Properties

Label 16.8.512...033.1
Degree $16$
Signature $[8, 4]$
Discriminant $5.129\times 10^{31}$
Root discriminant $95.91$
Ramified primes $3, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 239*x^13 + 696*x^12 + 3867*x^11 - 18073*x^10 - 9814*x^9 + 96124*x^8 + 66818*x^7 - 269519*x^6 - 788087*x^5 + 1399987*x^4 + 1685500*x^3 - 3041668*x^2 - 1014722*x + 2074829)
 
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 - 239*x^13 + 696*x^12 + 3867*x^11 - 18073*x^10 - 9814*x^9 + 96124*x^8 + 66818*x^7 - 269519*x^6 - 788087*x^5 + 1399987*x^4 + 1685500*x^3 - 3041668*x^2 - 1014722*x + 2074829, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2074829, -1014722, -3041668, 1685500, 1399987, -788087, -269519, 66818, 96124, -9814, -18073, 3867, 696, -239, 26, -7, 1]);
 

\( x^{16} - 7 x^{15} + 26 x^{14} - 239 x^{13} + 696 x^{12} + 3867 x^{11} - 18073 x^{10} - 9814 x^{9} + 96124 x^{8} + 66818 x^{7} - 269519 x^{6} - 788087 x^{5} + 1399987 x^{4} + 1685500 x^{3} - 3041668 x^{2} - 1014722 x + 2074829 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(51293346320079940269505352322033\)\(\medspace = 3^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $95.91$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 97$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{284034260049647522465209526156022081390583} a^{15} + \frac{131340948795977027324547144500420125151066}{284034260049647522465209526156022081390583} a^{14} - \frac{109175783117550375465126110772922736484102}{284034260049647522465209526156022081390583} a^{13} + \frac{43991419418218194186814998996799702264022}{284034260049647522465209526156022081390583} a^{12} - \frac{23046597840259715953508747749694855467960}{284034260049647522465209526156022081390583} a^{11} - \frac{128404387218755940392503551302250950188445}{284034260049647522465209526156022081390583} a^{10} - \frac{133426002705102781964101111581413045712590}{284034260049647522465209526156022081390583} a^{9} + \frac{8308380893220324564765267960460270909738}{284034260049647522465209526156022081390583} a^{8} - \frac{65722432704979654355773807266044944715184}{284034260049647522465209526156022081390583} a^{7} - \frac{25319813268384502382703614019009205664866}{284034260049647522465209526156022081390583} a^{6} - \frac{33450561588367229253968283530729594967929}{284034260049647522465209526156022081390583} a^{5} + \frac{309606012778940903452053776937213404826}{284034260049647522465209526156022081390583} a^{4} + \frac{35836256696768062011510582153891302171113}{284034260049647522465209526156022081390583} a^{3} + \frac{84927848755757475492826258593979896146619}{284034260049647522465209526156022081390583} a^{2} + \frac{25684078159961565144057385006229061759054}{284034260049647522465209526156022081390583} a - \frac{119503020987708022273378153573128712461937}{284034260049647522465209526156022081390583}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2334035612.49 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 2334035612.49 \cdot 2}{2\sqrt{51293346320079940269505352322033}}\approx 0.130027855661$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97Data not computed