Properties

Label 16.8.575...129.1
Degree $16$
Signature $[8, 4]$
Discriminant $5.757\times 10^{23}$
Root discriminant \(30.55\)
Ramified primes $17,43$
Class number $1$ (GRH)
Class group trivial (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 - x^15 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16)
 
gp: K = bnfinit(y^16 - y^15 + 7*y^14 + 15*y^13 - 62*y^12 + 43*y^11 - 310*y^10 - 629*y^9 - 268*y^8 - 1921*y^7 + 1573*y^6 - 776*y^5 + 7*y^4 + 1618*y^3 - 556*y^2 - 184*y + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16)
 

\( x^{16} - x^{15} + 7 x^{14} + 15 x^{13} - 62 x^{12} + 43 x^{11} - 310 x^{10} - 629 x^{9} - 268 x^{8} - 1921 x^{7} + 1573 x^{6} - 776 x^{5} + 7 x^{4} + 1618 x^{3} - 556 x^{2} - 184 x + 16 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(575650281819106455466129\) \(\medspace = 17^{14}\cdot 43^{4}\) 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:  \(30.55\)
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:  $17^{7/8}43^{1/2}\approx 78.23066716385114$
Ramified primes:   \(17\), \(43\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{11\!\cdots\!04}a^{15}+\frac{10\!\cdots\!29}{11\!\cdots\!04}a^{14}-\frac{14\!\cdots\!27}{11\!\cdots\!04}a^{13}-\frac{68\!\cdots\!31}{11\!\cdots\!04}a^{12}-\frac{75\!\cdots\!35}{29\!\cdots\!26}a^{11}+\frac{69\!\cdots\!19}{11\!\cdots\!04}a^{10}+\frac{45\!\cdots\!71}{29\!\cdots\!26}a^{9}+\frac{13\!\cdots\!87}{11\!\cdots\!04}a^{8}+\frac{14\!\cdots\!81}{58\!\cdots\!52}a^{7}-\frac{87\!\cdots\!09}{11\!\cdots\!04}a^{6}-\frac{51\!\cdots\!61}{11\!\cdots\!04}a^{5}-\frac{24\!\cdots\!41}{58\!\cdots\!52}a^{4}+\frac{35\!\cdots\!35}{11\!\cdots\!04}a^{3}-\frac{38\!\cdots\!90}{14\!\cdots\!63}a^{2}+\frac{18\!\cdots\!07}{14\!\cdots\!63}a-\frac{45\!\cdots\!60}{14\!\cdots\!63}$ 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

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 $2$) 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:   $\frac{15\!\cdots\!05}{58\!\cdots\!52}a^{15}-\frac{17\!\cdots\!15}{29\!\cdots\!26}a^{14}+\frac{24\!\cdots\!13}{14\!\cdots\!63}a^{13}+\frac{80\!\cdots\!71}{14\!\cdots\!63}a^{12}-\frac{79\!\cdots\!61}{58\!\cdots\!52}a^{11}-\frac{62\!\cdots\!97}{58\!\cdots\!52}a^{10}-\frac{42\!\cdots\!33}{58\!\cdots\!52}a^{9}-\frac{13\!\cdots\!27}{58\!\cdots\!52}a^{8}-\frac{11\!\cdots\!43}{58\!\cdots\!52}a^{7}-\frac{34\!\cdots\!55}{58\!\cdots\!52}a^{6}-\frac{79\!\cdots\!91}{29\!\cdots\!26}a^{5}+\frac{57\!\cdots\!01}{58\!\cdots\!52}a^{4}-\frac{18\!\cdots\!79}{58\!\cdots\!52}a^{3}+\frac{25\!\cdots\!01}{58\!\cdots\!52}a^{2}+\frac{29\!\cdots\!03}{29\!\cdots\!26}a-\frac{26\!\cdots\!50}{14\!\cdots\!63}$, $\frac{47\!\cdots\!19}{58\!\cdots\!52}a^{15}+\frac{12\!\cdots\!35}{58\!\cdots\!52}a^{14}+\frac{39\!\cdots\!79}{58\!\cdots\!52}a^{13}+\frac{10\!\cdots\!13}{58\!\cdots\!52}a^{12}-\frac{29\!\cdots\!16}{14\!\cdots\!63}a^{11}+\frac{74\!\cdots\!29}{58\!\cdots\!52}a^{10}-\frac{91\!\cdots\!67}{29\!\cdots\!26}a^{9}-\frac{46\!\cdots\!17}{58\!\cdots\!52}a^{8}-\frac{43\!\cdots\!93}{29\!\cdots\!26}a^{7}-\frac{20\!\cdots\!13}{58\!\cdots\!52}a^{6}-\frac{13\!\cdots\!39}{58\!\cdots\!52}a^{5}-\frac{58\!\cdots\!69}{14\!\cdots\!63}a^{4}-\frac{29\!\cdots\!89}{58\!\cdots\!52}a^{3}-\frac{91\!\cdots\!31}{29\!\cdots\!26}a^{2}-\frac{79\!\cdots\!89}{29\!\cdots\!26}a+\frac{26\!\cdots\!37}{14\!\cdots\!63}$, $\frac{11\!\cdots\!93}{11\!\cdots\!04}a^{15}-\frac{62\!\cdots\!53}{11\!\cdots\!04}a^{14}+\frac{58\!\cdots\!55}{11\!\cdots\!04}a^{13}+\frac{20\!\cdots\!55}{11\!\cdots\!04}a^{12}-\frac{35\!\cdots\!89}{58\!\cdots\!52}a^{11}-\frac{10\!\cdots\!69}{11\!\cdots\!04}a^{10}-\frac{12\!\cdots\!77}{58\!\cdots\!52}a^{9}-\frac{86\!\cdots\!89}{11\!\cdots\!04}a^{8}-\frac{33\!\cdots\!53}{29\!\cdots\!26}a^{7}-\frac{10\!\cdots\!49}{11\!\cdots\!04}a^{6}+\frac{19\!\cdots\!61}{11\!\cdots\!04}a^{5}+\frac{42\!\cdots\!27}{14\!\cdots\!63}a^{4}-\frac{62\!\cdots\!33}{11\!\cdots\!04}a^{3}+\frac{18\!\cdots\!91}{58\!\cdots\!52}a^{2}+\frac{18\!\cdots\!77}{29\!\cdots\!26}a-\frac{17\!\cdots\!16}{14\!\cdots\!63}$, $\frac{68\!\cdots\!93}{11\!\cdots\!04}a^{15}+\frac{14\!\cdots\!71}{11\!\cdots\!04}a^{14}+\frac{58\!\cdots\!19}{11\!\cdots\!04}a^{13}+\frac{24\!\cdots\!15}{11\!\cdots\!04}a^{12}+\frac{52\!\cdots\!73}{58\!\cdots\!52}a^{11}-\frac{34\!\cdots\!33}{11\!\cdots\!04}a^{10}-\frac{13\!\cdots\!77}{58\!\cdots\!52}a^{9}-\frac{10\!\cdots\!61}{11\!\cdots\!04}a^{8}-\frac{30\!\cdots\!30}{14\!\cdots\!63}a^{7}-\frac{47\!\cdots\!49}{11\!\cdots\!04}a^{6}-\frac{61\!\cdots\!39}{11\!\cdots\!04}a^{5}-\frac{14\!\cdots\!23}{29\!\cdots\!26}a^{4}-\frac{36\!\cdots\!77}{11\!\cdots\!04}a^{3}-\frac{12\!\cdots\!67}{58\!\cdots\!52}a^{2}-\frac{41\!\cdots\!23}{14\!\cdots\!63}a+\frac{19\!\cdots\!43}{14\!\cdots\!63}$, $\frac{21\!\cdots\!69}{29\!\cdots\!26}a^{15}+\frac{14\!\cdots\!49}{58\!\cdots\!52}a^{14}+\frac{17\!\cdots\!97}{58\!\cdots\!52}a^{13}+\frac{10\!\cdots\!17}{58\!\cdots\!52}a^{12}-\frac{22\!\cdots\!73}{58\!\cdots\!52}a^{11}-\frac{68\!\cdots\!23}{14\!\cdots\!63}a^{10}-\frac{72\!\cdots\!19}{58\!\cdots\!52}a^{9}-\frac{23\!\cdots\!69}{29\!\cdots\!26}a^{8}-\frac{24\!\cdots\!19}{58\!\cdots\!52}a^{7}-\frac{26\!\cdots\!07}{29\!\cdots\!26}a^{6}+\frac{18\!\cdots\!31}{58\!\cdots\!52}a^{5}+\frac{23\!\cdots\!19}{58\!\cdots\!52}a^{4}-\frac{38\!\cdots\!83}{29\!\cdots\!26}a^{3}+\frac{25\!\cdots\!07}{58\!\cdots\!52}a^{2}+\frac{13\!\cdots\!06}{14\!\cdots\!63}a-\frac{15\!\cdots\!71}{14\!\cdots\!63}$, $\frac{19\!\cdots\!65}{58\!\cdots\!52}a^{15}+\frac{14\!\cdots\!60}{14\!\cdots\!63}a^{14}+\frac{50\!\cdots\!85}{14\!\cdots\!63}a^{13}+\frac{20\!\cdots\!75}{14\!\cdots\!63}a^{12}+\frac{80\!\cdots\!25}{58\!\cdots\!52}a^{11}-\frac{11\!\cdots\!65}{58\!\cdots\!52}a^{10}-\frac{10\!\cdots\!09}{58\!\cdots\!52}a^{9}-\frac{37\!\cdots\!25}{58\!\cdots\!52}a^{8}-\frac{92\!\cdots\!59}{58\!\cdots\!52}a^{7}-\frac{17\!\cdots\!55}{58\!\cdots\!52}a^{6}-\frac{53\!\cdots\!97}{14\!\cdots\!63}a^{5}-\frac{17\!\cdots\!07}{58\!\cdots\!52}a^{4}-\frac{11\!\cdots\!75}{58\!\cdots\!52}a^{3}+\frac{10\!\cdots\!91}{58\!\cdots\!52}a^{2}+\frac{93\!\cdots\!39}{29\!\cdots\!26}a+\frac{54\!\cdots\!69}{14\!\cdots\!63}$, $\frac{56\!\cdots\!69}{11\!\cdots\!04}a^{15}-\frac{21\!\cdots\!07}{11\!\cdots\!04}a^{14}+\frac{38\!\cdots\!69}{11\!\cdots\!04}a^{13}+\frac{10\!\cdots\!81}{11\!\cdots\!04}a^{12}-\frac{35\!\cdots\!52}{14\!\cdots\!63}a^{11}+\frac{67\!\cdots\!19}{11\!\cdots\!04}a^{10}-\frac{21\!\cdots\!23}{14\!\cdots\!63}a^{9}-\frac{46\!\cdots\!33}{11\!\cdots\!04}a^{8}-\frac{22\!\cdots\!07}{58\!\cdots\!52}a^{7}-\frac{13\!\cdots\!49}{11\!\cdots\!04}a^{6}+\frac{48\!\cdots\!91}{11\!\cdots\!04}a^{5}-\frac{21\!\cdots\!49}{58\!\cdots\!52}a^{4}-\frac{24\!\cdots\!93}{11\!\cdots\!04}a^{3}+\frac{95\!\cdots\!39}{14\!\cdots\!63}a^{2}+\frac{36\!\cdots\!37}{29\!\cdots\!26}a-\frac{69\!\cdots\!11}{14\!\cdots\!63}$, $\frac{13\!\cdots\!47}{11\!\cdots\!04}a^{15}-\frac{11\!\cdots\!55}{11\!\cdots\!04}a^{14}+\frac{90\!\cdots\!93}{11\!\cdots\!04}a^{13}+\frac{20\!\cdots\!25}{11\!\cdots\!04}a^{12}-\frac{38\!\cdots\!65}{58\!\cdots\!52}a^{11}+\frac{45\!\cdots\!01}{11\!\cdots\!04}a^{10}-\frac{20\!\cdots\!61}{58\!\cdots\!52}a^{9}-\frac{88\!\cdots\!19}{11\!\cdots\!04}a^{8}-\frac{12\!\cdots\!31}{29\!\cdots\!26}a^{7}-\frac{26\!\cdots\!67}{11\!\cdots\!04}a^{6}+\frac{16\!\cdots\!87}{11\!\cdots\!04}a^{5}-\frac{11\!\cdots\!83}{14\!\cdots\!63}a^{4}-\frac{93\!\cdots\!55}{11\!\cdots\!04}a^{3}+\frac{10\!\cdots\!93}{58\!\cdots\!52}a^{2}-\frac{56\!\cdots\!02}{14\!\cdots\!63}a-\frac{24\!\cdots\!13}{14\!\cdots\!63}$, $\frac{41\!\cdots\!65}{58\!\cdots\!52}a^{15}-\frac{60\!\cdots\!15}{14\!\cdots\!63}a^{14}+\frac{59\!\cdots\!66}{14\!\cdots\!63}a^{13}+\frac{19\!\cdots\!44}{14\!\cdots\!63}a^{12}-\frac{26\!\cdots\!45}{58\!\cdots\!52}a^{11}+\frac{21\!\cdots\!79}{58\!\cdots\!52}a^{10}-\frac{10\!\cdots\!97}{58\!\cdots\!52}a^{9}-\frac{33\!\cdots\!73}{58\!\cdots\!52}a^{8}-\frac{91\!\cdots\!21}{58\!\cdots\!52}a^{7}-\frac{64\!\cdots\!67}{58\!\cdots\!52}a^{6}+\frac{23\!\cdots\!05}{29\!\cdots\!26}a^{5}+\frac{77\!\cdots\!87}{58\!\cdots\!52}a^{4}-\frac{70\!\cdots\!45}{58\!\cdots\!52}a^{3}+\frac{15\!\cdots\!43}{58\!\cdots\!52}a^{2}-\frac{20\!\cdots\!27}{29\!\cdots\!26}a-\frac{53\!\cdots\!32}{14\!\cdots\!63}$, $\frac{27\!\cdots\!99}{11\!\cdots\!04}a^{15}-\frac{30\!\cdots\!69}{11\!\cdots\!04}a^{14}+\frac{19\!\cdots\!19}{11\!\cdots\!04}a^{13}+\frac{38\!\cdots\!19}{11\!\cdots\!04}a^{12}-\frac{43\!\cdots\!81}{29\!\cdots\!26}a^{11}+\frac{13\!\cdots\!45}{11\!\cdots\!04}a^{10}-\frac{21\!\cdots\!23}{29\!\cdots\!26}a^{9}-\frac{16\!\cdots\!55}{11\!\cdots\!04}a^{8}-\frac{28\!\cdots\!07}{58\!\cdots\!52}a^{7}-\frac{51\!\cdots\!07}{11\!\cdots\!04}a^{6}+\frac{44\!\cdots\!65}{11\!\cdots\!04}a^{5}-\frac{13\!\cdots\!25}{58\!\cdots\!52}a^{4}-\frac{89\!\cdots\!55}{11\!\cdots\!04}a^{3}+\frac{60\!\cdots\!38}{14\!\cdots\!63}a^{2}-\frac{35\!\cdots\!93}{14\!\cdots\!63}a+\frac{60\!\cdots\!27}{14\!\cdots\!63}$, $\frac{61\!\cdots\!45}{29\!\cdots\!26}a^{15}-\frac{21\!\cdots\!65}{29\!\cdots\!26}a^{14}+\frac{47\!\cdots\!43}{29\!\cdots\!26}a^{13}+\frac{11\!\cdots\!29}{29\!\cdots\!26}a^{12}-\frac{12\!\cdots\!44}{14\!\cdots\!63}a^{11}+\frac{66\!\cdots\!65}{14\!\cdots\!63}a^{10}-\frac{11\!\cdots\!25}{14\!\cdots\!63}a^{9}-\frac{45\!\cdots\!59}{29\!\cdots\!26}a^{8}-\frac{35\!\cdots\!05}{14\!\cdots\!63}a^{7}-\frac{86\!\cdots\!22}{14\!\cdots\!63}a^{6}-\frac{11\!\cdots\!25}{29\!\cdots\!26}a^{5}-\frac{17\!\cdots\!33}{29\!\cdots\!26}a^{4}+\frac{71\!\cdots\!50}{14\!\cdots\!63}a^{3}-\frac{11\!\cdots\!79}{29\!\cdots\!26}a^{2}+\frac{70\!\cdots\!87}{14\!\cdots\!63}a-\frac{25\!\cdots\!65}{14\!\cdots\!63}$ 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:  \( 549740.860993 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 549740.860993 \cdot 1}{2\cdot\sqrt{575650281819106455466129}}\cr\approx \mathstrut & 0.144546673168 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16)
 
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 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16, 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 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16);
 
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 + 7*x^14 + 15*x^13 - 62*x^12 + 43*x^11 - 310*x^10 - 629*x^9 - 268*x^8 - 1921*x^7 + 1573*x^6 - 776*x^5 + 7*x^4 + 1618*x^3 - 556*x^2 - 184*x + 16);
 
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, 4.2.12427.1, 4.2.211259.1, \(\Q(\zeta_{17})^+\), 8.4.758716206377.1, 8.4.44630365081.1

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: 16.0.1968033759133442969054057291329.4
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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ R ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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}$
\(43\) Copy content Toggle raw display 43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.8.4.1$x^{8} + 182 x^{6} + 84 x^{5} + 11555 x^{4} - 6804 x^{3} + 301934 x^{2} - 447636 x + 2755621$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$