Properties

Label 16.16.104...369.2
Degree $16$
Signature $[16, 0]$
Discriminant $1.048\times 10^{31}$
Root discriminant \(86.85\)
Ramified primes $17,53$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259)
 
gp: K = bnfinit(y^16 - 6*y^15 - 54*y^14 + 316*y^13 + 1129*y^12 - 5904*y^11 - 12476*y^10 + 49215*y^9 + 76870*y^8 - 187170*y^7 - 241469*y^6 + 304908*y^5 + 315322*y^4 - 220035*y^3 - 165824*y^2 + 54341*y + 30259, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259)
 

\( x^{16} - 6 x^{15} - 54 x^{14} + 316 x^{13} + 1129 x^{12} - 5904 x^{11} - 12476 x^{10} + 49215 x^{9} + 76870 x^{8} - 187170 x^{7} - 241469 x^{6} + 304908 x^{5} + \cdots + 30259 \) 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:   \(10483151353726139536553735554369\) \(\medspace = 17^{14}\cdot 53^{8}\) 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:  \(86.85\)
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:  $17^{7/8}53^{1/2}\approx 86.8521834484431$
Ramified primes:   \(17\), \(53\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{26}a^{14}+\frac{2}{13}a^{13}-\frac{3}{13}a^{12}-\frac{11}{26}a^{11}-\frac{2}{13}a^{10}+\frac{11}{26}a^{8}-\frac{5}{13}a^{7}+\frac{1}{13}a^{6}+\frac{9}{26}a^{5}+\frac{4}{13}a^{4}+\frac{1}{13}a^{3}-\frac{1}{2}a^{2}+\frac{3}{13}a+\frac{6}{13}$, $\frac{1}{59\!\cdots\!18}a^{15}+\frac{24\!\cdots\!35}{29\!\cdots\!09}a^{14}-\frac{32\!\cdots\!91}{59\!\cdots\!18}a^{13}+\frac{13\!\cdots\!77}{59\!\cdots\!18}a^{12}+\frac{76\!\cdots\!21}{29\!\cdots\!09}a^{11}-\frac{22\!\cdots\!49}{59\!\cdots\!18}a^{10}-\frac{28\!\cdots\!09}{59\!\cdots\!18}a^{9}-\frac{11\!\cdots\!31}{29\!\cdots\!09}a^{8}+\frac{12\!\cdots\!05}{59\!\cdots\!18}a^{7}-\frac{37\!\cdots\!49}{59\!\cdots\!18}a^{6}+\frac{12\!\cdots\!90}{29\!\cdots\!09}a^{5}-\frac{23\!\cdots\!31}{59\!\cdots\!18}a^{4}-\frac{12\!\cdots\!61}{59\!\cdots\!18}a^{3}-\frac{51\!\cdots\!35}{29\!\cdots\!09}a^{2}-\frac{23\!\cdots\!31}{59\!\cdots\!18}a+\frac{76\!\cdots\!69}{29\!\cdots\!09}$ Copy content Toggle raw display

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 $2$) 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:   $\frac{11\!\cdots\!10}{42\!\cdots\!09}a^{15}-\frac{55\!\cdots\!76}{42\!\cdots\!09}a^{14}-\frac{64\!\cdots\!94}{42\!\cdots\!09}a^{13}+\frac{28\!\cdots\!99}{42\!\cdots\!09}a^{12}+\frac{15\!\cdots\!66}{42\!\cdots\!09}a^{11}-\frac{39\!\cdots\!06}{32\!\cdots\!93}a^{10}-\frac{18\!\cdots\!77}{42\!\cdots\!09}a^{9}+\frac{36\!\cdots\!34}{42\!\cdots\!09}a^{8}+\frac{11\!\cdots\!12}{42\!\cdots\!09}a^{7}-\frac{94\!\cdots\!97}{42\!\cdots\!09}a^{6}-\frac{34\!\cdots\!40}{42\!\cdots\!09}a^{5}+\frac{13\!\cdots\!89}{42\!\cdots\!09}a^{4}+\frac{23\!\cdots\!43}{32\!\cdots\!93}a^{3}+\frac{47\!\cdots\!76}{42\!\cdots\!09}a^{2}-\frac{66\!\cdots\!03}{42\!\cdots\!09}a-\frac{15\!\cdots\!27}{32\!\cdots\!93}$, $\frac{10\!\cdots\!92}{42\!\cdots\!09}a^{15}-\frac{53\!\cdots\!85}{42\!\cdots\!09}a^{14}-\frac{64\!\cdots\!97}{42\!\cdots\!09}a^{13}+\frac{28\!\cdots\!80}{42\!\cdots\!09}a^{12}+\frac{15\!\cdots\!13}{42\!\cdots\!09}a^{11}-\frac{38\!\cdots\!30}{32\!\cdots\!93}a^{10}-\frac{19\!\cdots\!31}{42\!\cdots\!09}a^{9}+\frac{37\!\cdots\!26}{42\!\cdots\!09}a^{8}+\frac{12\!\cdots\!03}{42\!\cdots\!09}a^{7}-\frac{10\!\cdots\!31}{42\!\cdots\!09}a^{6}-\frac{37\!\cdots\!04}{42\!\cdots\!09}a^{5}+\frac{42\!\cdots\!41}{42\!\cdots\!09}a^{4}+\frac{30\!\cdots\!90}{32\!\cdots\!93}a^{3}+\frac{61\!\cdots\!92}{42\!\cdots\!09}a^{2}-\frac{12\!\cdots\!65}{42\!\cdots\!09}a-\frac{30\!\cdots\!18}{32\!\cdots\!93}$, $\frac{26\!\cdots\!18}{42\!\cdots\!09}a^{15}-\frac{19\!\cdots\!91}{42\!\cdots\!09}a^{14}-\frac{60\!\cdots\!97}{42\!\cdots\!09}a^{13}+\frac{66\!\cdots\!19}{42\!\cdots\!09}a^{12}-\frac{12\!\cdots\!47}{42\!\cdots\!09}a^{11}-\frac{12\!\cdots\!76}{32\!\cdots\!93}a^{10}+\frac{47\!\cdots\!54}{42\!\cdots\!09}a^{9}-\frac{12\!\cdots\!92}{42\!\cdots\!09}a^{8}-\frac{58\!\cdots\!91}{42\!\cdots\!09}a^{7}+\frac{12\!\cdots\!34}{42\!\cdots\!09}a^{6}+\frac{35\!\cdots\!64}{42\!\cdots\!09}a^{5}-\frac{29\!\cdots\!52}{42\!\cdots\!09}a^{4}-\frac{74\!\cdots\!47}{32\!\cdots\!93}a^{3}-\frac{14\!\cdots\!16}{42\!\cdots\!09}a^{2}+\frac{63\!\cdots\!62}{42\!\cdots\!09}a+\frac{21\!\cdots\!77}{32\!\cdots\!93}$, $\frac{76\!\cdots\!87}{45\!\cdots\!86}a^{15}-\frac{24\!\cdots\!45}{22\!\cdots\!93}a^{14}-\frac{39\!\cdots\!61}{45\!\cdots\!86}a^{13}+\frac{12\!\cdots\!57}{22\!\cdots\!93}a^{12}+\frac{37\!\cdots\!96}{22\!\cdots\!93}a^{11}-\frac{45\!\cdots\!97}{45\!\cdots\!86}a^{10}-\frac{37\!\cdots\!49}{22\!\cdots\!93}a^{9}+\frac{18\!\cdots\!92}{22\!\cdots\!93}a^{8}+\frac{40\!\cdots\!93}{45\!\cdots\!86}a^{7}-\frac{67\!\cdots\!59}{22\!\cdots\!93}a^{6}-\frac{52\!\cdots\!28}{22\!\cdots\!93}a^{5}+\frac{19\!\cdots\!21}{45\!\cdots\!86}a^{4}+\frac{37\!\cdots\!55}{22\!\cdots\!93}a^{3}-\frac{58\!\cdots\!64}{22\!\cdots\!93}a^{2}+\frac{43\!\cdots\!91}{45\!\cdots\!86}a+\frac{11\!\cdots\!23}{45\!\cdots\!86}$, $\frac{65\!\cdots\!29}{59\!\cdots\!18}a^{15}-\frac{36\!\cdots\!77}{59\!\cdots\!18}a^{14}-\frac{29\!\cdots\!29}{45\!\cdots\!86}a^{13}+\frac{97\!\cdots\!78}{29\!\cdots\!09}a^{12}+\frac{86\!\cdots\!05}{59\!\cdots\!18}a^{11}-\frac{37\!\cdots\!71}{59\!\cdots\!18}a^{10}-\frac{52\!\cdots\!29}{29\!\cdots\!09}a^{9}+\frac{24\!\cdots\!11}{45\!\cdots\!86}a^{8}+\frac{52\!\cdots\!35}{45\!\cdots\!86}a^{7}-\frac{58\!\cdots\!98}{29\!\cdots\!09}a^{6}-\frac{22\!\cdots\!81}{59\!\cdots\!18}a^{5}+\frac{16\!\cdots\!49}{59\!\cdots\!18}a^{4}+\frac{15\!\cdots\!34}{29\!\cdots\!09}a^{3}-\frac{63\!\cdots\!51}{59\!\cdots\!18}a^{2}-\frac{13\!\cdots\!63}{59\!\cdots\!18}a-\frac{19\!\cdots\!25}{59\!\cdots\!18}$, $\frac{43\!\cdots\!10}{29\!\cdots\!09}a^{15}-\frac{47\!\cdots\!27}{59\!\cdots\!18}a^{14}-\frac{19\!\cdots\!70}{22\!\cdots\!93}a^{13}+\frac{12\!\cdots\!68}{29\!\cdots\!09}a^{12}+\frac{11\!\cdots\!09}{59\!\cdots\!18}a^{11}-\frac{21\!\cdots\!18}{29\!\cdots\!09}a^{10}-\frac{65\!\cdots\!89}{29\!\cdots\!09}a^{9}+\frac{25\!\cdots\!91}{45\!\cdots\!86}a^{8}+\frac{31\!\cdots\!85}{22\!\cdots\!93}a^{7}-\frac{48\!\cdots\!80}{29\!\cdots\!09}a^{6}-\frac{24\!\cdots\!75}{59\!\cdots\!18}a^{5}+\frac{33\!\cdots\!62}{29\!\cdots\!09}a^{4}+\frac{11\!\cdots\!66}{29\!\cdots\!09}a^{3}+\frac{11\!\cdots\!05}{59\!\cdots\!18}a^{2}-\frac{31\!\cdots\!73}{29\!\cdots\!09}a-\frac{75\!\cdots\!43}{29\!\cdots\!09}$, $\frac{72\!\cdots\!01}{59\!\cdots\!18}a^{15}-\frac{47\!\cdots\!89}{59\!\cdots\!18}a^{14}-\frac{35\!\cdots\!39}{59\!\cdots\!18}a^{13}+\frac{12\!\cdots\!98}{29\!\cdots\!09}a^{12}+\frac{61\!\cdots\!89}{59\!\cdots\!18}a^{11}-\frac{42\!\cdots\!83}{59\!\cdots\!18}a^{10}-\frac{27\!\cdots\!97}{29\!\cdots\!09}a^{9}+\frac{32\!\cdots\!65}{59\!\cdots\!18}a^{8}+\frac{25\!\cdots\!41}{59\!\cdots\!18}a^{7}-\frac{52\!\cdots\!46}{29\!\cdots\!09}a^{6}-\frac{54\!\cdots\!43}{59\!\cdots\!18}a^{5}+\frac{11\!\cdots\!45}{59\!\cdots\!18}a^{4}+\frac{43\!\cdots\!02}{29\!\cdots\!09}a^{3}-\frac{51\!\cdots\!81}{59\!\cdots\!18}a^{2}+\frac{35\!\cdots\!97}{59\!\cdots\!18}a-\frac{28\!\cdots\!35}{59\!\cdots\!18}$, $\frac{29\!\cdots\!37}{59\!\cdots\!18}a^{15}-\frac{67\!\cdots\!51}{29\!\cdots\!09}a^{14}-\frac{17\!\cdots\!95}{59\!\cdots\!18}a^{13}+\frac{34\!\cdots\!99}{29\!\cdots\!09}a^{12}+\frac{22\!\cdots\!50}{29\!\cdots\!09}a^{11}-\frac{94\!\cdots\!31}{45\!\cdots\!86}a^{10}-\frac{28\!\cdots\!62}{29\!\cdots\!09}a^{9}+\frac{43\!\cdots\!20}{29\!\cdots\!09}a^{8}+\frac{37\!\cdots\!43}{59\!\cdots\!18}a^{7}-\frac{99\!\cdots\!04}{29\!\cdots\!09}a^{6}-\frac{58\!\cdots\!92}{29\!\cdots\!09}a^{5}-\frac{12\!\cdots\!65}{59\!\cdots\!18}a^{4}+\frac{47\!\cdots\!65}{22\!\cdots\!93}a^{3}+\frac{18\!\cdots\!11}{29\!\cdots\!09}a^{2}-\frac{39\!\cdots\!51}{59\!\cdots\!18}a-\frac{11\!\cdots\!09}{45\!\cdots\!86}$, $\frac{36\!\cdots\!93}{59\!\cdots\!18}a^{15}-\frac{87\!\cdots\!76}{29\!\cdots\!09}a^{14}-\frac{21\!\cdots\!51}{59\!\cdots\!18}a^{13}+\frac{45\!\cdots\!19}{29\!\cdots\!09}a^{12}+\frac{26\!\cdots\!83}{29\!\cdots\!09}a^{11}-\frac{12\!\cdots\!07}{45\!\cdots\!86}a^{10}-\frac{32\!\cdots\!24}{29\!\cdots\!09}a^{9}+\frac{56\!\cdots\!93}{29\!\cdots\!09}a^{8}+\frac{42\!\cdots\!95}{59\!\cdots\!18}a^{7}-\frac{13\!\cdots\!74}{29\!\cdots\!09}a^{6}-\frac{64\!\cdots\!29}{29\!\cdots\!09}a^{5}-\frac{12\!\cdots\!31}{59\!\cdots\!18}a^{4}+\frac{49\!\cdots\!51}{22\!\cdots\!93}a^{3}+\frac{21\!\cdots\!93}{29\!\cdots\!09}a^{2}-\frac{37\!\cdots\!95}{59\!\cdots\!18}a-\frac{12\!\cdots\!15}{45\!\cdots\!86}$, $\frac{41\!\cdots\!87}{59\!\cdots\!18}a^{15}-\frac{11\!\cdots\!74}{29\!\cdots\!09}a^{14}-\frac{23\!\cdots\!87}{59\!\cdots\!18}a^{13}+\frac{59\!\cdots\!49}{29\!\cdots\!09}a^{12}+\frac{25\!\cdots\!15}{29\!\cdots\!09}a^{11}-\frac{16\!\cdots\!05}{45\!\cdots\!86}a^{10}-\frac{29\!\cdots\!39}{29\!\cdots\!09}a^{9}+\frac{80\!\cdots\!51}{29\!\cdots\!09}a^{8}+\frac{36\!\cdots\!35}{59\!\cdots\!18}a^{7}-\frac{23\!\cdots\!29}{29\!\cdots\!09}a^{6}-\frac{51\!\cdots\!96}{29\!\cdots\!09}a^{5}+\frac{36\!\cdots\!67}{59\!\cdots\!18}a^{4}+\frac{34\!\cdots\!61}{22\!\cdots\!93}a^{3}-\frac{36\!\cdots\!02}{29\!\cdots\!09}a^{2}-\frac{18\!\cdots\!51}{59\!\cdots\!18}a-\frac{37\!\cdots\!17}{45\!\cdots\!86}$, $\frac{33\!\cdots\!03}{59\!\cdots\!18}a^{15}-\frac{20\!\cdots\!39}{59\!\cdots\!18}a^{14}-\frac{18\!\cdots\!27}{59\!\cdots\!18}a^{13}+\frac{54\!\cdots\!24}{29\!\cdots\!09}a^{12}+\frac{36\!\cdots\!81}{59\!\cdots\!18}a^{11}-\frac{15\!\cdots\!21}{45\!\cdots\!86}a^{10}-\frac{19\!\cdots\!28}{29\!\cdots\!09}a^{9}+\frac{16\!\cdots\!07}{59\!\cdots\!18}a^{8}+\frac{22\!\cdots\!47}{59\!\cdots\!18}a^{7}-\frac{31\!\cdots\!92}{29\!\cdots\!09}a^{6}-\frac{65\!\cdots\!17}{59\!\cdots\!18}a^{5}+\frac{98\!\cdots\!13}{59\!\cdots\!18}a^{4}+\frac{25\!\cdots\!66}{22\!\cdots\!93}a^{3}-\frac{70\!\cdots\!47}{59\!\cdots\!18}a^{2}-\frac{18\!\cdots\!21}{59\!\cdots\!18}a+\frac{12\!\cdots\!31}{45\!\cdots\!86}$, $\frac{55\!\cdots\!45}{29\!\cdots\!09}a^{15}-\frac{37\!\cdots\!72}{29\!\cdots\!09}a^{14}-\frac{54\!\cdots\!77}{59\!\cdots\!18}a^{13}+\frac{19\!\cdots\!87}{29\!\cdots\!09}a^{12}+\frac{47\!\cdots\!31}{29\!\cdots\!09}a^{11}-\frac{73\!\cdots\!87}{59\!\cdots\!18}a^{10}-\frac{41\!\cdots\!10}{29\!\cdots\!09}a^{9}+\frac{31\!\cdots\!27}{29\!\cdots\!09}a^{8}+\frac{38\!\cdots\!23}{59\!\cdots\!18}a^{7}-\frac{12\!\cdots\!77}{29\!\cdots\!09}a^{6}-\frac{43\!\cdots\!29}{29\!\cdots\!09}a^{5}+\frac{33\!\cdots\!45}{45\!\cdots\!86}a^{4}+\frac{20\!\cdots\!60}{29\!\cdots\!09}a^{3}-\frac{15\!\cdots\!39}{29\!\cdots\!09}a^{2}+\frac{29\!\cdots\!73}{59\!\cdots\!18}a+\frac{27\!\cdots\!67}{29\!\cdots\!09}$, $\frac{31\!\cdots\!49}{45\!\cdots\!86}a^{15}-\frac{10\!\cdots\!74}{29\!\cdots\!09}a^{14}-\frac{23\!\cdots\!97}{59\!\cdots\!18}a^{13}+\frac{54\!\cdots\!01}{29\!\cdots\!09}a^{12}+\frac{26\!\cdots\!48}{29\!\cdots\!09}a^{11}-\frac{19\!\cdots\!89}{59\!\cdots\!18}a^{10}-\frac{24\!\cdots\!29}{22\!\cdots\!93}a^{9}+\frac{71\!\cdots\!04}{29\!\cdots\!09}a^{8}+\frac{40\!\cdots\!43}{59\!\cdots\!18}a^{7}-\frac{19\!\cdots\!64}{29\!\cdots\!09}a^{6}-\frac{59\!\cdots\!69}{29\!\cdots\!09}a^{5}+\frac{16\!\cdots\!61}{59\!\cdots\!18}a^{4}+\frac{57\!\cdots\!69}{29\!\cdots\!09}a^{3}+\frac{60\!\cdots\!75}{22\!\cdots\!93}a^{2}-\frac{31\!\cdots\!23}{59\!\cdots\!18}a-\frac{93\!\cdots\!11}{59\!\cdots\!18}$, $\frac{59\!\cdots\!53}{59\!\cdots\!18}a^{15}-\frac{24\!\cdots\!83}{29\!\cdots\!09}a^{14}-\frac{22\!\cdots\!77}{59\!\cdots\!18}a^{13}+\frac{24\!\cdots\!19}{59\!\cdots\!18}a^{12}+\frac{10\!\cdots\!25}{29\!\cdots\!09}a^{11}-\frac{41\!\cdots\!61}{59\!\cdots\!18}a^{10}+\frac{77\!\cdots\!23}{59\!\cdots\!18}a^{9}+\frac{15\!\cdots\!25}{29\!\cdots\!09}a^{8}-\frac{16\!\cdots\!41}{59\!\cdots\!18}a^{7}-\frac{10\!\cdots\!11}{59\!\cdots\!18}a^{6}+\frac{32\!\cdots\!07}{29\!\cdots\!09}a^{5}+\frac{14\!\cdots\!95}{59\!\cdots\!18}a^{4}-\frac{82\!\cdots\!55}{59\!\cdots\!18}a^{3}-\frac{37\!\cdots\!74}{29\!\cdots\!09}a^{2}+\frac{27\!\cdots\!63}{59\!\cdots\!18}a+\frac{76\!\cdots\!14}{29\!\cdots\!09}$, $\frac{10\!\cdots\!97}{59\!\cdots\!18}a^{15}-\frac{31\!\cdots\!78}{29\!\cdots\!09}a^{14}-\frac{57\!\cdots\!21}{59\!\cdots\!18}a^{13}+\frac{16\!\cdots\!86}{29\!\cdots\!09}a^{12}+\frac{45\!\cdots\!86}{22\!\cdots\!93}a^{11}-\frac{63\!\cdots\!55}{59\!\cdots\!18}a^{10}-\frac{63\!\cdots\!62}{29\!\cdots\!09}a^{9}+\frac{26\!\cdots\!46}{29\!\cdots\!09}a^{8}+\frac{73\!\cdots\!75}{59\!\cdots\!18}a^{7}-\frac{76\!\cdots\!70}{22\!\cdots\!93}a^{6}-\frac{75\!\cdots\!25}{22\!\cdots\!93}a^{5}+\frac{30\!\cdots\!01}{59\!\cdots\!18}a^{4}+\frac{88\!\cdots\!33}{29\!\cdots\!09}a^{3}-\frac{93\!\cdots\!05}{29\!\cdots\!09}a^{2}-\frac{38\!\cdots\!39}{59\!\cdots\!18}a+\frac{29\!\cdots\!35}{59\!\cdots\!18}$ 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:  \( 7359535457.67 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 7359535457.67 \cdot 2}{2\cdot\sqrt{10483151353726139536553735554369}}\cr\approx \mathstrut & 0.148965056137 \end{aligned}\] (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 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259)
 
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 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259, 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 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259);
 
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 - 54*x^14 + 316*x^13 + 1129*x^12 - 5904*x^11 - 12476*x^10 + 49215*x^9 + 76870*x^8 - 187170*x^7 - 241469*x^6 + 304908*x^5 + 315322*x^4 - 220035*x^3 - 165824*x^2 + 54341*x + 30259);
 
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

$C_2^2:C_8$ (as 16T24):

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 $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.15317.1, 4.4.260389.1, 8.8.3237769502871713.1, 8.8.1152641332457.1, 8.8.67802431321.1

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
Degree 16 sibling: 16.16.1328582041288248401656849.1
Minimal sibling: 16.16.1328582041288248401656849.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ R ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(17\) Copy content Toggle raw display 17.16.14.1$x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$
\(53\) Copy content Toggle raw display 53.4.2.2$x^{4} - 2597 x^{2} + 5618$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.2.2$x^{4} - 2597 x^{2} + 5618$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.8.4.1$x^{8} + 9752 x^{7} + 35663294 x^{6} + 57966048984 x^{5} + 35333436724405 x^{4} + 3595233169984 x^{3} + 320278224174124 x^{2} + 1356456509257952 x + 99990743929156$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$