}a^{6}+\frac{30\!\cdots\!37}{14\!\cdots\!92}a^{5}+\frac{67\!\cdots\!07}{14\!\cdots\!92}a^{4}-\frac{27\!\cdots\!37}{73\!\cdots\!96}a^{3}-\frac{35\!\cdots\!75}{13\!\cdots\!72}a^{2}-\frac{31\!\cdots\!33}{73\!\cdots\!96}a-\frac{46\!\cdots\!79}{14\!\cdots\!92}$, $\frac{30\!\cdots\!45}{73\!\cdots\!96}a^{15}-\frac{36\!\cdots\!21}{73\!\cdots\!96}a^{14}-\frac{19\!\cdots\!90}{18\!\cdots\!49}a^{13}+\frac{13\!\cdots\!03}{73\!\cdots\!96}a^{12}+\frac{18\!\cdots\!03}{36\!\cdots\!98}a^{11}+\frac{12\!\cdots\!79}{73\!\cdots\!96}a^{10}-\frac{14\!\cdots\!59}{36\!\cdots\!98}a^{9}-\frac{25\!\cdots\!25}{18\!\cdots\!49}a^{8}+\frac{30\!\cdots\!17}{36\!\cdots\!98}a^{7}+\frac{47\!\cdots\!34}{18\!\cdots\!49}a^{6}-\frac{21\!\cdots\!87}{73\!\cdots\!96}a^{5}-\frac{19\!\cdots\!89}{73\!\cdots\!96}a^{4}-\frac{50\!\cdots\!83}{18\!\cdots\!49}a^{3}+\frac{51\!\cdots\!77}{67\!\cdots\!36}a^{2}+\frac{64\!\cdots\!95}{36\!\cdots\!98}a+\frac{14\!\cdots\!91}{73\!\cdots\!96}$, $\frac{21\!\cdots\!11}{73\!\cdots\!96}a^{15}-\frac{11\!\cdots\!49}{36\!\cdots\!98}a^{14}-\frac{11\!\cdots\!59}{36\!\cdots\!98}a^{13}+\frac{48\!\cdots\!93}{73\!\cdots\!96}a^{12}+\frac{21\!\cdots\!93}{73\!\cdots\!96}a^{11}+\frac{24\!\cdots\!28}{18\!\cdots\!49}a^{10}-\frac{35\!\cdots\!56}{18\!\cdots\!49}a^{9}-\frac{19\!\cdots\!47}{36\!\cdots\!98}a^{8}+\frac{12\!\cdots\!03}{36\!\cdots\!98}a^{7}+\frac{32\!\cdots\!79}{36\!\cdots\!98}a^{6}-\frac{19\!\cdots\!15}{73\!\cdots\!96}a^{5}-\frac{33\!\cdots\!11}{36\!\cdots\!98}a^{4}+\frac{11\!\cdots\!03}{18\!\cdots\!49}a^{3}+\frac{34\!\cdots\!55}{67\!\cdots\!36}a^{2}+\frac{19\!\cdots\!21}{73\!\cdots\!96}a-\frac{19\!\cdots\!99}{18\!\cdots\!49}$
| 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: | \( 507241262351 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 507241262351 \cdot 1}{2\cdot\sqrt{92534813992455271438265413802888409}}\cr\approx \mathstrut & 0.332652996301041 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649)
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 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649, 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 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649);
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 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649);
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
| \(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.3582738126537849.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{8}$ | $16$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.16.12.3 | $x^{16} - 6 x^{12} + 162$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |
|
\(89\)
| 89.16.15.1 | $x^{16} + 979$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |