Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152)
gp: K = bnfinit(y^16 - y^15 - 234*y^14 + 509*y^13 + 18337*y^12 - 57176*y^11 - 592421*y^10 + 2200433*y^9 + 8770762*y^8 - 37263645*y^7 - 58997921*y^6 + 293141240*y^5 + 166860260*y^4 - 1019696592*y^3 - 220707968*y^2 + 1184846848*y + 504881152, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152)
\( x^{16} - x^{15} - 234 x^{14} + 509 x^{13} + 18337 x^{12} - 57176 x^{11} - 592421 x^{10} + \cdots + 504881152 \)
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: |
\(6123007382888435990757129497254763904689\)
\(\medspace = 13^{14}\cdot 41^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(306.68\) | 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: | $13^{7/8}41^{15/16}\approx 306.6798900930961$ | ||
| Ramified primes: |
\(13\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{368}a^{12}+\frac{1}{184}a^{11}-\frac{5}{184}a^{10}-\frac{37}{368}a^{9}+\frac{21}{184}a^{8}+\frac{1}{23}a^{7}+\frac{15}{368}a^{6}-\frac{37}{184}a^{5}+\frac{45}{184}a^{4}+\frac{53}{368}a^{3}-\frac{89}{184}a^{2}-\frac{8}{23}a-\frac{8}{23}$, $\frac{1}{736}a^{13}+\frac{1}{23}a^{11}+\frac{29}{736}a^{10}+\frac{35}{368}a^{9}+\frac{3}{92}a^{8}-\frac{17}{736}a^{7}+\frac{5}{46}a^{6}-\frac{11}{46}a^{5}-\frac{173}{736}a^{4}-\frac{27}{368}a^{3}-\frac{29}{92}a^{2}+\frac{4}{23}a+\frac{8}{23}$, $\frac{1}{29440}a^{14}+\frac{3}{5888}a^{13}-\frac{13}{14720}a^{12}-\frac{67}{29440}a^{11}-\frac{1583}{29440}a^{10}-\frac{3}{32}a^{9}-\frac{7}{256}a^{8}+\frac{5153}{29440}a^{7}+\frac{89}{2944}a^{6}-\frac{7261}{29440}a^{5}-\frac{2257}{29440}a^{4}+\frac{249}{736}a^{3}+\frac{1881}{7360}a^{2}+\frac{807}{1840}a-\frac{73}{230}$, $\frac{1}{22\!\cdots\!60}a^{15}-\frac{59\!\cdots\!77}{22\!\cdots\!60}a^{14}+\frac{32\!\cdots\!47}{11\!\cdots\!80}a^{13}-\frac{14\!\cdots\!79}{45\!\cdots\!92}a^{12}-\frac{31\!\cdots\!13}{98\!\cdots\!20}a^{11}+\frac{21\!\cdots\!99}{35\!\cdots\!40}a^{10}+\frac{53\!\cdots\!35}{45\!\cdots\!92}a^{9}+\frac{15\!\cdots\!33}{22\!\cdots\!60}a^{8}+\frac{61\!\cdots\!97}{11\!\cdots\!80}a^{7}+\frac{19\!\cdots\!79}{22\!\cdots\!60}a^{6}-\frac{77\!\cdots\!45}{45\!\cdots\!92}a^{5}+\frac{84\!\cdots\!47}{71\!\cdots\!80}a^{4}+\frac{28\!\cdots\!21}{56\!\cdots\!40}a^{3}+\frac{41\!\cdots\!09}{14\!\cdots\!60}a^{2}+\frac{12\!\cdots\!77}{88\!\cdots\!60}a+\frac{93\!\cdots\!07}{22\!\cdots\!40}$
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
$C_{2}$, 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: |
$\frac{94\!\cdots\!55}{38\!\cdots\!58}a^{15}+\frac{27\!\cdots\!35}{16\!\cdots\!46}a^{14}-\frac{21\!\cdots\!65}{38\!\cdots\!58}a^{13}+\frac{55\!\cdots\!20}{19\!\cdots\!79}a^{12}+\frac{87\!\cdots\!85}{19\!\cdots\!79}a^{11}-\frac{11\!\cdots\!15}{19\!\cdots\!79}a^{10}-\frac{25\!\cdots\!95}{16\!\cdots\!46}a^{9}+\frac{10\!\cdots\!65}{38\!\cdots\!58}a^{8}+\frac{42\!\cdots\!85}{16\!\cdots\!46}a^{7}-\frac{81\!\cdots\!25}{19\!\cdots\!79}a^{6}-\frac{41\!\cdots\!90}{19\!\cdots\!79}a^{5}+\frac{53\!\cdots\!80}{19\!\cdots\!79}a^{4}+\frac{16\!\cdots\!40}{19\!\cdots\!79}a^{3}-\frac{10\!\cdots\!40}{19\!\cdots\!79}a^{2}-\frac{27\!\cdots\!40}{19\!\cdots\!79}a-\frac{81\!\cdots\!77}{19\!\cdots\!79}$, $\frac{29\!\cdots\!49}{11\!\cdots\!80}a^{15}-\frac{35\!\cdots\!65}{22\!\cdots\!96}a^{14}-\frac{32\!\cdots\!37}{56\!\cdots\!40}a^{13}+\frac{47\!\cdots\!97}{11\!\cdots\!80}a^{12}+\frac{43\!\cdots\!33}{11\!\cdots\!80}a^{11}-\frac{25\!\cdots\!93}{71\!\cdots\!28}a^{10}-\frac{14\!\cdots\!17}{22\!\cdots\!96}a^{9}+\frac{12\!\cdots\!77}{11\!\cdots\!80}a^{8}-\frac{75\!\cdots\!95}{11\!\cdots\!48}a^{7}-\frac{16\!\cdots\!69}{11\!\cdots\!80}a^{6}+\frac{25\!\cdots\!27}{11\!\cdots\!80}a^{5}+\frac{25\!\cdots\!35}{35\!\cdots\!64}a^{4}-\frac{39\!\cdots\!51}{28\!\cdots\!20}a^{3}-\frac{10\!\cdots\!47}{71\!\cdots\!80}a^{2}+\frac{97\!\cdots\!19}{44\!\cdots\!80}a+\frac{42\!\cdots\!79}{22\!\cdots\!04}$, $\frac{22\!\cdots\!53}{11\!\cdots\!80}a^{15}+\frac{86\!\cdots\!03}{11\!\cdots\!80}a^{14}-\frac{25\!\cdots\!49}{56\!\cdots\!40}a^{13}-\frac{13\!\cdots\!79}{11\!\cdots\!80}a^{12}+\frac{34\!\cdots\!59}{98\!\cdots\!52}a^{11}+\frac{45\!\cdots\!47}{88\!\cdots\!60}a^{10}-\frac{27\!\cdots\!81}{22\!\cdots\!96}a^{9}-\frac{10\!\cdots\!71}{11\!\cdots\!80}a^{8}+\frac{12\!\cdots\!17}{56\!\cdots\!40}a^{7}+\frac{77\!\cdots\!47}{11\!\cdots\!80}a^{6}-\frac{21\!\cdots\!49}{11\!\cdots\!80}a^{5}-\frac{32\!\cdots\!23}{35\!\cdots\!40}a^{4}+\frac{22\!\cdots\!53}{28\!\cdots\!20}a^{3}+\frac{88\!\cdots\!13}{71\!\cdots\!80}a^{2}-\frac{46\!\cdots\!11}{44\!\cdots\!80}a-\frac{46\!\cdots\!53}{11\!\cdots\!20}$, $\frac{40\!\cdots\!31}{11\!\cdots\!80}a^{15}-\frac{74\!\cdots\!47}{11\!\cdots\!80}a^{14}-\frac{43\!\cdots\!83}{56\!\cdots\!40}a^{13}+\frac{23\!\cdots\!69}{98\!\cdots\!52}a^{12}+\frac{57\!\cdots\!71}{11\!\cdots\!80}a^{11}-\frac{76\!\cdots\!07}{35\!\cdots\!40}a^{10}-\frac{24\!\cdots\!47}{22\!\cdots\!96}a^{9}+\frac{66\!\cdots\!83}{11\!\cdots\!80}a^{8}+\frac{38\!\cdots\!47}{56\!\cdots\!40}a^{7}-\frac{65\!\cdots\!51}{11\!\cdots\!80}a^{6}+\frac{12\!\cdots\!09}{22\!\cdots\!96}a^{5}+\frac{39\!\cdots\!21}{17\!\cdots\!20}a^{4}-\frac{31\!\cdots\!29}{28\!\cdots\!20}a^{3}-\frac{19\!\cdots\!61}{71\!\cdots\!80}a^{2}+\frac{37\!\cdots\!97}{44\!\cdots\!80}a+\frac{89\!\cdots\!17}{11\!\cdots\!20}$, $\frac{15\!\cdots\!33}{11\!\cdots\!80}a^{15}-\frac{33\!\cdots\!93}{11\!\cdots\!80}a^{14}-\frac{18\!\cdots\!89}{56\!\cdots\!40}a^{13}+\frac{12\!\cdots\!77}{11\!\cdots\!80}a^{12}+\frac{25\!\cdots\!57}{11\!\cdots\!80}a^{11}-\frac{79\!\cdots\!33}{77\!\cdots\!40}a^{10}-\frac{13\!\cdots\!97}{22\!\cdots\!96}a^{9}+\frac{39\!\cdots\!69}{11\!\cdots\!80}a^{8}+\frac{26\!\cdots\!93}{56\!\cdots\!40}a^{7}-\frac{53\!\cdots\!53}{11\!\cdots\!80}a^{6}+\frac{17\!\cdots\!87}{11\!\cdots\!80}a^{5}+\frac{38\!\cdots\!01}{15\!\cdots\!80}a^{4}-\frac{69\!\cdots\!27}{28\!\cdots\!20}a^{3}-\frac{28\!\cdots\!91}{71\!\cdots\!80}a^{2}+\frac{15\!\cdots\!37}{44\!\cdots\!80}a+\frac{34\!\cdots\!23}{11\!\cdots\!20}$, $\frac{20\!\cdots\!11}{11\!\cdots\!80}a^{15}+\frac{24\!\cdots\!41}{11\!\cdots\!80}a^{14}-\frac{23\!\cdots\!83}{56\!\cdots\!40}a^{13}+\frac{62\!\cdots\!47}{11\!\cdots\!80}a^{12}+\frac{32\!\cdots\!93}{98\!\cdots\!52}a^{11}-\frac{14\!\cdots\!83}{44\!\cdots\!80}a^{10}-\frac{25\!\cdots\!51}{22\!\cdots\!96}a^{9}+\frac{18\!\cdots\!03}{11\!\cdots\!80}a^{8}+\frac{99\!\cdots\!59}{56\!\cdots\!40}a^{7}-\frac{31\!\cdots\!91}{11\!\cdots\!80}a^{6}-\frac{15\!\cdots\!03}{11\!\cdots\!80}a^{5}+\frac{59\!\cdots\!19}{35\!\cdots\!40}a^{4}+\frac{14\!\cdots\!91}{28\!\cdots\!20}a^{3}-\frac{19\!\cdots\!89}{71\!\cdots\!80}a^{2}-\frac{36\!\cdots\!37}{44\!\cdots\!80}a-\frac{31\!\cdots\!71}{11\!\cdots\!20}$, $\frac{10\!\cdots\!91}{11\!\cdots\!80}a^{15}+\frac{31\!\cdots\!65}{22\!\cdots\!96}a^{14}-\frac{12\!\cdots\!63}{56\!\cdots\!40}a^{13}-\frac{60\!\cdots\!37}{11\!\cdots\!80}a^{12}+\frac{19\!\cdots\!27}{11\!\cdots\!80}a^{11}-\frac{83\!\cdots\!09}{71\!\cdots\!28}a^{10}-\frac{13\!\cdots\!07}{22\!\cdots\!96}a^{9}+\frac{70\!\cdots\!23}{11\!\cdots\!80}a^{8}+\frac{11\!\cdots\!47}{11\!\cdots\!48}a^{7}-\frac{53\!\cdots\!37}{49\!\cdots\!60}a^{6}-\frac{94\!\cdots\!87}{11\!\cdots\!80}a^{5}+\frac{12\!\cdots\!53}{17\!\cdots\!32}a^{4}+\frac{95\!\cdots\!91}{28\!\cdots\!20}a^{3}-\frac{90\!\cdots\!33}{71\!\cdots\!80}a^{2}-\frac{23\!\cdots\!99}{44\!\cdots\!80}a-\frac{40\!\cdots\!95}{22\!\cdots\!04}$, $\frac{34\!\cdots\!13}{11\!\cdots\!80}a^{15}+\frac{72\!\cdots\!87}{11\!\cdots\!80}a^{14}-\frac{38\!\cdots\!09}{56\!\cdots\!40}a^{13}-\frac{30\!\cdots\!21}{49\!\cdots\!60}a^{12}+\frac{61\!\cdots\!77}{11\!\cdots\!80}a^{11}-\frac{16\!\cdots\!91}{44\!\cdots\!80}a^{10}-\frac{42\!\cdots\!65}{22\!\cdots\!96}a^{9}+\frac{10\!\cdots\!49}{11\!\cdots\!80}a^{8}+\frac{17\!\cdots\!33}{56\!\cdots\!40}a^{7}-\frac{22\!\cdots\!33}{11\!\cdots\!80}a^{6}-\frac{30\!\cdots\!13}{11\!\cdots\!80}a^{5}+\frac{50\!\cdots\!93}{35\!\cdots\!40}a^{4}+\frac{30\!\cdots\!73}{28\!\cdots\!20}a^{3}-\frac{17\!\cdots\!51}{71\!\cdots\!80}a^{2}-\frac{68\!\cdots\!03}{44\!\cdots\!80}a-\frac{63\!\cdots\!57}{11\!\cdots\!20}$, $\frac{12\!\cdots\!23}{11\!\cdots\!80}a^{15}+\frac{73\!\cdots\!49}{11\!\cdots\!80}a^{14}-\frac{14\!\cdots\!39}{56\!\cdots\!40}a^{13}-\frac{26\!\cdots\!37}{22\!\cdots\!96}a^{12}+\frac{24\!\cdots\!83}{11\!\cdots\!80}a^{11}+\frac{23\!\cdots\!29}{35\!\cdots\!40}a^{10}-\frac{20\!\cdots\!03}{22\!\cdots\!96}a^{9}-\frac{17\!\cdots\!41}{11\!\cdots\!80}a^{8}+\frac{11\!\cdots\!11}{56\!\cdots\!40}a^{7}+\frac{14\!\cdots\!57}{11\!\cdots\!80}a^{6}-\frac{47\!\cdots\!79}{22\!\cdots\!96}a^{5}-\frac{96\!\cdots\!77}{17\!\cdots\!20}a^{4}+\frac{26\!\cdots\!83}{28\!\cdots\!20}a^{3}-\frac{51\!\cdots\!33}{71\!\cdots\!80}a^{2}-\frac{62\!\cdots\!39}{44\!\cdots\!80}a-\frac{61\!\cdots\!19}{11\!\cdots\!20}$, $\frac{12\!\cdots\!59}{56\!\cdots\!40}a^{15}+\frac{24\!\cdots\!37}{56\!\cdots\!40}a^{14}-\frac{14\!\cdots\!87}{28\!\cdots\!20}a^{13}-\frac{42\!\cdots\!05}{11\!\cdots\!48}a^{12}+\frac{23\!\cdots\!19}{56\!\cdots\!40}a^{11}-\frac{16\!\cdots\!63}{17\!\cdots\!20}a^{10}-\frac{16\!\cdots\!03}{11\!\cdots\!48}a^{9}+\frac{49\!\cdots\!07}{56\!\cdots\!40}a^{8}+\frac{70\!\cdots\!83}{28\!\cdots\!20}a^{7}-\frac{90\!\cdots\!39}{56\!\cdots\!40}a^{6}-\frac{24\!\cdots\!39}{11\!\cdots\!48}a^{5}+\frac{81\!\cdots\!99}{88\!\cdots\!60}a^{4}+\frac{13\!\cdots\!59}{14\!\cdots\!60}a^{3}+\frac{24\!\cdots\!91}{35\!\cdots\!40}a^{2}-\frac{36\!\cdots\!27}{22\!\cdots\!40}a-\frac{23\!\cdots\!79}{24\!\cdots\!70}$, $\frac{24\!\cdots\!53}{28\!\cdots\!20}a^{15}+\frac{13\!\cdots\!23}{28\!\cdots\!20}a^{14}-\frac{26\!\cdots\!49}{14\!\cdots\!60}a^{13}-\frac{23\!\cdots\!19}{28\!\cdots\!20}a^{12}+\frac{78\!\cdots\!49}{56\!\cdots\!24}a^{11}+\frac{20\!\cdots\!49}{44\!\cdots\!80}a^{10}-\frac{26\!\cdots\!41}{56\!\cdots\!24}a^{9}-\frac{35\!\cdots\!91}{28\!\cdots\!20}a^{8}+\frac{11\!\cdots\!57}{14\!\cdots\!60}a^{7}+\frac{51\!\cdots\!27}{28\!\cdots\!20}a^{6}-\frac{19\!\cdots\!49}{28\!\cdots\!20}a^{5}-\frac{12\!\cdots\!93}{88\!\cdots\!60}a^{4}+\frac{22\!\cdots\!33}{71\!\cdots\!80}a^{3}+\frac{42\!\cdots\!11}{77\!\cdots\!40}a^{2}-\frac{66\!\cdots\!81}{11\!\cdots\!20}a-\frac{21\!\cdots\!68}{27\!\cdots\!05}$, $\frac{62\!\cdots\!01}{11\!\cdots\!80}a^{15}+\frac{11\!\cdots\!47}{11\!\cdots\!80}a^{14}-\frac{71\!\cdots\!73}{56\!\cdots\!40}a^{13}-\frac{98\!\cdots\!39}{11\!\cdots\!80}a^{12}+\frac{48\!\cdots\!31}{49\!\cdots\!60}a^{11}-\frac{96\!\cdots\!73}{35\!\cdots\!40}a^{10}-\frac{76\!\cdots\!49}{22\!\cdots\!96}a^{9}+\frac{26\!\cdots\!73}{11\!\cdots\!80}a^{8}+\frac{31\!\cdots\!33}{56\!\cdots\!40}a^{7}-\frac{49\!\cdots\!01}{11\!\cdots\!80}a^{6}-\frac{52\!\cdots\!89}{11\!\cdots\!80}a^{5}+\frac{50\!\cdots\!09}{17\!\cdots\!20}a^{4}+\frac{51\!\cdots\!41}{28\!\cdots\!20}a^{3}-\frac{61\!\cdots\!43}{14\!\cdots\!56}a^{2}-\frac{23\!\cdots\!05}{88\!\cdots\!16}a-\frac{11\!\cdots\!37}{11\!\cdots\!20}$, $\frac{28\!\cdots\!87}{11\!\cdots\!80}a^{15}-\frac{28\!\cdots\!63}{22\!\cdots\!96}a^{14}-\frac{29\!\cdots\!91}{56\!\cdots\!40}a^{13}+\frac{39\!\cdots\!31}{11\!\cdots\!80}a^{12}+\frac{35\!\cdots\!19}{11\!\cdots\!80}a^{11}-\frac{19\!\cdots\!63}{71\!\cdots\!28}a^{10}-\frac{78\!\cdots\!75}{22\!\cdots\!96}a^{9}+\frac{78\!\cdots\!31}{11\!\cdots\!80}a^{8}-\frac{77\!\cdots\!93}{11\!\cdots\!48}a^{7}-\frac{72\!\cdots\!27}{11\!\cdots\!80}a^{6}+\frac{13\!\cdots\!41}{11\!\cdots\!80}a^{5}+\frac{81\!\cdots\!73}{35\!\cdots\!64}a^{4}-\frac{15\!\cdots\!33}{28\!\cdots\!20}a^{3}-\frac{20\!\cdots\!41}{71\!\cdots\!80}a^{2}+\frac{28\!\cdots\!77}{44\!\cdots\!80}a+\frac{66\!\cdots\!21}{22\!\cdots\!04}$, $\frac{89\!\cdots\!71}{11\!\cdots\!80}a^{15}-\frac{17\!\cdots\!87}{11\!\cdots\!80}a^{14}-\frac{10\!\cdots\!43}{56\!\cdots\!40}a^{13}+\frac{12\!\cdots\!31}{22\!\cdots\!96}a^{12}+\frac{15\!\cdots\!91}{11\!\cdots\!80}a^{11}-\frac{51\!\cdots\!73}{88\!\cdots\!60}a^{10}-\frac{94\!\cdots\!51}{22\!\cdots\!96}a^{9}+\frac{23\!\cdots\!83}{11\!\cdots\!80}a^{8}+\frac{28\!\cdots\!47}{56\!\cdots\!40}a^{7}-\frac{16\!\cdots\!57}{49\!\cdots\!60}a^{6}-\frac{33\!\cdots\!27}{22\!\cdots\!96}a^{5}+\frac{86\!\cdots\!67}{35\!\cdots\!40}a^{4}-\frac{29\!\cdots\!69}{28\!\cdots\!20}a^{3}-\frac{49\!\cdots\!01}{71\!\cdots\!80}a^{2}+\frac{22\!\cdots\!87}{44\!\cdots\!80}a+\frac{44\!\cdots\!37}{11\!\cdots\!20}$, $\frac{50\!\cdots\!27}{11\!\cdots\!80}a^{15}-\frac{35\!\cdots\!87}{11\!\cdots\!80}a^{14}-\frac{58\!\cdots\!51}{56\!\cdots\!40}a^{13}+\frac{93\!\cdots\!23}{11\!\cdots\!80}a^{12}+\frac{81\!\cdots\!23}{11\!\cdots\!80}a^{11}-\frac{53\!\cdots\!87}{77\!\cdots\!40}a^{10}-\frac{30\!\cdots\!87}{22\!\cdots\!96}a^{9}+\frac{24\!\cdots\!31}{11\!\cdots\!80}a^{8}-\frac{64\!\cdots\!33}{56\!\cdots\!40}a^{7}-\frac{31\!\cdots\!47}{11\!\cdots\!80}a^{6}+\frac{51\!\cdots\!53}{11\!\cdots\!80}a^{5}+\frac{44\!\cdots\!37}{35\!\cdots\!40}a^{4}-\frac{75\!\cdots\!73}{28\!\cdots\!20}a^{3}-\frac{11\!\cdots\!69}{71\!\cdots\!80}a^{2}+\frac{14\!\cdots\!43}{44\!\cdots\!80}a+\frac{18\!\cdots\!37}{11\!\cdots\!20}$
| 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: | \( 222610131226000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 222610131226000000 \cdot 2}{2\cdot\sqrt{6123007382888435990757129497254763904689}}\cr\approx \mathstrut & 186.441446929653 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152)
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 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152, 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 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152);
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 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152);
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
| \(\Q(\sqrt{41}) \), 4.4.11647649.1, 8.8.940041681957275729.2 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ | $16$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | $16$ | $16$ | R | $16$ | $16$ | ${\href{/padicField/23.1.0.1}{1} }^{16}$ | $16$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.16.14.5 | $x^{16} - 156 x^{8} + 338$ | $8$ | $2$ | $14$ | $C_{16} : C_2$ | $[\ ]_{8}^{4}$ |
|
\(41\)
| 41.16.15.2 | $x^{16} + 205$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |