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Properties

Label 16.8.258...017.5
Degree $16$
Signature $[8, 4]$
Discriminant $2.589\times 10^{29}$
Root discriminant \(68.92\)
Ramified primes $17,67$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_2^7.C_8$ (as 16T1155)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312)
 
gp: K = bnfinit(y^16 - 7*y^15 + 15*y^14 + 14*y^13 - 64*y^12 - 776*y^11 + 3052*y^10 + 226*y^9 - 23529*y^8 + 77527*y^7 - 46867*y^6 - 251376*y^5 + 485040*y^4 - 142024*y^3 - 237040*y^2 + 96640*y + 13312, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312)
 

\( x^{16} - 7 x^{15} + 15 x^{14} + 14 x^{13} - 64 x^{12} - 776 x^{11} + 3052 x^{10} + 226 x^{9} + \cdots + 13312 \) 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:   \(258930158322830089457985795017\) \(\medspace = 17^{15}\cdot 67^{6}\) 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:  \(68.92\)
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:  $17^{15/16}67^{1/2}\approx 116.56905215178652$
Ramified primes:   \(17\), \(67\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{3}{16}a^{5}-\frac{1}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{8}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{128}a^{12}+\frac{1}{64}a^{11}-\frac{1}{64}a^{10}+\frac{1}{128}a^{9}-\frac{3}{128}a^{8}-\frac{3}{32}a^{7}+\frac{1}{32}a^{6}-\frac{31}{128}a^{5}+\frac{1}{16}a^{4}-\frac{7}{16}a^{3}-\frac{3}{16}a^{2}+\frac{1}{8}a$, $\frac{1}{128}a^{13}+\frac{1}{64}a^{11}-\frac{3}{128}a^{10}+\frac{3}{128}a^{9}-\frac{3}{64}a^{8}-\frac{3}{32}a^{7}+\frac{1}{128}a^{6}+\frac{7}{64}a^{5}+\frac{3}{16}a^{4}-\frac{3}{16}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{206848}a^{14}-\frac{329}{206848}a^{13}+\frac{49}{12928}a^{12}-\frac{4761}{206848}a^{11}+\frac{33}{103424}a^{10}+\frac{11997}{206848}a^{9}-\frac{189}{12928}a^{8}+\frac{19413}{206848}a^{7}-\frac{18339}{206848}a^{6}+\frac{3935}{25856}a^{5}+\frac{1143}{12928}a^{4}+\frac{5955}{25856}a^{3}+\frac{369}{12928}a^{2}-\frac{41}{1616}a+\frac{69}{202}$, $\frac{1}{72\!\cdots\!92}a^{15}-\frac{28\!\cdots\!71}{13\!\cdots\!96}a^{14}-\frac{17\!\cdots\!33}{72\!\cdots\!92}a^{13}-\frac{16\!\cdots\!49}{72\!\cdots\!92}a^{12}-\frac{17\!\cdots\!99}{72\!\cdots\!92}a^{11}+\frac{35\!\cdots\!07}{72\!\cdots\!92}a^{10}-\frac{22\!\cdots\!47}{72\!\cdots\!92}a^{9}-\frac{23\!\cdots\!87}{72\!\cdots\!92}a^{8}+\frac{33\!\cdots\!99}{36\!\cdots\!96}a^{7}-\frac{72\!\cdots\!19}{72\!\cdots\!92}a^{6}+\frac{19\!\cdots\!49}{69\!\cdots\!48}a^{5}+\frac{14\!\cdots\!85}{90\!\cdots\!24}a^{4}-\frac{41\!\cdots\!83}{90\!\cdots\!24}a^{3}+\frac{15\!\cdots\!73}{45\!\cdots\!12}a^{2}-\frac{27\!\cdots\!33}{56\!\cdots\!64}a-\frac{13\!\cdots\!75}{54\!\cdots\!16}$ 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

$C_{4}$, 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:  $11$
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:   $\frac{12\!\cdots\!45}{71\!\cdots\!44}a^{15}-\frac{13\!\cdots\!43}{13\!\cdots\!72}a^{14}+\frac{10\!\cdots\!43}{71\!\cdots\!44}a^{13}+\frac{25\!\cdots\!63}{71\!\cdots\!44}a^{12}-\frac{51\!\cdots\!15}{71\!\cdots\!44}a^{11}-\frac{93\!\cdots\!09}{71\!\cdots\!44}a^{10}+\frac{24\!\cdots\!01}{71\!\cdots\!44}a^{9}+\frac{25\!\cdots\!41}{71\!\cdots\!44}a^{8}-\frac{12\!\cdots\!33}{35\!\cdots\!72}a^{7}+\frac{73\!\cdots\!01}{71\!\cdots\!44}a^{6}+\frac{46\!\cdots\!17}{68\!\cdots\!36}a^{5}-\frac{31\!\cdots\!39}{89\!\cdots\!68}a^{4}+\frac{50\!\cdots\!85}{89\!\cdots\!68}a^{3}-\frac{98\!\cdots\!31}{44\!\cdots\!84}a^{2}-\frac{14\!\cdots\!21}{55\!\cdots\!48}a+\frac{96\!\cdots\!17}{53\!\cdots\!12}$, $\frac{18\!\cdots\!19}{14\!\cdots\!88}a^{15}-\frac{23\!\cdots\!81}{27\!\cdots\!44}a^{14}+\frac{23\!\cdots\!49}{14\!\cdots\!88}a^{13}+\frac{38\!\cdots\!01}{14\!\cdots\!88}a^{12}-\frac{12\!\cdots\!57}{14\!\cdots\!88}a^{11}-\frac{14\!\cdots\!67}{14\!\cdots\!88}a^{10}+\frac{52\!\cdots\!63}{14\!\cdots\!88}a^{9}+\frac{27\!\cdots\!19}{14\!\cdots\!88}a^{8}-\frac{22\!\cdots\!27}{71\!\cdots\!44}a^{7}+\frac{13\!\cdots\!07}{14\!\cdots\!88}a^{6}-\frac{42\!\cdots\!09}{13\!\cdots\!72}a^{5}-\frac{66\!\cdots\!13}{17\!\cdots\!36}a^{4}+\frac{95\!\cdots\!79}{17\!\cdots\!36}a^{3}+\frac{87\!\cdots\!59}{89\!\cdots\!68}a^{2}-\frac{24\!\cdots\!95}{11\!\cdots\!96}a+\frac{11\!\cdots\!35}{10\!\cdots\!24}$, $\frac{28\!\cdots\!59}{14\!\cdots\!88}a^{15}-\frac{34\!\cdots\!29}{27\!\cdots\!44}a^{14}+\frac{31\!\cdots\!97}{14\!\cdots\!88}a^{13}+\frac{59\!\cdots\!97}{14\!\cdots\!88}a^{12}-\frac{16\!\cdots\!73}{14\!\cdots\!88}a^{11}-\frac{22\!\cdots\!71}{14\!\cdots\!88}a^{10}+\frac{71\!\cdots\!03}{14\!\cdots\!88}a^{9}+\frac{49\!\cdots\!83}{14\!\cdots\!88}a^{8}-\frac{32\!\cdots\!55}{71\!\cdots\!44}a^{7}+\frac{19\!\cdots\!03}{14\!\cdots\!88}a^{6}-\frac{37\!\cdots\!45}{13\!\cdots\!72}a^{5}-\frac{90\!\cdots\!29}{17\!\cdots\!36}a^{4}+\frac{13\!\cdots\!67}{17\!\cdots\!36}a^{3}-\frac{88\!\cdots\!29}{89\!\cdots\!68}a^{2}-\frac{35\!\cdots\!03}{11\!\cdots\!96}a+\frac{89\!\cdots\!03}{10\!\cdots\!24}$, $\frac{35\!\cdots\!53}{72\!\cdots\!92}a^{15}-\frac{43\!\cdots\!15}{13\!\cdots\!96}a^{14}+\frac{41\!\cdots\!07}{72\!\cdots\!92}a^{13}+\frac{61\!\cdots\!03}{72\!\cdots\!92}a^{12}-\frac{16\!\cdots\!63}{72\!\cdots\!92}a^{11}-\frac{28\!\cdots\!45}{72\!\cdots\!92}a^{10}+\frac{90\!\cdots\!89}{72\!\cdots\!92}a^{9}+\frac{47\!\cdots\!73}{72\!\cdots\!92}a^{8}-\frac{37\!\cdots\!93}{36\!\cdots\!96}a^{7}+\frac{23\!\cdots\!37}{72\!\cdots\!92}a^{6}-\frac{73\!\cdots\!71}{69\!\cdots\!48}a^{5}-\frac{99\!\cdots\!63}{90\!\cdots\!24}a^{4}+\frac{14\!\cdots\!97}{90\!\cdots\!24}a^{3}-\frac{20\!\cdots\!39}{45\!\cdots\!12}a^{2}-\frac{12\!\cdots\!77}{56\!\cdots\!64}a+\frac{36\!\cdots\!89}{54\!\cdots\!16}$, $\frac{28\!\cdots\!47}{72\!\cdots\!92}a^{15}-\frac{38\!\cdots\!57}{13\!\cdots\!96}a^{14}+\frac{42\!\cdots\!93}{72\!\cdots\!92}a^{13}+\frac{42\!\cdots\!65}{72\!\cdots\!92}a^{12}-\frac{17\!\cdots\!41}{72\!\cdots\!92}a^{11}-\frac{22\!\cdots\!51}{72\!\cdots\!92}a^{10}+\frac{87\!\cdots\!35}{72\!\cdots\!92}a^{9}+\frac{12\!\cdots\!67}{72\!\cdots\!92}a^{8}-\frac{33\!\cdots\!83}{36\!\cdots\!96}a^{7}+\frac{21\!\cdots\!87}{72\!\cdots\!92}a^{6}-\frac{11\!\cdots\!05}{69\!\cdots\!48}a^{5}-\frac{91\!\cdots\!17}{90\!\cdots\!24}a^{4}+\frac{16\!\cdots\!75}{90\!\cdots\!24}a^{3}-\frac{12\!\cdots\!93}{45\!\cdots\!12}a^{2}-\frac{37\!\cdots\!99}{56\!\cdots\!64}a-\frac{28\!\cdots\!33}{54\!\cdots\!16}$, $\frac{96\!\cdots\!51}{72\!\cdots\!92}a^{15}-\frac{10\!\cdots\!89}{13\!\cdots\!96}a^{14}+\frac{64\!\cdots\!61}{72\!\cdots\!92}a^{13}+\frac{25\!\cdots\!25}{72\!\cdots\!92}a^{12}-\frac{33\!\cdots\!29}{72\!\cdots\!92}a^{11}-\frac{78\!\cdots\!39}{72\!\cdots\!92}a^{10}+\frac{17\!\cdots\!87}{72\!\cdots\!92}a^{9}+\frac{30\!\cdots\!71}{72\!\cdots\!92}a^{8}-\frac{97\!\cdots\!95}{36\!\cdots\!96}a^{7}+\frac{49\!\cdots\!83}{72\!\cdots\!92}a^{6}+\frac{24\!\cdots\!87}{69\!\cdots\!48}a^{5}-\frac{27\!\cdots\!13}{90\!\cdots\!24}a^{4}+\frac{24\!\cdots\!63}{90\!\cdots\!24}a^{3}+\frac{50\!\cdots\!83}{45\!\cdots\!12}a^{2}-\frac{65\!\cdots\!03}{56\!\cdots\!64}a+\frac{34\!\cdots\!43}{54\!\cdots\!16}$, $\frac{64\!\cdots\!31}{72\!\cdots\!92}a^{15}-\frac{12\!\cdots\!21}{13\!\cdots\!96}a^{14}+\frac{28\!\cdots\!81}{72\!\cdots\!92}a^{13}-\frac{63\!\cdots\!71}{72\!\cdots\!92}a^{12}+\frac{97\!\cdots\!39}{72\!\cdots\!92}a^{11}-\frac{68\!\cdots\!31}{72\!\cdots\!92}a^{10}+\frac{39\!\cdots\!79}{72\!\cdots\!92}a^{9}-\frac{10\!\cdots\!09}{72\!\cdots\!92}a^{8}+\frac{45\!\cdots\!69}{36\!\cdots\!96}a^{7}+\frac{41\!\cdots\!03}{72\!\cdots\!92}a^{6}-\frac{15\!\cdots\!25}{69\!\cdots\!48}a^{5}+\frac{30\!\cdots\!79}{90\!\cdots\!24}a^{4}-\frac{11\!\cdots\!37}{90\!\cdots\!24}a^{3}-\frac{75\!\cdots\!97}{45\!\cdots\!12}a^{2}+\frac{67\!\cdots\!09}{56\!\cdots\!64}a+\frac{73\!\cdots\!23}{54\!\cdots\!16}$, $\frac{22\!\cdots\!91}{72\!\cdots\!92}a^{15}-\frac{28\!\cdots\!81}{13\!\cdots\!96}a^{14}+\frac{28\!\cdots\!97}{72\!\cdots\!92}a^{13}+\frac{39\!\cdots\!37}{72\!\cdots\!92}a^{12}-\frac{11\!\cdots\!61}{72\!\cdots\!92}a^{11}-\frac{18\!\cdots\!15}{72\!\cdots\!92}a^{10}+\frac{61\!\cdots\!03}{72\!\cdots\!92}a^{9}+\frac{26\!\cdots\!03}{72\!\cdots\!92}a^{8}-\frac{24\!\cdots\!43}{36\!\cdots\!96}a^{7}+\frac{14\!\cdots\!03}{72\!\cdots\!92}a^{6}-\frac{41\!\cdots\!41}{69\!\cdots\!48}a^{5}-\frac{69\!\cdots\!29}{90\!\cdots\!24}a^{4}+\frac{96\!\cdots\!03}{90\!\cdots\!24}a^{3}+\frac{65\!\cdots\!39}{45\!\cdots\!12}a^{2}-\frac{36\!\cdots\!51}{56\!\cdots\!64}a-\frac{37\!\cdots\!89}{54\!\cdots\!16}$, $\frac{24\!\cdots\!65}{72\!\cdots\!92}a^{15}-\frac{33\!\cdots\!07}{13\!\cdots\!96}a^{14}+\frac{40\!\cdots\!71}{72\!\cdots\!92}a^{13}+\frac{13\!\cdots\!83}{72\!\cdots\!92}a^{12}-\frac{11\!\cdots\!47}{72\!\cdots\!92}a^{11}-\frac{18\!\cdots\!97}{72\!\cdots\!92}a^{10}+\frac{77\!\cdots\!37}{72\!\cdots\!92}a^{9}-\frac{23\!\cdots\!15}{72\!\cdots\!92}a^{8}-\frac{24\!\cdots\!81}{36\!\cdots\!96}a^{7}+\frac{18\!\cdots\!93}{72\!\cdots\!92}a^{6}-\frac{15\!\cdots\!15}{69\!\cdots\!48}a^{5}-\frac{60\!\cdots\!71}{90\!\cdots\!24}a^{4}+\frac{13\!\cdots\!21}{90\!\cdots\!24}a^{3}-\frac{27\!\cdots\!27}{45\!\cdots\!12}a^{2}-\frac{30\!\cdots\!53}{56\!\cdots\!64}a+\frac{15\!\cdots\!69}{54\!\cdots\!16}$, $\frac{32\!\cdots\!39}{36\!\cdots\!96}a^{15}-\frac{56\!\cdots\!89}{69\!\cdots\!48}a^{14}+\frac{10\!\cdots\!01}{36\!\cdots\!96}a^{13}-\frac{17\!\cdots\!43}{36\!\cdots\!96}a^{12}+\frac{16\!\cdots\!51}{36\!\cdots\!96}a^{11}-\frac{29\!\cdots\!87}{36\!\cdots\!96}a^{10}+\frac{16\!\cdots\!27}{36\!\cdots\!96}a^{9}-\frac{32\!\cdots\!41}{36\!\cdots\!96}a^{8}-\frac{41\!\cdots\!35}{18\!\cdots\!48}a^{7}+\frac{25\!\cdots\!15}{36\!\cdots\!96}a^{6}-\frac{59\!\cdots\!97}{34\!\cdots\!24}a^{5}+\frac{47\!\cdots\!35}{45\!\cdots\!12}a^{4}+\frac{77\!\cdots\!67}{45\!\cdots\!12}a^{3}-\frac{56\!\cdots\!45}{22\!\cdots\!56}a^{2}+\frac{65\!\cdots\!09}{28\!\cdots\!32}a+\frac{16\!\cdots\!91}{27\!\cdots\!08}$, $\frac{92\!\cdots\!31}{27\!\cdots\!92}a^{15}-\frac{17\!\cdots\!89}{69\!\cdots\!48}a^{14}+\frac{19\!\cdots\!97}{27\!\cdots\!92}a^{13}-\frac{20\!\cdots\!63}{27\!\cdots\!92}a^{12}-\frac{52\!\cdots\!57}{27\!\cdots\!92}a^{11}-\frac{68\!\cdots\!99}{27\!\cdots\!92}a^{10}+\frac{33\!\cdots\!39}{27\!\cdots\!92}a^{9}-\frac{23\!\cdots\!29}{27\!\cdots\!92}a^{8}-\frac{96\!\cdots\!35}{13\!\cdots\!96}a^{7}+\frac{83\!\cdots\!15}{27\!\cdots\!92}a^{6}-\frac{12\!\cdots\!77}{34\!\cdots\!24}a^{5}-\frac{18\!\cdots\!49}{34\!\cdots\!24}a^{4}+\frac{66\!\cdots\!71}{34\!\cdots\!24}a^{3}-\frac{29\!\cdots\!81}{17\!\cdots\!12}a^{2}+\frac{81\!\cdots\!37}{21\!\cdots\!64}a+\frac{17\!\cdots\!03}{27\!\cdots\!08}$ 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:  \( 494529828.491 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 494529828.491 \cdot 4}{2\cdot\sqrt{258930158322830089457985795017}}\cr\approx \mathstrut & 0.775515724324 \end{aligned}\] (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 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312)
 
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 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312, 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 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312);
 
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 + 15*x^14 + 14*x^13 - 64*x^12 - 776*x^11 + 3052*x^10 + 226*x^9 - 23529*x^8 + 77527*x^7 - 46867*x^6 - 251376*x^5 + 485040*x^4 - 142024*x^3 - 237040*x^2 + 96640*x + 13312);
 
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

$C_2^7.C_8$ (as 16T1155):

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 1024
The 40 conjugacy class representatives for $C_2^7.C_8$
Character table for $C_2^7.C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.1842010303097.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.57681033264163530732453953.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ R ${\href{/padicField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(17\) Copy content Toggle raw display 17.16.15.5$x^{16} + 17$$16$$1$$15$$C_{16}$$[\ ]_{16}$
\(67\) Copy content Toggle raw display 67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$