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Properties

Label 16.16.446...161.1
Degree $16$
Signature $[16, 0]$
Discriminant $4.466\times 10^{27}$
Root discriminant \(53.47\)
Ramified primes $13,61$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $QD_{16}$ (as 16T12)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 47*y^14 + 84*y^13 + 830*y^12 - 1285*y^11 - 6962*y^10 + 8567*y^9 + 29036*y^8 - 23444*y^7 - 56582*y^6 + 17003*y^5 + 42505*y^4 + 2494*y^3 - 9018*y^2 - 1896*y + 107, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107)
 

\( x^{16} - 2 x^{15} - 47 x^{14} + 84 x^{13} + 830 x^{12} - 1285 x^{11} - 6962 x^{10} + 8567 x^{9} + \cdots + 107 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[16, 0]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(4466413296812760910104974161\) \(\medspace = 13^{12}\cdot 61^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(53.47\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $13^{3/4}61^{1/2}\approx 53.471507940612106$
Ramified primes:   \(13\), \(61\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $\SD_{16}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{2703}a^{14}-\frac{20}{159}a^{13}+\frac{82}{901}a^{12}+\frac{253}{2703}a^{11}+\frac{365}{2703}a^{10}+\frac{440}{2703}a^{9}+\frac{1073}{2703}a^{8}+\frac{395}{901}a^{7}-\frac{148}{901}a^{6}-\frac{150}{901}a^{5}-\frac{1288}{2703}a^{4}-\frac{55}{2703}a^{3}+\frac{214}{901}a^{2}+\frac{124}{2703}a-\frac{31}{2703}$, $\frac{1}{51\cdots 41}a^{15}-\frac{27\cdots 96}{17\cdots 47}a^{14}-\frac{77\cdots 19}{51\cdots 41}a^{13}-\frac{68\cdots 33}{17\cdots 47}a^{12}+\frac{85\cdots 74}{51\cdots 41}a^{11}-\frac{24\cdots 59}{51\cdots 41}a^{10}-\frac{11\cdots 94}{51\cdots 41}a^{9}-\frac{21\cdots 59}{51\cdots 41}a^{8}+\frac{23\cdots 72}{51\cdots 41}a^{7}-\frac{25\cdots 87}{51\cdots 41}a^{6}-\frac{26\cdots 34}{17\cdots 47}a^{5}+\frac{23\cdots 03}{51\cdots 41}a^{4}-\frac{80\cdots 67}{51\cdots 41}a^{3}+\frac{85\cdots 61}{17\cdots 47}a^{2}+\frac{11\cdots 44}{51\cdots 41}a+\frac{23\cdots 98}{51\cdots 41}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $15$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{10\cdots 63}{65\cdots 79}a^{15}-\frac{21\cdots 80}{65\cdots 79}a^{14}-\frac{15\cdots 60}{21\cdots 93}a^{13}+\frac{91\cdots 15}{65\cdots 79}a^{12}+\frac{81\cdots 55}{65\cdots 79}a^{11}-\frac{14\cdots 84}{65\cdots 79}a^{10}-\frac{65\cdots 10}{65\cdots 79}a^{9}+\frac{32\cdots 87}{21\cdots 93}a^{8}+\frac{84\cdots 07}{21\cdots 93}a^{7}-\frac{29\cdots 96}{65\cdots 79}a^{6}-\frac{44\cdots 09}{65\cdots 79}a^{5}+\frac{31\cdots 17}{65\cdots 79}a^{4}+\frac{29\cdots 40}{65\cdots 79}a^{3}-\frac{12\cdots 82}{65\cdots 79}a^{2}-\frac{20\cdots 56}{21\cdots 93}a+\frac{10\cdots 73}{65\cdots 79}$, $\frac{29\cdots 51}{51\cdots 41}a^{15}-\frac{73\cdots 08}{51\cdots 41}a^{14}-\frac{45\cdots 59}{17\cdots 47}a^{13}+\frac{31\cdots 22}{51\cdots 41}a^{12}+\frac{23\cdots 45}{51\cdots 41}a^{11}-\frac{48\cdots 08}{51\cdots 41}a^{10}-\frac{18\cdots 15}{51\cdots 41}a^{9}+\frac{11\cdots 17}{17\cdots 47}a^{8}+\frac{25\cdots 12}{17\cdots 47}a^{7}-\frac{99\cdots 66}{51\cdots 41}a^{6}-\frac{13\cdots 56}{51\cdots 41}a^{5}+\frac{96\cdots 38}{51\cdots 41}a^{4}+\frac{90\cdots 08}{51\cdots 41}a^{3}-\frac{19\cdots 65}{51\cdots 41}a^{2}-\frac{55\cdots 09}{17\cdots 47}a-\frac{10\cdots 62}{51\cdots 41}$, $\frac{44\cdots 41}{51\cdots 41}a^{15}-\frac{11\cdots 97}{51\cdots 41}a^{14}-\frac{20\cdots 84}{51\cdots 41}a^{13}+\frac{16\cdots 43}{17\cdots 47}a^{12}+\frac{33\cdots 96}{51\cdots 41}a^{11}-\frac{79\cdots 91}{51\cdots 41}a^{10}-\frac{26\cdots 84}{51\cdots 41}a^{9}+\frac{56\cdots 66}{51\cdots 41}a^{8}+\frac{95\cdots 60}{51\cdots 41}a^{7}-\frac{17\cdots 83}{51\cdots 41}a^{6}-\frac{15\cdots 07}{51\cdots 41}a^{5}+\frac{63\cdots 13}{17\cdots 47}a^{4}+\frac{51\cdots 70}{30\cdots 73}a^{3}-\frac{20\cdots 10}{17\cdots 47}a^{2}-\frac{49\cdots 09}{17\cdots 47}a+\frac{35\cdots 18}{51\cdots 41}$, $\frac{29\cdots 51}{51\cdots 41}a^{15}-\frac{73\cdots 08}{51\cdots 41}a^{14}-\frac{45\cdots 59}{17\cdots 47}a^{13}+\frac{31\cdots 22}{51\cdots 41}a^{12}+\frac{23\cdots 45}{51\cdots 41}a^{11}-\frac{48\cdots 08}{51\cdots 41}a^{10}-\frac{18\cdots 15}{51\cdots 41}a^{9}+\frac{11\cdots 17}{17\cdots 47}a^{8}+\frac{25\cdots 12}{17\cdots 47}a^{7}-\frac{99\cdots 66}{51\cdots 41}a^{6}-\frac{13\cdots 56}{51\cdots 41}a^{5}+\frac{96\cdots 38}{51\cdots 41}a^{4}+\frac{90\cdots 08}{51\cdots 41}a^{3}-\frac{19\cdots 65}{51\cdots 41}a^{2}-\frac{54\cdots 62}{17\cdots 47}a-\frac{10\cdots 62}{51\cdots 41}$, $\frac{14\cdots 82}{51\cdots 41}a^{15}-\frac{36\cdots 32}{51\cdots 41}a^{14}-\frac{67\cdots 58}{51\cdots 41}a^{13}+\frac{15\cdots 18}{51\cdots 41}a^{12}+\frac{37\cdots 36}{17\cdots 47}a^{11}-\frac{81\cdots 69}{17\cdots 47}a^{10}-\frac{30\cdots 21}{17\cdots 47}a^{9}+\frac{16\cdots 05}{51\cdots 41}a^{8}+\frac{34\cdots 17}{51\cdots 41}a^{7}-\frac{16\cdots 01}{17\cdots 47}a^{6}-\frac{58\cdots 75}{51\cdots 41}a^{5}+\frac{53\cdots 89}{51\cdots 41}a^{4}+\frac{12\cdots 10}{17\cdots 47}a^{3}-\frac{14\cdots 32}{51\cdots 41}a^{2}-\frac{21\cdots 61}{17\cdots 47}a+\frac{26\cdots 97}{51\cdots 41}$, $\frac{58\cdots 48}{51\cdots 41}a^{15}-\frac{19\cdots 60}{51\cdots 41}a^{14}-\frac{24\cdots 47}{51\cdots 41}a^{13}+\frac{28\cdots 56}{17\cdots 47}a^{12}+\frac{37\cdots 41}{51\cdots 41}a^{11}-\frac{13\cdots 78}{51\cdots 41}a^{10}-\frac{79\cdots 79}{17\cdots 47}a^{9}+\frac{10\cdots 94}{51\cdots 41}a^{8}+\frac{18\cdots 75}{17\cdots 47}a^{7}-\frac{11\cdots 64}{17\cdots 47}a^{6}+\frac{74\cdots 88}{51\cdots 41}a^{5}+\frac{49\cdots 79}{51\cdots 41}a^{4}-\frac{78\cdots 98}{51\cdots 41}a^{3}-\frac{25\cdots 72}{51\cdots 41}a^{2}+\frac{16\cdots 20}{51\cdots 41}a+\frac{32\cdots 58}{51\cdots 41}$, $\frac{85\cdots 17}{65\cdots 79}a^{15}-\frac{71\cdots 87}{21\cdots 93}a^{14}-\frac{13\cdots 38}{21\cdots 93}a^{13}+\frac{91\cdots 27}{65\cdots 79}a^{12}+\frac{66\cdots 98}{65\cdots 79}a^{11}-\frac{47\cdots 10}{21\cdots 93}a^{10}-\frac{52\cdots 98}{65\cdots 79}a^{9}+\frac{32\cdots 68}{21\cdots 93}a^{8}+\frac{66\cdots 72}{21\cdots 93}a^{7}-\frac{29\cdots 53}{65\cdots 79}a^{6}-\frac{11\cdots 08}{21\cdots 93}a^{5}+\frac{30\cdots 58}{65\cdots 79}a^{4}+\frac{20\cdots 17}{65\cdots 79}a^{3}-\frac{25\cdots 58}{21\cdots 93}a^{2}-\frac{34\cdots 71}{65\cdots 79}a+\frac{18\cdots 98}{65\cdots 79}$, $\frac{10\cdots 63}{65\cdots 79}a^{15}-\frac{21\cdots 80}{65\cdots 79}a^{14}-\frac{15\cdots 60}{21\cdots 93}a^{13}+\frac{91\cdots 15}{65\cdots 79}a^{12}+\frac{81\cdots 55}{65\cdots 79}a^{11}-\frac{14\cdots 84}{65\cdots 79}a^{10}-\frac{65\cdots 10}{65\cdots 79}a^{9}+\frac{32\cdots 87}{21\cdots 93}a^{8}+\frac{84\cdots 07}{21\cdots 93}a^{7}-\frac{29\cdots 96}{65\cdots 79}a^{6}-\frac{44\cdots 09}{65\cdots 79}a^{5}+\frac{31\cdots 17}{65\cdots 79}a^{4}+\frac{29\cdots 40}{65\cdots 79}a^{3}-\frac{12\cdots 82}{65\cdots 79}a^{2}-\frac{20\cdots 56}{21\cdots 93}a+\frac{17\cdots 52}{65\cdots 79}$, $\frac{29\cdots 51}{51\cdots 41}a^{15}-\frac{73\cdots 08}{51\cdots 41}a^{14}-\frac{45\cdots 59}{17\cdots 47}a^{13}+\frac{31\cdots 22}{51\cdots 41}a^{12}+\frac{23\cdots 45}{51\cdots 41}a^{11}-\frac{48\cdots 08}{51\cdots 41}a^{10}-\frac{18\cdots 15}{51\cdots 41}a^{9}+\frac{11\cdots 17}{17\cdots 47}a^{8}+\frac{25\cdots 12}{17\cdots 47}a^{7}-\frac{99\cdots 66}{51\cdots 41}a^{6}-\frac{13\cdots 56}{51\cdots 41}a^{5}+\frac{96\cdots 38}{51\cdots 41}a^{4}+\frac{90\cdots 08}{51\cdots 41}a^{3}-\frac{19\cdots 65}{51\cdots 41}a^{2}-\frac{55\cdots 09}{17\cdots 47}a-\frac{53\cdots 21}{51\cdots 41}$, $\frac{33\cdots 43}{51\cdots 41}a^{15}-\frac{24\cdots 76}{51\cdots 41}a^{14}-\frac{16\cdots 39}{51\cdots 41}a^{13}+\frac{78\cdots 50}{51\cdots 41}a^{12}+\frac{30\cdots 20}{51\cdots 41}a^{11}-\frac{60\cdots 31}{51\cdots 41}a^{10}-\frac{91\cdots 72}{17\cdots 47}a^{9}-\frac{35\cdots 69}{51\cdots 41}a^{8}+\frac{39\cdots 10}{17\cdots 47}a^{7}+\frac{19\cdots 68}{17\cdots 47}a^{6}-\frac{21\cdots 83}{51\cdots 41}a^{5}-\frac{64\cdots 01}{17\cdots 47}a^{4}+\frac{25\cdots 46}{51\cdots 41}a^{3}+\frac{11\cdots 26}{51\cdots 41}a^{2}+\frac{31\cdots 78}{17\cdots 47}a+\frac{23\cdots 57}{51\cdots 41}$, $\frac{10\cdots 33}{17\cdots 47}a^{15}-\frac{44\cdots 03}{51\cdots 41}a^{14}-\frac{14\cdots 54}{51\cdots 41}a^{13}+\frac{18\cdots 97}{51\cdots 41}a^{12}+\frac{25\cdots 04}{51\cdots 41}a^{11}-\frac{27\cdots 83}{51\cdots 41}a^{10}-\frac{20\cdots 67}{51\cdots 41}a^{9}+\frac{59\cdots 98}{17\cdots 47}a^{8}+\frac{81\cdots 34}{51\cdots 41}a^{7}-\frac{46\cdots 14}{51\cdots 41}a^{6}-\frac{49\cdots 27}{17\cdots 47}a^{5}+\frac{26\cdots 20}{51\cdots 41}a^{4}+\frac{92\cdots 47}{51\cdots 41}a^{3}+\frac{77\cdots 20}{51\cdots 41}a^{2}-\frac{93\cdots 68}{51\cdots 41}a+\frac{28\cdots 43}{17\cdots 47}$, $\frac{82\cdots 42}{17\cdots 47}a^{15}-\frac{28\cdots 15}{51\cdots 41}a^{14}-\frac{11\cdots 86}{51\cdots 41}a^{13}+\frac{22\cdots 96}{10\cdots 91}a^{12}+\frac{19\cdots 64}{51\cdots 41}a^{11}-\frac{10\cdots 62}{30\cdots 73}a^{10}-\frac{54\cdots 61}{17\cdots 47}a^{9}+\frac{11\cdots 16}{51\cdots 41}a^{8}+\frac{66\cdots 02}{51\cdots 41}a^{7}-\frac{18\cdots 53}{30\cdots 73}a^{6}-\frac{12\cdots 03}{51\cdots 41}a^{5}+\frac{74\cdots 70}{17\cdots 47}a^{4}+\frac{32\cdots 10}{17\cdots 47}a^{3}+\frac{69\cdots 81}{51\cdots 41}a^{2}-\frac{40\cdots 50}{10\cdots 91}a-\frac{42\cdots 37}{51\cdots 41}$, $\frac{21\cdots 58}{51\cdots 41}a^{15}-\frac{56\cdots 61}{51\cdots 41}a^{14}-\frac{32\cdots 97}{17\cdots 47}a^{13}+\frac{24\cdots 63}{51\cdots 41}a^{12}+\frac{16\cdots 32}{51\cdots 41}a^{11}-\frac{12\cdots 32}{17\cdots 47}a^{10}-\frac{41\cdots 49}{17\cdots 47}a^{9}+\frac{26\cdots 32}{51\cdots 41}a^{8}+\frac{15\cdots 37}{17\cdots 47}a^{7}-\frac{25\cdots 35}{17\cdots 47}a^{6}-\frac{25\cdots 79}{17\cdots 47}a^{5}+\frac{75\cdots 89}{51\cdots 41}a^{4}+\frac{46\cdots 95}{51\cdots 41}a^{3}-\frac{19\cdots 41}{51\cdots 41}a^{2}-\frac{77\cdots 76}{51\cdots 41}a+\frac{70\cdots 13}{51\cdots 41}$, $\frac{21\cdots 18}{17\cdots 47}a^{15}-\frac{20\cdots 35}{17\cdots 47}a^{14}-\frac{28\cdots 97}{51\cdots 41}a^{13}+\frac{24\cdots 45}{51\cdots 41}a^{12}+\frac{46\cdots 88}{51\cdots 41}a^{11}-\frac{36\cdots 45}{51\cdots 41}a^{10}-\frac{33\cdots 80}{51\cdots 41}a^{9}+\frac{23\cdots 45}{51\cdots 41}a^{8}+\frac{35\cdots 32}{17\cdots 47}a^{7}-\frac{58\cdots 77}{51\cdots 41}a^{6}-\frac{12\cdots 55}{51\cdots 41}a^{5}+\frac{11\cdots 94}{17\cdots 47}a^{4}+\frac{17\cdots 84}{57\cdots 41}a^{3}-\frac{41\cdots 40}{51\cdots 41}a^{2}+\frac{41\cdots 18}{51\cdots 41}a+\frac{97\cdots 05}{51\cdots 41}$, $\frac{91\cdots 24}{51\cdots 41}a^{15}-\frac{21\cdots 14}{51\cdots 41}a^{14}-\frac{42\cdots 82}{51\cdots 41}a^{13}+\frac{90\cdots 50}{51\cdots 41}a^{12}+\frac{24\cdots 58}{17\cdots 47}a^{11}-\frac{46\cdots 63}{17\cdots 47}a^{10}-\frac{58\cdots 28}{51\cdots 41}a^{9}+\frac{96\cdots 43}{51\cdots 41}a^{8}+\frac{23\cdots 74}{51\cdots 41}a^{7}-\frac{28\cdots 99}{51\cdots 41}a^{6}-\frac{13\cdots 10}{17\cdots 47}a^{5}+\frac{28\cdots 51}{51\cdots 41}a^{4}+\frac{28\cdots 57}{51\cdots 41}a^{3}-\frac{61\cdots 76}{51\cdots 41}a^{2}-\frac{53\cdots 36}{51\cdots 41}a-\frac{41\cdots 58}{17\cdots 47}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 410433815.678 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 410433815.678 \cdot 1}{2\cdot\sqrt{4466413296812760910104974161}}\cr\approx \mathstrut & 0.201239679143 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 47*x^14 + 84*x^13 + 830*x^12 - 1285*x^11 - 6962*x^10 + 8567*x^9 + 29036*x^8 - 23444*x^7 - 56582*x^6 + 17003*x^5 + 42505*x^4 + 2494*x^3 - 9018*x^2 - 1896*x + 107); 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{793}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{13}, \sqrt{61})\), 4.4.10309.1 x2, 4.4.48373.1 x2, 8.8.395451064801.1, 8.8.1095593933629.1 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 8 sibling: 8.8.1095593933629.1
Minimal sibling: 8.8.1095593933629.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(61\) Copy content Toggle raw display 61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)