Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452)
gp: K = bnfinit(y^16 - 5*y^15 - 146*y^14 + 723*y^13 + 7442*y^12 - 35545*y^11 - 159987*y^10 + 700390*y^9 + 1461579*y^8 - 5189481*y^7 - 6310742*y^6 + 14524949*y^5 + 10383990*y^4 - 12357337*y^3 - 1117197*y^2 + 2153042*y - 297452, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452)
\( x^{16} - 5 x^{15} - 146 x^{14} + 723 x^{13} + 7442 x^{12} - 35545 x^{11} - 159987 x^{10} + 700390 x^{9} + \cdots - 297452 \)
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: |
\(1171598758708107367475386427203165009\)
\(\medspace = 13^{14}\cdot 29^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(179.60\) | 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: | $13^{7/8}29^{7/8}\approx 179.59683604141384$ | ||
| Ramified primes: |
\(13\), \(29\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{32}a^{12}+\frac{1}{32}a^{11}+\frac{1}{32}a^{9}-\frac{1}{32}a^{8}+\frac{1}{16}a^{7}+\frac{7}{32}a^{6}-\frac{5}{32}a^{5}-\frac{1}{4}a^{4}+\frac{3}{32}a^{3}-\frac{11}{32}a^{2}+\frac{7}{16}a-\frac{1}{8}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{11}+\frac{1}{64}a^{10}-\frac{1}{32}a^{9}+\frac{3}{64}a^{8}-\frac{11}{64}a^{7}-\frac{3}{16}a^{6}+\frac{13}{64}a^{5}-\frac{5}{64}a^{4}+\frac{9}{32}a^{3}+\frac{9}{64}a^{2}+\frac{7}{32}a-\frac{7}{16}$, $\frac{1}{7936}a^{14}+\frac{33}{7936}a^{13}-\frac{49}{7936}a^{12}+\frac{45}{992}a^{11}+\frac{439}{7936}a^{10}+\frac{105}{7936}a^{9}-\frac{17}{496}a^{8}+\frac{153}{7936}a^{7}+\frac{47}{256}a^{6}+\frac{1}{248}a^{5}-\frac{1739}{7936}a^{4}+\frac{3123}{7936}a^{3}+\frac{2591}{7936}a^{2}+\frac{401}{3968}a-\frac{607}{1984}$, $\frac{1}{52\!\cdots\!84}a^{15}-\frac{96\!\cdots\!43}{16\!\cdots\!12}a^{14}+\frac{11\!\cdots\!35}{26\!\cdots\!92}a^{13}-\frac{39\!\cdots\!75}{52\!\cdots\!84}a^{12}+\frac{16\!\cdots\!15}{52\!\cdots\!84}a^{11}+\frac{46\!\cdots\!97}{26\!\cdots\!92}a^{10}-\frac{13\!\cdots\!37}{52\!\cdots\!84}a^{9}-\frac{40\!\cdots\!11}{52\!\cdots\!84}a^{8}-\frac{28\!\cdots\!17}{65\!\cdots\!48}a^{7}-\frac{11\!\cdots\!81}{52\!\cdots\!84}a^{6}-\frac{79\!\cdots\!39}{52\!\cdots\!84}a^{5}-\frac{49\!\cdots\!25}{26\!\cdots\!92}a^{4}+\frac{42\!\cdots\!59}{13\!\cdots\!96}a^{3}-\frac{25\!\cdots\!53}{52\!\cdots\!84}a^{2}-\frac{23\!\cdots\!03}{70\!\cdots\!16}a+\frac{14\!\cdots\!67}{13\!\cdots\!96}$
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
$C_{4}$, which has order $4$ (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: |
$\frac{34\!\cdots\!95}{11\!\cdots\!64}a^{15}-\frac{18\!\cdots\!67}{87\!\cdots\!88}a^{14}-\frac{22\!\cdots\!03}{55\!\cdots\!32}a^{13}+\frac{32\!\cdots\!75}{11\!\cdots\!64}a^{12}+\frac{19\!\cdots\!21}{11\!\cdots\!64}a^{11}-\frac{77\!\cdots\!69}{55\!\cdots\!32}a^{10}-\frac{27\!\cdots\!75}{11\!\cdots\!64}a^{9}+\frac{28\!\cdots\!39}{11\!\cdots\!64}a^{8}+\frac{29\!\cdots\!41}{13\!\cdots\!08}a^{7}-\frac{17\!\cdots\!31}{11\!\cdots\!64}a^{6}+\frac{82\!\cdots\!35}{11\!\cdots\!64}a^{5}+\frac{17\!\cdots\!81}{55\!\cdots\!32}a^{4}-\frac{54\!\cdots\!87}{27\!\cdots\!16}a^{3}-\frac{51\!\cdots\!31}{11\!\cdots\!64}a^{2}+\frac{63\!\cdots\!67}{15\!\cdots\!36}a-\frac{99\!\cdots\!15}{27\!\cdots\!16}$, $\frac{17\!\cdots\!01}{31\!\cdots\!56}a^{15}-\frac{10\!\cdots\!15}{39\!\cdots\!32}a^{14}-\frac{13\!\cdots\!65}{15\!\cdots\!28}a^{13}+\frac{11\!\cdots\!57}{31\!\cdots\!56}a^{12}+\frac{13\!\cdots\!75}{31\!\cdots\!56}a^{11}-\frac{29\!\cdots\!51}{15\!\cdots\!28}a^{10}-\frac{30\!\cdots\!49}{31\!\cdots\!56}a^{9}+\frac{11\!\cdots\!49}{31\!\cdots\!56}a^{8}+\frac{95\!\cdots\!53}{97\!\cdots\!08}a^{7}-\frac{84\!\cdots\!37}{31\!\cdots\!56}a^{6}-\frac{14\!\cdots\!07}{31\!\cdots\!56}a^{5}+\frac{11\!\cdots\!51}{15\!\cdots\!28}a^{4}+\frac{66\!\cdots\!45}{78\!\cdots\!64}a^{3}-\frac{17\!\cdots\!41}{31\!\cdots\!56}a^{2}-\frac{10\!\cdots\!71}{42\!\cdots\!44}a+\frac{47\!\cdots\!79}{78\!\cdots\!64}$, $\frac{95\!\cdots\!09}{13\!\cdots\!96}a^{15}-\frac{60\!\cdots\!31}{16\!\cdots\!12}a^{14}-\frac{69\!\cdots\!33}{65\!\cdots\!48}a^{13}+\frac{69\!\cdots\!21}{13\!\cdots\!96}a^{12}+\frac{70\!\cdots\!95}{13\!\cdots\!96}a^{11}-\frac{16\!\cdots\!27}{65\!\cdots\!48}a^{10}-\frac{15\!\cdots\!13}{13\!\cdots\!96}a^{9}+\frac{65\!\cdots\!69}{13\!\cdots\!96}a^{8}+\frac{17\!\cdots\!59}{16\!\cdots\!12}a^{7}-\frac{47\!\cdots\!21}{13\!\cdots\!96}a^{6}-\frac{59\!\cdots\!23}{13\!\cdots\!96}a^{5}+\frac{60\!\cdots\!71}{65\!\cdots\!48}a^{4}+\frac{26\!\cdots\!09}{32\!\cdots\!24}a^{3}-\frac{79\!\cdots\!97}{13\!\cdots\!96}a^{2}-\frac{43\!\cdots\!87}{17\!\cdots\!04}a+\frac{19\!\cdots\!95}{32\!\cdots\!24}$, $\frac{12\!\cdots\!55}{52\!\cdots\!84}a^{15}-\frac{90\!\cdots\!57}{40\!\cdots\!28}a^{14}-\frac{72\!\cdots\!23}{26\!\cdots\!92}a^{13}+\frac{16\!\cdots\!71}{52\!\cdots\!84}a^{12}+\frac{42\!\cdots\!09}{52\!\cdots\!84}a^{11}-\frac{37\!\cdots\!01}{26\!\cdots\!92}a^{10}+\frac{35\!\cdots\!05}{52\!\cdots\!84}a^{9}+\frac{13\!\cdots\!71}{52\!\cdots\!84}a^{8}-\frac{30\!\cdots\!19}{65\!\cdots\!48}a^{7}-\frac{71\!\cdots\!03}{52\!\cdots\!84}a^{6}+\frac{17\!\cdots\!83}{52\!\cdots\!84}a^{5}+\frac{45\!\cdots\!13}{26\!\cdots\!92}a^{4}-\frac{85\!\cdots\!03}{13\!\cdots\!96}a^{3}+\frac{77\!\cdots\!21}{52\!\cdots\!84}a^{2}+\frac{11\!\cdots\!59}{70\!\cdots\!16}a-\frac{77\!\cdots\!71}{13\!\cdots\!96}$, $\frac{39\!\cdots\!77}{52\!\cdots\!84}a^{15}-\frac{11\!\cdots\!87}{32\!\cdots\!24}a^{14}-\frac{29\!\cdots\!65}{26\!\cdots\!92}a^{13}+\frac{26\!\cdots\!81}{52\!\cdots\!84}a^{12}+\frac{30\!\cdots\!31}{52\!\cdots\!84}a^{11}-\frac{64\!\cdots\!87}{26\!\cdots\!92}a^{10}-\frac{69\!\cdots\!89}{52\!\cdots\!84}a^{9}+\frac{25\!\cdots\!29}{52\!\cdots\!84}a^{8}+\frac{85\!\cdots\!89}{65\!\cdots\!48}a^{7}-\frac{18\!\cdots\!81}{52\!\cdots\!84}a^{6}-\frac{32\!\cdots\!15}{52\!\cdots\!84}a^{5}+\frac{22\!\cdots\!91}{26\!\cdots\!92}a^{4}+\frac{14\!\cdots\!27}{13\!\cdots\!96}a^{3}-\frac{84\!\cdots\!83}{16\!\cdots\!64}a^{2}-\frac{19\!\cdots\!55}{70\!\cdots\!16}a+\frac{79\!\cdots\!51}{13\!\cdots\!96}$, $\frac{11\!\cdots\!01}{52\!\cdots\!84}a^{15}-\frac{69\!\cdots\!71}{65\!\cdots\!48}a^{14}-\frac{86\!\cdots\!93}{26\!\cdots\!92}a^{13}+\frac{79\!\cdots\!13}{52\!\cdots\!84}a^{12}+\frac{89\!\cdots\!59}{52\!\cdots\!84}a^{11}-\frac{19\!\cdots\!15}{26\!\cdots\!92}a^{10}-\frac{19\!\cdots\!17}{52\!\cdots\!84}a^{9}+\frac{76\!\cdots\!25}{52\!\cdots\!84}a^{8}+\frac{59\!\cdots\!65}{16\!\cdots\!12}a^{7}-\frac{54\!\cdots\!89}{52\!\cdots\!84}a^{6}-\frac{88\!\cdots\!79}{52\!\cdots\!84}a^{5}+\frac{70\!\cdots\!67}{26\!\cdots\!92}a^{4}+\frac{38\!\cdots\!33}{13\!\cdots\!96}a^{3}-\frac{90\!\cdots\!69}{52\!\cdots\!84}a^{2}-\frac{36\!\cdots\!19}{70\!\cdots\!16}a+\frac{31\!\cdots\!63}{13\!\cdots\!96}$, $\frac{11\!\cdots\!43}{13\!\cdots\!96}a^{15}-\frac{25\!\cdots\!83}{65\!\cdots\!48}a^{14}-\frac{41\!\cdots\!23}{32\!\cdots\!24}a^{13}+\frac{73\!\cdots\!57}{13\!\cdots\!96}a^{12}+\frac{86\!\cdots\!61}{13\!\cdots\!96}a^{11}-\frac{28\!\cdots\!39}{10\!\cdots\!04}a^{10}-\frac{19\!\cdots\!29}{13\!\cdots\!96}a^{9}+\frac{69\!\cdots\!55}{13\!\cdots\!96}a^{8}+\frac{98\!\cdots\!05}{65\!\cdots\!48}a^{7}-\frac{49\!\cdots\!33}{13\!\cdots\!96}a^{6}-\frac{95\!\cdots\!45}{13\!\cdots\!96}a^{5}+\frac{29\!\cdots\!87}{32\!\cdots\!24}a^{4}+\frac{28\!\cdots\!91}{21\!\cdots\!08}a^{3}-\frac{50\!\cdots\!65}{13\!\cdots\!96}a^{2}-\frac{50\!\cdots\!35}{17\!\cdots\!04}a+\frac{20\!\cdots\!75}{32\!\cdots\!24}$, $\frac{64\!\cdots\!89}{13\!\cdots\!96}a^{15}-\frac{65\!\cdots\!35}{32\!\cdots\!24}a^{14}-\frac{48\!\cdots\!19}{65\!\cdots\!48}a^{13}+\frac{37\!\cdots\!77}{13\!\cdots\!96}a^{12}+\frac{51\!\cdots\!71}{13\!\cdots\!96}a^{11}-\frac{89\!\cdots\!61}{65\!\cdots\!48}a^{10}-\frac{11\!\cdots\!57}{13\!\cdots\!96}a^{9}+\frac{33\!\cdots\!41}{13\!\cdots\!96}a^{8}+\frac{31\!\cdots\!05}{32\!\cdots\!24}a^{7}-\frac{20\!\cdots\!61}{13\!\cdots\!96}a^{6}-\frac{62\!\cdots\!15}{13\!\cdots\!96}a^{5}+\frac{16\!\cdots\!33}{65\!\cdots\!48}a^{4}+\frac{14\!\cdots\!97}{16\!\cdots\!12}a^{3}+\frac{21\!\cdots\!71}{13\!\cdots\!96}a^{2}-\frac{10\!\cdots\!53}{57\!\cdots\!84}a+\frac{70\!\cdots\!55}{32\!\cdots\!24}$, $\frac{80\!\cdots\!49}{52\!\cdots\!84}a^{15}-\frac{25\!\cdots\!07}{32\!\cdots\!24}a^{14}-\frac{58\!\cdots\!29}{26\!\cdots\!92}a^{13}+\frac{59\!\cdots\!41}{52\!\cdots\!84}a^{12}+\frac{60\!\cdots\!51}{52\!\cdots\!84}a^{11}-\frac{14\!\cdots\!43}{26\!\cdots\!92}a^{10}-\frac{13\!\cdots\!65}{52\!\cdots\!84}a^{9}+\frac{61\!\cdots\!61}{52\!\cdots\!84}a^{8}+\frac{14\!\cdots\!33}{65\!\cdots\!48}a^{7}-\frac{50\!\cdots\!85}{52\!\cdots\!84}a^{6}-\frac{43\!\cdots\!99}{52\!\cdots\!84}a^{5}+\frac{80\!\cdots\!87}{26\!\cdots\!92}a^{4}+\frac{72\!\cdots\!95}{13\!\cdots\!96}a^{3}-\frac{15\!\cdots\!45}{52\!\cdots\!84}a^{2}+\frac{90\!\cdots\!85}{70\!\cdots\!16}a-\frac{18\!\cdots\!49}{13\!\cdots\!96}$, $\frac{24\!\cdots\!31}{52\!\cdots\!84}a^{15}-\frac{69\!\cdots\!21}{32\!\cdots\!24}a^{14}-\frac{17\!\cdots\!43}{26\!\cdots\!92}a^{13}+\frac{16\!\cdots\!07}{52\!\cdots\!84}a^{12}+\frac{18\!\cdots\!73}{52\!\cdots\!84}a^{11}-\frac{39\!\cdots\!33}{26\!\cdots\!92}a^{10}-\frac{41\!\cdots\!19}{52\!\cdots\!84}a^{9}+\frac{15\!\cdots\!35}{52\!\cdots\!84}a^{8}+\frac{51\!\cdots\!03}{65\!\cdots\!48}a^{7}-\frac{11\!\cdots\!15}{52\!\cdots\!84}a^{6}-\frac{19\!\cdots\!09}{52\!\cdots\!84}a^{5}+\frac{14\!\cdots\!81}{26\!\cdots\!92}a^{4}+\frac{88\!\cdots\!17}{13\!\cdots\!96}a^{3}-\frac{17\!\cdots\!15}{52\!\cdots\!84}a^{2}-\frac{11\!\cdots\!29}{70\!\cdots\!16}a+\frac{49\!\cdots\!61}{13\!\cdots\!96}$, $\frac{22\!\cdots\!29}{52\!\cdots\!84}a^{15}-\frac{15\!\cdots\!23}{81\!\cdots\!56}a^{14}-\frac{16\!\cdots\!65}{26\!\cdots\!92}a^{13}+\frac{14\!\cdots\!17}{52\!\cdots\!84}a^{12}+\frac{17\!\cdots\!23}{52\!\cdots\!84}a^{11}-\frac{34\!\cdots\!63}{26\!\cdots\!92}a^{10}-\frac{39\!\cdots\!69}{52\!\cdots\!84}a^{9}+\frac{13\!\cdots\!73}{52\!\cdots\!84}a^{8}+\frac{49\!\cdots\!91}{65\!\cdots\!48}a^{7}-\frac{92\!\cdots\!65}{52\!\cdots\!84}a^{6}-\frac{18\!\cdots\!39}{52\!\cdots\!84}a^{5}+\frac{10\!\cdots\!79}{26\!\cdots\!92}a^{4}+\frac{84\!\cdots\!75}{13\!\cdots\!96}a^{3}-\frac{73\!\cdots\!41}{52\!\cdots\!84}a^{2}-\frac{68\!\cdots\!15}{70\!\cdots\!16}a+\frac{24\!\cdots\!51}{13\!\cdots\!96}$, $\frac{11\!\cdots\!67}{52\!\cdots\!84}a^{15}-\frac{10\!\cdots\!47}{81\!\cdots\!56}a^{14}-\frac{85\!\cdots\!83}{26\!\cdots\!92}a^{13}+\frac{94\!\cdots\!79}{52\!\cdots\!84}a^{12}+\frac{84\!\cdots\!21}{52\!\cdots\!84}a^{11}-\frac{23\!\cdots\!93}{26\!\cdots\!92}a^{10}-\frac{16\!\cdots\!35}{52\!\cdots\!84}a^{9}+\frac{92\!\cdots\!95}{52\!\cdots\!84}a^{8}+\frac{52\!\cdots\!11}{21\!\cdots\!08}a^{7}-\frac{68\!\cdots\!59}{52\!\cdots\!84}a^{6}-\frac{42\!\cdots\!57}{52\!\cdots\!84}a^{5}+\frac{97\!\cdots\!69}{26\!\cdots\!92}a^{4}+\frac{74\!\cdots\!01}{13\!\cdots\!96}a^{3}-\frac{16\!\cdots\!11}{52\!\cdots\!84}a^{2}+\frac{88\!\cdots\!87}{70\!\cdots\!16}a-\frac{57\!\cdots\!01}{42\!\cdots\!16}$, $\frac{36\!\cdots\!23}{52\!\cdots\!84}a^{15}-\frac{99\!\cdots\!21}{32\!\cdots\!24}a^{14}-\frac{26\!\cdots\!71}{26\!\cdots\!92}a^{13}+\frac{22\!\cdots\!27}{52\!\cdots\!84}a^{12}+\frac{28\!\cdots\!01}{52\!\cdots\!84}a^{11}-\frac{55\!\cdots\!89}{26\!\cdots\!92}a^{10}-\frac{64\!\cdots\!95}{52\!\cdots\!84}a^{9}+\frac{21\!\cdots\!95}{52\!\cdots\!84}a^{8}+\frac{81\!\cdots\!07}{65\!\cdots\!48}a^{7}-\frac{14\!\cdots\!47}{52\!\cdots\!84}a^{6}-\frac{31\!\cdots\!33}{52\!\cdots\!84}a^{5}+\frac{17\!\cdots\!57}{26\!\cdots\!92}a^{4}+\frac{14\!\cdots\!29}{13\!\cdots\!96}a^{3}-\frac{11\!\cdots\!39}{52\!\cdots\!84}a^{2}-\frac{11\!\cdots\!53}{70\!\cdots\!16}a+\frac{39\!\cdots\!61}{13\!\cdots\!96}$, $\frac{82\!\cdots\!43}{74\!\cdots\!28}a^{15}-\frac{65\!\cdots\!43}{93\!\cdots\!16}a^{14}-\frac{56\!\cdots\!67}{37\!\cdots\!64}a^{13}+\frac{75\!\cdots\!95}{74\!\cdots\!28}a^{12}+\frac{50\!\cdots\!33}{74\!\cdots\!28}a^{11}-\frac{18\!\cdots\!49}{37\!\cdots\!64}a^{10}-\frac{80\!\cdots\!47}{74\!\cdots\!28}a^{9}+\frac{68\!\cdots\!59}{74\!\cdots\!28}a^{8}+\frac{16\!\cdots\!91}{46\!\cdots\!08}a^{7}-\frac{45\!\cdots\!55}{74\!\cdots\!28}a^{6}+\frac{48\!\cdots\!95}{74\!\cdots\!28}a^{5}+\frac{55\!\cdots\!53}{37\!\cdots\!64}a^{4}-\frac{73\!\cdots\!41}{18\!\cdots\!32}a^{3}-\frac{25\!\cdots\!37}{24\!\cdots\!88}a^{2}+\frac{20\!\cdots\!91}{37\!\cdots\!64}a-\frac{11\!\cdots\!95}{18\!\cdots\!32}$, $\frac{59\!\cdots\!47}{28\!\cdots\!68}a^{15}-\frac{16\!\cdots\!73}{17\!\cdots\!48}a^{14}-\frac{44\!\cdots\!39}{14\!\cdots\!84}a^{13}+\frac{38\!\cdots\!59}{28\!\cdots\!68}a^{12}+\frac{46\!\cdots\!13}{28\!\cdots\!68}a^{11}-\frac{95\!\cdots\!69}{14\!\cdots\!84}a^{10}-\frac{10\!\cdots\!63}{28\!\cdots\!68}a^{9}+\frac{37\!\cdots\!55}{28\!\cdots\!68}a^{8}+\frac{13\!\cdots\!67}{35\!\cdots\!96}a^{7}-\frac{28\!\cdots\!87}{28\!\cdots\!68}a^{6}-\frac{53\!\cdots\!29}{28\!\cdots\!68}a^{5}+\frac{38\!\cdots\!05}{14\!\cdots\!84}a^{4}+\frac{25\!\cdots\!29}{71\!\cdots\!92}a^{3}-\frac{61\!\cdots\!35}{28\!\cdots\!68}a^{2}-\frac{39\!\cdots\!01}{38\!\cdots\!32}a+\frac{32\!\cdots\!01}{71\!\cdots\!92}$
| 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: | \( 15637438683500 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 15637438683500 \cdot 4}{2\cdot\sqrt{1171598758708107367475386427203165009}}\cr\approx \mathstrut & 1.89359062435542 \end{aligned}\] (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 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452)
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 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452, 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 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452);
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 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452);
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.8.7.1 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 13.8.7.1 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(29\)
| 29.8.7.2 | $x^{8} + 29$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 29.8.7.2 | $x^{8} + 29$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |