Properties

Label 16.16.685...361.1
Degree $16$
Signature $[16, 0]$
Discriminant $6.856\times 10^{44}$
Root discriminant \(634.24\)
Ramified primes $13,89$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699)
 
gp: K = bnfinit(y^16 - 7*y^15 - 486*y^14 + 3640*y^13 + 85482*y^12 - 680369*y^11 - 6731288*y^10 + 57906718*y^9 + 234236862*y^8 - 2308745163*y^7 - 3167386332*y^6 + 41364170537*y^5 + 15994510202*y^4 - 316660788899*y^3 - 107062029196*y^2 + 853361887750*y + 670030778699, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699)
 

\( x^{16} - 7 x^{15} - 486 x^{14} + 3640 x^{13} + 85482 x^{12} - 680369 x^{11} - 6731288 x^{10} + 57906718 x^{9} + 234236862 x^{8} - 2308745163 x^{7} + \cdots + 670030778699 \) Copy content Toggle raw display

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:   \(685578251337344185848967768724165145970351361\) \(\medspace = 13^{14}\cdot 89^{15}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(634.24\)
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:  $13^{7/8}89^{15/16}\approx 634.2399065771889$
Ramified primes:   \(13\), \(89\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{12049}a^{14}-\frac{5289}{12049}a^{13}-\frac{756}{12049}a^{12}+\frac{772}{12049}a^{11}-\frac{5486}{12049}a^{10}+\frac{579}{12049}a^{9}+\frac{4339}{12049}a^{8}-\frac{5965}{12049}a^{7}-\frac{5994}{12049}a^{6}+\frac{1607}{12049}a^{5}-\frac{1377}{12049}a^{4}+\frac{4540}{12049}a^{3}+\frac{5030}{12049}a^{2}-\frac{3946}{12049}a-\frac{1677}{12049}$, $\frac{1}{72\!\cdots\!89}a^{15}+\frac{46\!\cdots\!87}{72\!\cdots\!89}a^{14}+\frac{17\!\cdots\!18}{72\!\cdots\!89}a^{13}-\frac{19\!\cdots\!20}{72\!\cdots\!89}a^{12}-\frac{22\!\cdots\!40}{72\!\cdots\!89}a^{11}-\frac{23\!\cdots\!87}{72\!\cdots\!89}a^{10}-\frac{10\!\cdots\!73}{72\!\cdots\!89}a^{9}+\frac{28\!\cdots\!47}{72\!\cdots\!89}a^{8}-\frac{10\!\cdots\!30}{72\!\cdots\!89}a^{7}+\frac{12\!\cdots\!44}{72\!\cdots\!89}a^{6}+\frac{15\!\cdots\!37}{72\!\cdots\!89}a^{5}+\frac{44\!\cdots\!98}{72\!\cdots\!89}a^{4}+\frac{27\!\cdots\!46}{72\!\cdots\!89}a^{3}-\frac{48\!\cdots\!93}{72\!\cdots\!89}a^{2}-\frac{62\!\cdots\!08}{72\!\cdots\!89}a+\frac{15\!\cdots\!63}{72\!\cdots\!89}$ Copy content Toggle raw display

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 $2$) Copy content Toggle raw display
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:   $\frac{82\!\cdots\!50}{20\!\cdots\!39}a^{15}-\frac{66\!\cdots\!91}{20\!\cdots\!39}a^{14}-\frac{39\!\cdots\!31}{20\!\cdots\!39}a^{13}+\frac{34\!\cdots\!13}{20\!\cdots\!39}a^{12}+\frac{66\!\cdots\!87}{20\!\cdots\!39}a^{11}-\frac{63\!\cdots\!62}{20\!\cdots\!39}a^{10}-\frac{47\!\cdots\!97}{20\!\cdots\!39}a^{9}+\frac{53\!\cdots\!81}{20\!\cdots\!39}a^{8}+\frac{12\!\cdots\!69}{20\!\cdots\!39}a^{7}-\frac{20\!\cdots\!78}{20\!\cdots\!39}a^{6}-\frac{16\!\cdots\!26}{20\!\cdots\!39}a^{5}+\frac{34\!\cdots\!44}{20\!\cdots\!39}a^{4}-\frac{28\!\cdots\!47}{20\!\cdots\!39}a^{3}-\frac{23\!\cdots\!06}{20\!\cdots\!39}a^{2}+\frac{23\!\cdots\!00}{20\!\cdots\!39}a+\frac{40\!\cdots\!74}{20\!\cdots\!39}$, $\frac{61\!\cdots\!87}{20\!\cdots\!39}a^{15}-\frac{40\!\cdots\!31}{20\!\cdots\!39}a^{14}-\frac{30\!\cdots\!03}{20\!\cdots\!39}a^{13}+\frac{32\!\cdots\!86}{20\!\cdots\!39}a^{12}+\frac{54\!\cdots\!98}{20\!\cdots\!39}a^{11}-\frac{71\!\cdots\!94}{20\!\cdots\!39}a^{10}-\frac{45\!\cdots\!51}{20\!\cdots\!39}a^{9}+\frac{63\!\cdots\!26}{20\!\cdots\!39}a^{8}+\frac{18\!\cdots\!58}{20\!\cdots\!39}a^{7}-\frac{24\!\cdots\!61}{20\!\cdots\!39}a^{6}-\frac{34\!\cdots\!01}{20\!\cdots\!39}a^{5}+\frac{31\!\cdots\!57}{20\!\cdots\!39}a^{4}+\frac{29\!\cdots\!92}{20\!\cdots\!39}a^{3}-\frac{42\!\cdots\!84}{20\!\cdots\!39}a^{2}-\frac{91\!\cdots\!21}{20\!\cdots\!39}a-\frac{63\!\cdots\!16}{20\!\cdots\!39}$, $\frac{40\!\cdots\!82}{20\!\cdots\!39}a^{15}-\frac{56\!\cdots\!16}{20\!\cdots\!39}a^{14}-\frac{20\!\cdots\!57}{20\!\cdots\!39}a^{13}+\frac{34\!\cdots\!15}{20\!\cdots\!39}a^{12}+\frac{37\!\cdots\!34}{20\!\cdots\!39}a^{11}-\frac{67\!\cdots\!94}{20\!\cdots\!39}a^{10}-\frac{33\!\cdots\!40}{20\!\cdots\!39}a^{9}+\frac{55\!\cdots\!65}{20\!\cdots\!39}a^{8}+\frac{14\!\cdots\!86}{20\!\cdots\!39}a^{7}-\frac{18\!\cdots\!51}{20\!\cdots\!39}a^{6}-\frac{32\!\cdots\!94}{20\!\cdots\!39}a^{5}+\frac{17\!\cdots\!48}{20\!\cdots\!39}a^{4}+\frac{34\!\cdots\!48}{20\!\cdots\!39}a^{3}+\frac{11\!\cdots\!96}{20\!\cdots\!39}a^{2}-\frac{13\!\cdots\!66}{20\!\cdots\!39}a-\frac{13\!\cdots\!52}{20\!\cdots\!39}$, $\frac{17\!\cdots\!11}{72\!\cdots\!89}a^{15}-\frac{28\!\cdots\!37}{72\!\cdots\!89}a^{14}-\frac{84\!\cdots\!54}{72\!\cdots\!89}a^{13}+\frac{17\!\cdots\!48}{72\!\cdots\!89}a^{12}+\frac{15\!\cdots\!28}{72\!\cdots\!89}a^{11}-\frac{33\!\cdots\!00}{72\!\cdots\!89}a^{10}-\frac{13\!\cdots\!09}{72\!\cdots\!89}a^{9}+\frac{28\!\cdots\!18}{72\!\cdots\!89}a^{8}+\frac{54\!\cdots\!00}{72\!\cdots\!89}a^{7}-\frac{10\!\cdots\!64}{72\!\cdots\!89}a^{6}-\frac{10\!\cdots\!47}{72\!\cdots\!89}a^{5}+\frac{13\!\cdots\!01}{72\!\cdots\!89}a^{4}+\frac{97\!\cdots\!99}{72\!\cdots\!89}a^{3}-\frac{24\!\cdots\!95}{72\!\cdots\!89}a^{2}-\frac{31\!\cdots\!33}{72\!\cdots\!89}a-\frac{21\!\cdots\!49}{72\!\cdots\!89}$, $\frac{33\!\cdots\!49}{72\!\cdots\!89}a^{15}-\frac{56\!\cdots\!74}{72\!\cdots\!89}a^{14}-\frac{16\!\cdots\!47}{72\!\cdots\!89}a^{13}+\frac{34\!\cdots\!83}{72\!\cdots\!89}a^{12}+\frac{30\!\cdots\!89}{72\!\cdots\!89}a^{11}-\frac{66\!\cdots\!34}{72\!\cdots\!89}a^{10}-\frac{25\!\cdots\!22}{72\!\cdots\!89}a^{9}+\frac{55\!\cdots\!60}{72\!\cdots\!89}a^{8}+\frac{10\!\cdots\!11}{72\!\cdots\!89}a^{7}-\frac{20\!\cdots\!01}{72\!\cdots\!89}a^{6}-\frac{21\!\cdots\!69}{72\!\cdots\!89}a^{5}+\frac{25\!\cdots\!55}{72\!\cdots\!89}a^{4}+\frac{19\!\cdots\!12}{72\!\cdots\!89}a^{3}-\frac{47\!\cdots\!58}{72\!\cdots\!89}a^{2}-\frac{61\!\cdots\!38}{72\!\cdots\!89}a-\frac{42\!\cdots\!14}{72\!\cdots\!89}$, $\frac{69\!\cdots\!40}{72\!\cdots\!89}a^{15}+\frac{29\!\cdots\!39}{72\!\cdots\!89}a^{14}-\frac{35\!\cdots\!72}{72\!\cdots\!89}a^{13}-\frac{13\!\cdots\!69}{72\!\cdots\!89}a^{12}+\frac{68\!\cdots\!87}{72\!\cdots\!89}a^{11}+\frac{22\!\cdots\!34}{72\!\cdots\!89}a^{10}-\frac{64\!\cdots\!37}{72\!\cdots\!89}a^{9}-\frac{18\!\cdots\!24}{72\!\cdots\!89}a^{8}+\frac{31\!\cdots\!19}{72\!\cdots\!89}a^{7}+\frac{71\!\cdots\!26}{72\!\cdots\!89}a^{6}-\frac{77\!\cdots\!02}{72\!\cdots\!89}a^{5}-\frac{13\!\cdots\!58}{72\!\cdots\!89}a^{4}+\frac{89\!\cdots\!14}{72\!\cdots\!89}a^{3}+\frac{11\!\cdots\!28}{72\!\cdots\!89}a^{2}-\frac{35\!\cdots\!07}{72\!\cdots\!89}a-\frac{37\!\cdots\!62}{72\!\cdots\!89}$, $\frac{75\!\cdots\!90}{72\!\cdots\!89}a^{15}-\frac{13\!\cdots\!65}{72\!\cdots\!89}a^{14}-\frac{37\!\cdots\!98}{72\!\cdots\!89}a^{13}+\frac{79\!\cdots\!03}{72\!\cdots\!89}a^{12}+\frac{68\!\cdots\!21}{72\!\cdots\!89}a^{11}-\frac{15\!\cdots\!15}{72\!\cdots\!89}a^{10}-\frac{59\!\cdots\!69}{72\!\cdots\!89}a^{9}+\frac{12\!\cdots\!29}{72\!\cdots\!89}a^{8}+\frac{24\!\cdots\!94}{72\!\cdots\!89}a^{7}-\frac{46\!\cdots\!27}{72\!\cdots\!89}a^{6}-\frac{49\!\cdots\!68}{72\!\cdots\!89}a^{5}+\frac{59\!\cdots\!16}{72\!\cdots\!89}a^{4}+\frac{45\!\cdots\!30}{72\!\cdots\!89}a^{3}-\frac{10\!\cdots\!61}{72\!\cdots\!89}a^{2}-\frac{14\!\cdots\!38}{72\!\cdots\!89}a-\frac{10\!\cdots\!37}{72\!\cdots\!89}$, $\frac{22\!\cdots\!55}{72\!\cdots\!89}a^{15}-\frac{38\!\cdots\!21}{72\!\cdots\!89}a^{14}-\frac{11\!\cdots\!47}{72\!\cdots\!89}a^{13}+\frac{23\!\cdots\!95}{72\!\cdots\!89}a^{12}+\frac{20\!\cdots\!97}{72\!\cdots\!89}a^{11}-\frac{45\!\cdots\!15}{72\!\cdots\!89}a^{10}-\frac{17\!\cdots\!68}{72\!\cdots\!89}a^{9}+\frac{37\!\cdots\!07}{72\!\cdots\!89}a^{8}+\frac{73\!\cdots\!76}{72\!\cdots\!89}a^{7}-\frac{13\!\cdots\!15}{72\!\cdots\!89}a^{6}-\frac{14\!\cdots\!21}{72\!\cdots\!89}a^{5}+\frac{17\!\cdots\!29}{72\!\cdots\!89}a^{4}+\frac{12\!\cdots\!18}{72\!\cdots\!89}a^{3}-\frac{33\!\cdots\!35}{72\!\cdots\!89}a^{2}-\frac{41\!\cdots\!23}{72\!\cdots\!89}a-\frac{28\!\cdots\!68}{72\!\cdots\!89}$, $\frac{61\!\cdots\!44}{72\!\cdots\!89}a^{15}-\frac{10\!\cdots\!21}{72\!\cdots\!89}a^{14}-\frac{30\!\cdots\!54}{72\!\cdots\!89}a^{13}+\frac{63\!\cdots\!75}{72\!\cdots\!89}a^{12}+\frac{56\!\cdots\!06}{72\!\cdots\!89}a^{11}-\frac{12\!\cdots\!30}{72\!\cdots\!89}a^{10}-\frac{48\!\cdots\!94}{72\!\cdots\!89}a^{9}+\frac{10\!\cdots\!91}{72\!\cdots\!89}a^{8}+\frac{19\!\cdots\!33}{72\!\cdots\!89}a^{7}-\frac{37\!\cdots\!04}{72\!\cdots\!89}a^{6}-\frac{39\!\cdots\!87}{72\!\cdots\!89}a^{5}+\frac{48\!\cdots\!97}{72\!\cdots\!89}a^{4}+\frac{35\!\cdots\!74}{72\!\cdots\!89}a^{3}-\frac{91\!\cdots\!60}{72\!\cdots\!89}a^{2}-\frac{11\!\cdots\!62}{72\!\cdots\!89}a-\frac{78\!\cdots\!83}{72\!\cdots\!89}$, $\frac{18\!\cdots\!90}{15\!\cdots\!87}a^{15}-\frac{32\!\cdots\!29}{15\!\cdots\!87}a^{14}-\frac{92\!\cdots\!67}{15\!\cdots\!87}a^{13}+\frac{19\!\cdots\!92}{15\!\cdots\!87}a^{12}+\frac{17\!\cdots\!76}{15\!\cdots\!87}a^{11}-\frac{38\!\cdots\!20}{15\!\cdots\!87}a^{10}-\frac{14\!\cdots\!14}{15\!\cdots\!87}a^{9}+\frac{31\!\cdots\!86}{15\!\cdots\!87}a^{8}+\frac{60\!\cdots\!30}{15\!\cdots\!87}a^{7}-\frac{11\!\cdots\!60}{15\!\cdots\!87}a^{6}-\frac{11\!\cdots\!73}{15\!\cdots\!87}a^{5}+\frac{14\!\cdots\!19}{15\!\cdots\!87}a^{4}+\frac{10\!\cdots\!91}{15\!\cdots\!87}a^{3}-\frac{28\!\cdots\!34}{15\!\cdots\!87}a^{2}-\frac{34\!\cdots\!03}{15\!\cdots\!87}a-\frac{23\!\cdots\!06}{15\!\cdots\!87}$, $\frac{38\!\cdots\!29}{72\!\cdots\!89}a^{15}-\frac{66\!\cdots\!98}{72\!\cdots\!89}a^{14}-\frac{19\!\cdots\!13}{72\!\cdots\!89}a^{13}+\frac{39\!\cdots\!20}{72\!\cdots\!89}a^{12}+\frac{35\!\cdots\!75}{72\!\cdots\!89}a^{11}-\frac{77\!\cdots\!61}{72\!\cdots\!89}a^{10}-\frac{30\!\cdots\!56}{72\!\cdots\!89}a^{9}+\frac{65\!\cdots\!62}{72\!\cdots\!89}a^{8}+\frac{12\!\cdots\!00}{72\!\cdots\!89}a^{7}-\frac{23\!\cdots\!55}{72\!\cdots\!89}a^{6}-\frac{24\!\cdots\!57}{72\!\cdots\!89}a^{5}+\frac{30\!\cdots\!45}{72\!\cdots\!89}a^{4}+\frac{22\!\cdots\!96}{72\!\cdots\!89}a^{3}-\frac{57\!\cdots\!35}{72\!\cdots\!89}a^{2}-\frac{72\!\cdots\!41}{72\!\cdots\!89}a-\frac{49\!\cdots\!53}{72\!\cdots\!89}$, $\frac{30\!\cdots\!12}{72\!\cdots\!89}a^{15}-\frac{26\!\cdots\!26}{72\!\cdots\!89}a^{14}-\frac{14\!\cdots\!51}{72\!\cdots\!89}a^{13}+\frac{13\!\cdots\!21}{72\!\cdots\!89}a^{12}+\frac{23\!\cdots\!37}{72\!\cdots\!89}a^{11}-\frac{24\!\cdots\!60}{72\!\cdots\!89}a^{10}-\frac{16\!\cdots\!38}{72\!\cdots\!89}a^{9}+\frac{20\!\cdots\!06}{72\!\cdots\!89}a^{8}+\frac{37\!\cdots\!53}{72\!\cdots\!89}a^{7}-\frac{77\!\cdots\!98}{72\!\cdots\!89}a^{6}+\frac{29\!\cdots\!72}{72\!\cdots\!89}a^{5}+\frac{12\!\cdots\!28}{72\!\cdots\!89}a^{4}-\frac{15\!\cdots\!53}{72\!\cdots\!89}a^{3}-\frac{76\!\cdots\!25}{72\!\cdots\!89}a^{2}+\frac{86\!\cdots\!61}{72\!\cdots\!89}a+\frac{13\!\cdots\!92}{72\!\cdots\!89}$, $\frac{12\!\cdots\!12}{72\!\cdots\!89}a^{15}-\frac{91\!\cdots\!86}{72\!\cdots\!89}a^{14}-\frac{62\!\cdots\!85}{72\!\cdots\!89}a^{13}+\frac{71\!\cdots\!91}{72\!\cdots\!89}a^{12}+\frac{11\!\cdots\!90}{72\!\cdots\!89}a^{11}-\frac{15\!\cdots\!96}{72\!\cdots\!89}a^{10}-\frac{94\!\cdots\!26}{72\!\cdots\!89}a^{9}+\frac{14\!\cdots\!69}{72\!\cdots\!89}a^{8}+\frac{37\!\cdots\!33}{72\!\cdots\!89}a^{7}-\frac{53\!\cdots\!59}{72\!\cdots\!89}a^{6}-\frac{71\!\cdots\!63}{72\!\cdots\!89}a^{5}+\frac{69\!\cdots\!83}{72\!\cdots\!89}a^{4}+\frac{61\!\cdots\!88}{72\!\cdots\!89}a^{3}-\frac{10\!\cdots\!74}{72\!\cdots\!89}a^{2}-\frac{18\!\cdots\!63}{72\!\cdots\!89}a-\frac{13\!\cdots\!36}{72\!\cdots\!89}$, $\frac{42\!\cdots\!86}{72\!\cdots\!89}a^{15}-\frac{73\!\cdots\!76}{72\!\cdots\!89}a^{14}-\frac{20\!\cdots\!69}{72\!\cdots\!89}a^{13}+\frac{43\!\cdots\!82}{72\!\cdots\!89}a^{12}+\frac{38\!\cdots\!30}{72\!\cdots\!89}a^{11}-\frac{85\!\cdots\!65}{72\!\cdots\!89}a^{10}-\frac{32\!\cdots\!26}{72\!\cdots\!89}a^{9}+\frac{71\!\cdots\!79}{72\!\cdots\!89}a^{8}+\frac{13\!\cdots\!57}{72\!\cdots\!89}a^{7}-\frac{25\!\cdots\!67}{72\!\cdots\!89}a^{6}-\frac{26\!\cdots\!50}{72\!\cdots\!89}a^{5}+\frac{33\!\cdots\!38}{72\!\cdots\!89}a^{4}+\frac{24\!\cdots\!08}{72\!\cdots\!89}a^{3}-\frac{65\!\cdots\!42}{72\!\cdots\!89}a^{2}-\frac{78\!\cdots\!65}{72\!\cdots\!89}a-\frac{53\!\cdots\!70}{72\!\cdots\!89}$, $\frac{79\!\cdots\!58}{72\!\cdots\!89}a^{15}-\frac{77\!\cdots\!57}{72\!\cdots\!89}a^{14}-\frac{39\!\cdots\!50}{72\!\cdots\!89}a^{13}+\frac{53\!\cdots\!65}{72\!\cdots\!89}a^{12}+\frac{73\!\cdots\!30}{72\!\cdots\!89}a^{11}-\frac{11\!\cdots\!73}{72\!\cdots\!89}a^{10}-\frac{64\!\cdots\!44}{72\!\cdots\!89}a^{9}+\frac{98\!\cdots\!76}{72\!\cdots\!89}a^{8}+\frac{27\!\cdots\!46}{72\!\cdots\!89}a^{7}-\frac{37\!\cdots\!22}{72\!\cdots\!89}a^{6}-\frac{55\!\cdots\!69}{72\!\cdots\!89}a^{5}+\frac{52\!\cdots\!32}{72\!\cdots\!89}a^{4}+\frac{49\!\cdots\!18}{72\!\cdots\!89}a^{3}-\frac{66\!\cdots\!05}{72\!\cdots\!89}a^{2}-\frac{15\!\cdots\!89}{72\!\cdots\!89}a-\frac{11\!\cdots\!15}{72\!\cdots\!89}$ Copy content Toggle raw display (assuming GRH)
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:  \( 55517458750200000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 55517458750200000 \cdot 2}{2\cdot\sqrt{685578251337344185848967768724165145970351361}}\cr\approx \mathstrut & 0.138957180399297 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699)
 
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 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699, 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 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699);
 
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 - 7*x^15 - 486*x^14 + 3640*x^13 + 85482*x^12 - 680369*x^11 - 6731288*x^10 + 57906718*x^9 + 234236862*x^8 - 2308745163*x^7 - 3167386332*x^6 + 41364170537*x^5 + 15994510202*x^4 - 316660788899*x^3 - 107062029196*x^2 + 853361887750*x + 670030778699);
 
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

\(\Q(\sqrt{89}) \), 4.4.119139761.1, 8.8.213496205355753436961.1

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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ $16$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ R ${\href{/padicField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ $16$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(13\) Copy content Toggle raw display 13.16.14.5$x^{16} - 156 x^{8} + 338$$8$$2$$14$$C_{16} : C_2$$[\ ]_{8}^{4}$
\(89\) Copy content Toggle raw display 89.16.15.4$x^{16} + 801$$16$$1$$15$$C_{16} : C_2$$[\ ]_{16}^{2}$