Properties

Label 16.16.440...625.1
Degree $16$
Signature $[16, 0]$
Discriminant $4.402\times 10^{29}$
Root discriminant \(71.24\)
Ramified primes $5,101$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $QD_{16}$ (as 16T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375)
 
gp: K = bnfinit(y^16 - 2*y^15 - 78*y^14 + 159*y^13 + 2362*y^12 - 5016*y^11 - 34929*y^10 + 79762*y^9 + 256088*y^8 - 659347*y^7 - 785803*y^6 + 2560624*y^5 + 268374*y^4 - 3315295*y^3 + 864770*y^2 + 891400*y - 317375, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375)
 

\( x^{16} - 2 x^{15} - 78 x^{14} + 159 x^{13} + 2362 x^{12} - 5016 x^{11} - 34929 x^{10} + 79762 x^{9} + 256088 x^{8} - 659347 x^{7} - 785803 x^{6} + 2560624 x^{5} + \cdots - 317375 \) 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:   \(440166027395300672133281640625\) \(\medspace = 5^{8}\cdot 101^{12}\) 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:  \(71.24\)
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:  $5^{1/2}101^{3/4}\approx 71.24034803863643$
Ramified primes:   \(5\), \(101\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{8}+\frac{1}{5}a^{4}$, $\frac{1}{65}a^{13}+\frac{6}{65}a^{12}-\frac{1}{13}a^{11}-\frac{27}{65}a^{9}+\frac{1}{5}a^{8}-\frac{2}{13}a^{7}-\frac{5}{13}a^{6}+\frac{6}{65}a^{5}-\frac{19}{65}a^{4}+\frac{2}{13}a^{3}+\frac{4}{13}a^{2}-\frac{3}{13}a-\frac{1}{13}$, $\frac{1}{325}a^{14}+\frac{2}{325}a^{13}+\frac{2}{65}a^{12}-\frac{6}{325}a^{11}-\frac{27}{325}a^{10}+\frac{121}{325}a^{9}+\frac{24}{65}a^{8}+\frac{2}{325}a^{7}-\frac{24}{325}a^{6}-\frac{108}{325}a^{5}-\frac{1}{65}a^{4}+\frac{149}{325}a^{3}-\frac{19}{65}a^{2}+\frac{24}{65}a+\frac{6}{13}$, $\frac{1}{39\!\cdots\!75}a^{15}+\frac{40\!\cdots\!47}{39\!\cdots\!75}a^{14}+\frac{53\!\cdots\!98}{79\!\cdots\!75}a^{13}-\frac{33\!\cdots\!01}{39\!\cdots\!75}a^{12}-\frac{35\!\cdots\!37}{39\!\cdots\!75}a^{11}+\frac{33\!\cdots\!81}{39\!\cdots\!75}a^{10}+\frac{21\!\cdots\!47}{79\!\cdots\!75}a^{9}+\frac{54\!\cdots\!17}{39\!\cdots\!75}a^{8}+\frac{59\!\cdots\!21}{39\!\cdots\!75}a^{7}+\frac{18\!\cdots\!62}{39\!\cdots\!75}a^{6}-\frac{87\!\cdots\!29}{31\!\cdots\!23}a^{5}+\frac{18\!\cdots\!54}{39\!\cdots\!75}a^{4}+\frac{17\!\cdots\!09}{79\!\cdots\!75}a^{3}-\frac{15\!\cdots\!01}{79\!\cdots\!75}a^{2}+\frac{29\!\cdots\!38}{15\!\cdots\!15}a+\frac{10\!\cdots\!19}{31\!\cdots\!23}$ 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:  $5$

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:  $15$
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:   $\frac{71\!\cdots\!68}{32\!\cdots\!25}a^{15}-\frac{33\!\cdots\!28}{24\!\cdots\!25}a^{14}-\frac{17\!\cdots\!73}{98\!\cdots\!45}a^{13}+\frac{51\!\cdots\!57}{32\!\cdots\!25}a^{12}+\frac{17\!\cdots\!94}{32\!\cdots\!25}a^{11}-\frac{27\!\cdots\!72}{32\!\cdots\!25}a^{10}-\frac{51\!\cdots\!86}{64\!\cdots\!25}a^{9}+\frac{71\!\cdots\!56}{32\!\cdots\!25}a^{8}+\frac{19\!\cdots\!58}{32\!\cdots\!25}a^{7}-\frac{84\!\cdots\!94}{32\!\cdots\!25}a^{6}-\frac{14\!\cdots\!39}{64\!\cdots\!25}a^{5}+\frac{30\!\cdots\!94}{24\!\cdots\!25}a^{4}+\frac{19\!\cdots\!79}{64\!\cdots\!25}a^{3}-\frac{89\!\cdots\!08}{64\!\cdots\!25}a^{2}-\frac{81\!\cdots\!28}{98\!\cdots\!45}a+\frac{10\!\cdots\!73}{25\!\cdots\!77}$, $\frac{16\!\cdots\!39}{39\!\cdots\!75}a^{15}+\frac{86\!\cdots\!78}{39\!\cdots\!75}a^{14}-\frac{99\!\cdots\!81}{31\!\cdots\!23}a^{13}-\frac{59\!\cdots\!14}{39\!\cdots\!75}a^{12}+\frac{37\!\cdots\!87}{39\!\cdots\!75}a^{11}+\frac{13\!\cdots\!19}{39\!\cdots\!75}a^{10}-\frac{10\!\cdots\!98}{79\!\cdots\!75}a^{9}-\frac{95\!\cdots\!87}{39\!\cdots\!75}a^{8}+\frac{31\!\cdots\!68}{30\!\cdots\!75}a^{7}-\frac{21\!\cdots\!99}{30\!\cdots\!75}a^{6}-\frac{29\!\cdots\!37}{79\!\cdots\!75}a^{5}+\frac{34\!\cdots\!62}{30\!\cdots\!75}a^{4}+\frac{29\!\cdots\!14}{61\!\cdots\!75}a^{3}-\frac{14\!\cdots\!59}{79\!\cdots\!75}a^{2}-\frac{20\!\cdots\!07}{15\!\cdots\!15}a+\frac{17\!\cdots\!40}{31\!\cdots\!23}$, $\frac{38\!\cdots\!39}{39\!\cdots\!75}a^{15}+\frac{11\!\cdots\!06}{30\!\cdots\!75}a^{14}-\frac{92\!\cdots\!07}{12\!\cdots\!55}a^{13}-\frac{92\!\cdots\!39}{39\!\cdots\!75}a^{12}+\frac{89\!\cdots\!37}{39\!\cdots\!75}a^{11}+\frac{17\!\cdots\!69}{39\!\cdots\!75}a^{10}-\frac{26\!\cdots\!88}{79\!\cdots\!75}a^{9}-\frac{12\!\cdots\!12}{39\!\cdots\!75}a^{8}+\frac{99\!\cdots\!09}{39\!\cdots\!75}a^{7}-\frac{22\!\cdots\!62}{39\!\cdots\!75}a^{6}-\frac{71\!\cdots\!17}{79\!\cdots\!75}a^{5}+\frac{11\!\cdots\!37}{30\!\cdots\!75}a^{4}+\frac{91\!\cdots\!42}{79\!\cdots\!75}a^{3}-\frac{40\!\cdots\!59}{79\!\cdots\!75}a^{2}-\frac{38\!\cdots\!44}{12\!\cdots\!55}a+\frac{43\!\cdots\!80}{31\!\cdots\!23}$, $\frac{24\!\cdots\!48}{39\!\cdots\!75}a^{15}+\frac{79\!\cdots\!67}{30\!\cdots\!75}a^{14}-\frac{58\!\cdots\!11}{12\!\cdots\!55}a^{13}-\frac{67\!\cdots\!73}{39\!\cdots\!75}a^{12}+\frac{56\!\cdots\!34}{39\!\cdots\!75}a^{11}+\frac{13\!\cdots\!08}{39\!\cdots\!75}a^{10}-\frac{16\!\cdots\!81}{79\!\cdots\!75}a^{9}-\frac{51\!\cdots\!84}{39\!\cdots\!75}a^{8}+\frac{62\!\cdots\!63}{39\!\cdots\!75}a^{7}-\frac{10\!\cdots\!09}{39\!\cdots\!75}a^{6}-\frac{45\!\cdots\!74}{79\!\cdots\!75}a^{5}+\frac{65\!\cdots\!34}{30\!\cdots\!75}a^{4}+\frac{58\!\cdots\!19}{79\!\cdots\!75}a^{3}-\frac{21\!\cdots\!88}{79\!\cdots\!75}a^{2}-\frac{24\!\cdots\!08}{12\!\cdots\!55}a+\frac{23\!\cdots\!58}{31\!\cdots\!23}$, $\frac{72\!\cdots\!07}{39\!\cdots\!75}a^{15}+\frac{91\!\cdots\!14}{39\!\cdots\!75}a^{14}-\frac{22\!\cdots\!54}{15\!\cdots\!15}a^{13}-\frac{50\!\cdots\!89}{30\!\cdots\!75}a^{12}+\frac{16\!\cdots\!81}{39\!\cdots\!75}a^{11}+\frac{16\!\cdots\!47}{39\!\cdots\!75}a^{10}-\frac{45\!\cdots\!84}{79\!\cdots\!75}a^{9}-\frac{18\!\cdots\!31}{39\!\cdots\!75}a^{8}+\frac{16\!\cdots\!42}{39\!\cdots\!75}a^{7}+\frac{76\!\cdots\!19}{39\!\cdots\!75}a^{6}-\frac{11\!\cdots\!76}{79\!\cdots\!75}a^{5}-\frac{28\!\cdots\!44}{30\!\cdots\!75}a^{4}+\frac{14\!\cdots\!61}{79\!\cdots\!75}a^{3}-\frac{36\!\cdots\!67}{79\!\cdots\!75}a^{2}-\frac{76\!\cdots\!71}{15\!\cdots\!15}a+\frac{42\!\cdots\!13}{31\!\cdots\!23}$, $\frac{55\!\cdots\!68}{79\!\cdots\!75}a^{15}+\frac{41\!\cdots\!51}{15\!\cdots\!15}a^{14}-\frac{42\!\cdots\!12}{79\!\cdots\!75}a^{13}-\frac{13\!\cdots\!73}{79\!\cdots\!75}a^{12}+\frac{50\!\cdots\!51}{31\!\cdots\!23}a^{11}+\frac{38\!\cdots\!04}{12\!\cdots\!55}a^{10}-\frac{18\!\cdots\!31}{79\!\cdots\!75}a^{9}-\frac{15\!\cdots\!74}{79\!\cdots\!75}a^{8}+\frac{13\!\cdots\!56}{79\!\cdots\!75}a^{7}-\frac{64\!\cdots\!18}{15\!\cdots\!15}a^{6}-\frac{49\!\cdots\!72}{79\!\cdots\!75}a^{5}+\frac{22\!\cdots\!82}{79\!\cdots\!75}a^{4}+\frac{64\!\cdots\!96}{79\!\cdots\!75}a^{3}-\frac{11\!\cdots\!20}{31\!\cdots\!23}a^{2}-\frac{35\!\cdots\!14}{15\!\cdots\!15}a+\frac{31\!\cdots\!88}{31\!\cdots\!23}$, $\frac{50\!\cdots\!92}{39\!\cdots\!75}a^{15}+\frac{86\!\cdots\!44}{39\!\cdots\!75}a^{14}-\frac{78\!\cdots\!16}{79\!\cdots\!75}a^{13}-\frac{48\!\cdots\!17}{39\!\cdots\!75}a^{12}+\frac{11\!\cdots\!01}{39\!\cdots\!75}a^{11}+\frac{35\!\cdots\!37}{39\!\cdots\!75}a^{10}-\frac{35\!\cdots\!62}{79\!\cdots\!75}a^{9}+\frac{21\!\cdots\!64}{39\!\cdots\!75}a^{8}+\frac{13\!\cdots\!22}{39\!\cdots\!75}a^{7}-\frac{42\!\cdots\!01}{39\!\cdots\!75}a^{6}-\frac{97\!\cdots\!97}{79\!\cdots\!75}a^{5}+\frac{17\!\cdots\!36}{30\!\cdots\!75}a^{4}+\frac{12\!\cdots\!69}{79\!\cdots\!75}a^{3}-\frac{56\!\cdots\!87}{79\!\cdots\!75}a^{2}-\frac{68\!\cdots\!53}{15\!\cdots\!15}a+\frac{59\!\cdots\!04}{31\!\cdots\!23}$, $\frac{71\!\cdots\!78}{30\!\cdots\!75}a^{15}-\frac{30\!\cdots\!27}{39\!\cdots\!75}a^{14}-\frac{14\!\cdots\!67}{79\!\cdots\!75}a^{13}+\frac{24\!\cdots\!36}{39\!\cdots\!75}a^{12}+\frac{22\!\cdots\!42}{39\!\cdots\!75}a^{11}-\frac{80\!\cdots\!96}{39\!\cdots\!75}a^{10}-\frac{69\!\cdots\!89}{79\!\cdots\!75}a^{9}+\frac{14\!\cdots\!88}{39\!\cdots\!75}a^{8}+\frac{27\!\cdots\!99}{39\!\cdots\!75}a^{7}-\frac{13\!\cdots\!67}{39\!\cdots\!75}a^{6}-\frac{20\!\cdots\!44}{79\!\cdots\!75}a^{5}+\frac{58\!\cdots\!31}{39\!\cdots\!75}a^{4}+\frac{27\!\cdots\!28}{79\!\cdots\!75}a^{3}-\frac{12\!\cdots\!89}{79\!\cdots\!75}a^{2}-\frac{15\!\cdots\!91}{15\!\cdots\!15}a+\frac{13\!\cdots\!30}{31\!\cdots\!23}$, $\frac{87\!\cdots\!58}{79\!\cdots\!75}a^{15}+\frac{34\!\cdots\!96}{79\!\cdots\!75}a^{14}-\frac{13\!\cdots\!87}{15\!\cdots\!15}a^{13}-\frac{22\!\cdots\!43}{79\!\cdots\!75}a^{12}+\frac{20\!\cdots\!54}{79\!\cdots\!75}a^{11}+\frac{42\!\cdots\!93}{79\!\cdots\!75}a^{10}-\frac{59\!\cdots\!32}{15\!\cdots\!15}a^{9}-\frac{82\!\cdots\!69}{79\!\cdots\!75}a^{8}+\frac{22\!\cdots\!93}{79\!\cdots\!75}a^{7}-\frac{46\!\cdots\!54}{79\!\cdots\!75}a^{6}-\frac{15\!\cdots\!41}{15\!\cdots\!15}a^{5}+\frac{33\!\cdots\!72}{79\!\cdots\!75}a^{4}+\frac{15\!\cdots\!14}{12\!\cdots\!55}a^{3}-\frac{88\!\cdots\!16}{15\!\cdots\!15}a^{2}-\frac{11\!\cdots\!73}{31\!\cdots\!23}a+\frac{47\!\cdots\!16}{31\!\cdots\!23}$, $\frac{14\!\cdots\!08}{10\!\cdots\!25}a^{15}+\frac{53\!\cdots\!99}{10\!\cdots\!25}a^{14}-\frac{11\!\cdots\!39}{10\!\cdots\!25}a^{13}-\frac{34\!\cdots\!03}{10\!\cdots\!25}a^{12}+\frac{33\!\cdots\!56}{10\!\cdots\!25}a^{11}+\frac{64\!\cdots\!97}{10\!\cdots\!25}a^{10}-\frac{49\!\cdots\!77}{10\!\cdots\!25}a^{9}-\frac{38\!\cdots\!29}{10\!\cdots\!25}a^{8}+\frac{37\!\cdots\!59}{10\!\cdots\!25}a^{7}-\frac{84\!\cdots\!21}{10\!\cdots\!25}a^{6}-\frac{10\!\cdots\!53}{77\!\cdots\!25}a^{5}+\frac{58\!\cdots\!17}{10\!\cdots\!25}a^{4}+\frac{17\!\cdots\!77}{10\!\cdots\!25}a^{3}-\frac{15\!\cdots\!66}{20\!\cdots\!85}a^{2}-\frac{93\!\cdots\!18}{20\!\cdots\!85}a+\frac{81\!\cdots\!53}{40\!\cdots\!37}$, $\frac{45\!\cdots\!86}{39\!\cdots\!75}a^{15}+\frac{18\!\cdots\!22}{39\!\cdots\!75}a^{14}-\frac{27\!\cdots\!45}{31\!\cdots\!23}a^{13}-\frac{12\!\cdots\!61}{39\!\cdots\!75}a^{12}+\frac{10\!\cdots\!88}{39\!\cdots\!75}a^{11}+\frac{25\!\cdots\!06}{39\!\cdots\!75}a^{10}-\frac{30\!\cdots\!52}{79\!\cdots\!75}a^{9}-\frac{93\!\cdots\!38}{39\!\cdots\!75}a^{8}+\frac{11\!\cdots\!16}{39\!\cdots\!75}a^{7}-\frac{19\!\cdots\!38}{39\!\cdots\!75}a^{6}-\frac{82\!\cdots\!13}{79\!\cdots\!75}a^{5}+\frac{12\!\cdots\!38}{30\!\cdots\!75}a^{4}+\frac{10\!\cdots\!53}{79\!\cdots\!75}a^{3}-\frac{39\!\cdots\!66}{79\!\cdots\!75}a^{2}-\frac{57\!\cdots\!58}{15\!\cdots\!15}a+\frac{42\!\cdots\!09}{31\!\cdots\!23}$, $\frac{58\!\cdots\!17}{79\!\cdots\!75}a^{15}+\frac{17\!\cdots\!26}{79\!\cdots\!75}a^{14}-\frac{45\!\cdots\!01}{79\!\cdots\!75}a^{13}-\frac{10\!\cdots\!47}{79\!\cdots\!75}a^{12}+\frac{13\!\cdots\!09}{79\!\cdots\!75}a^{11}+\frac{17\!\cdots\!28}{79\!\cdots\!75}a^{10}-\frac{20\!\cdots\!68}{79\!\cdots\!75}a^{9}+\frac{90\!\cdots\!54}{79\!\cdots\!75}a^{8}+\frac{15\!\cdots\!11}{79\!\cdots\!75}a^{7}-\frac{40\!\cdots\!29}{79\!\cdots\!75}a^{6}-\frac{57\!\cdots\!51}{79\!\cdots\!75}a^{5}+\frac{25\!\cdots\!33}{79\!\cdots\!75}a^{4}+\frac{75\!\cdots\!88}{79\!\cdots\!75}a^{3}-\frac{63\!\cdots\!29}{15\!\cdots\!15}a^{2}-\frac{41\!\cdots\!77}{15\!\cdots\!15}a+\frac{34\!\cdots\!90}{31\!\cdots\!23}$, $\frac{30\!\cdots\!13}{30\!\cdots\!75}a^{15}+\frac{12\!\cdots\!38}{39\!\cdots\!75}a^{14}-\frac{12\!\cdots\!13}{15\!\cdots\!15}a^{13}-\frac{59\!\cdots\!13}{30\!\cdots\!75}a^{12}+\frac{92\!\cdots\!02}{39\!\cdots\!75}a^{11}+\frac{13\!\cdots\!74}{39\!\cdots\!75}a^{10}-\frac{27\!\cdots\!03}{79\!\cdots\!75}a^{9}+\frac{22\!\cdots\!98}{39\!\cdots\!75}a^{8}+\frac{10\!\cdots\!89}{39\!\cdots\!75}a^{7}-\frac{23\!\cdots\!02}{39\!\cdots\!75}a^{6}-\frac{76\!\cdots\!92}{79\!\cdots\!75}a^{5}+\frac{15\!\cdots\!51}{39\!\cdots\!75}a^{4}+\frac{10\!\cdots\!77}{79\!\cdots\!75}a^{3}-\frac{37\!\cdots\!29}{79\!\cdots\!75}a^{2}-\frac{53\!\cdots\!22}{15\!\cdots\!15}a+\frac{41\!\cdots\!28}{31\!\cdots\!23}$, $\frac{22\!\cdots\!89}{39\!\cdots\!75}a^{15}+\frac{73\!\cdots\!13}{39\!\cdots\!75}a^{14}-\frac{36\!\cdots\!86}{79\!\cdots\!75}a^{13}-\frac{53\!\cdots\!39}{39\!\cdots\!75}a^{12}+\frac{58\!\cdots\!77}{39\!\cdots\!75}a^{11}+\frac{15\!\cdots\!24}{39\!\cdots\!75}a^{10}-\frac{18\!\cdots\!26}{79\!\cdots\!75}a^{9}-\frac{19\!\cdots\!87}{39\!\cdots\!75}a^{8}+\frac{80\!\cdots\!29}{39\!\cdots\!75}a^{7}+\frac{12\!\cdots\!23}{39\!\cdots\!75}a^{6}-\frac{76\!\cdots\!93}{79\!\cdots\!75}a^{5}-\frac{34\!\cdots\!94}{39\!\cdots\!75}a^{4}+\frac{34\!\cdots\!92}{15\!\cdots\!15}a^{3}+\frac{36\!\cdots\!22}{61\!\cdots\!75}a^{2}-\frac{25\!\cdots\!39}{15\!\cdots\!15}a+\frac{13\!\cdots\!71}{31\!\cdots\!23}$, $\frac{53\!\cdots\!49}{61\!\cdots\!75}a^{15}+\frac{21\!\cdots\!61}{79\!\cdots\!75}a^{14}-\frac{41\!\cdots\!02}{61\!\cdots\!75}a^{13}-\frac{13\!\cdots\!07}{79\!\cdots\!75}a^{12}+\frac{16\!\cdots\!89}{79\!\cdots\!75}a^{11}+\frac{20\!\cdots\!73}{79\!\cdots\!75}a^{10}-\frac{23\!\cdots\!68}{79\!\cdots\!75}a^{9}+\frac{13\!\cdots\!99}{79\!\cdots\!75}a^{8}+\frac{17\!\cdots\!66}{79\!\cdots\!75}a^{7}-\frac{40\!\cdots\!23}{61\!\cdots\!75}a^{6}-\frac{64\!\cdots\!76}{79\!\cdots\!75}a^{5}+\frac{32\!\cdots\!73}{79\!\cdots\!75}a^{4}+\frac{83\!\cdots\!08}{79\!\cdots\!75}a^{3}-\frac{84\!\cdots\!26}{15\!\cdots\!15}a^{2}-\frac{46\!\cdots\!22}{15\!\cdots\!15}a+\frac{43\!\cdots\!68}{31\!\cdots\!23}$ 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:  \( 18951197490.7 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 18951197490.7 \cdot 1}{2\cdot\sqrt{440166027395300672133281640625}}\cr\approx \mathstrut & 0.936005352737 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375)
 
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 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375, 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 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375);
 
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 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375);
 
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

$\SD_{16}$ (as 16T12):

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 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.51005.1 x2, 4.4.2525.1 x2, 8.8.65037750625.1, 8.8.132690018825125.1 x4

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 8 sibling: 8.8.132690018825125.1
Minimal sibling: 8.8.132690018825125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(5\) Copy content Toggle raw display 5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
\(101\) Copy content Toggle raw display 101.4.3.1$x^{4} + 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} + 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} + 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} + 404$$4$$1$$3$$C_4$$[\ ]_{4}$