Properties

Label 16.8.576...953.2
Degree $16$
Signature $[8, 4]$
Discriminant $5.768\times 10^{25}$
Root discriminant $40.74$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group trivial (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 - x^15 + x^14 + 16*x^13 - 118*x^12 + 152*x^11 - 254*x^10 - 1055*x^9 + 4234*x^8 - 9028*x^7 + 11255*x^6 + 8856*x^5 - 53600*x^4 + 108034*x^3 - 108306*x^2 + 39813*x - 4079)
 
gp: K = bnfinit(x^16 - x^15 + x^14 + 16*x^13 - 118*x^12 + 152*x^11 - 254*x^10 - 1055*x^9 + 4234*x^8 - 9028*x^7 + 11255*x^6 + 8856*x^5 - 53600*x^4 + 108034*x^3 - 108306*x^2 + 39813*x - 4079, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4079, 39813, -108306, 108034, -53600, 8856, 11255, -9028, 4234, -1055, -254, 152, -118, 16, 1, -1, 1]);
 

\( x^{16} - x^{15} + x^{14} + 16 x^{13} - 118 x^{12} + 152 x^{11} - 254 x^{10} - 1055 x^{9} + 4234 x^{8} - 9028 x^{7} + 11255 x^{6} + 8856 x^{5} - 53600 x^{4} + 108034 x^{3} - 108306 x^{2} + 39813 x - 4079 \)

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:  \(57681033264163530732453953\)\(\medspace = 17^{15}\cdot 67^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $40.74$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{883} a^{14} - \frac{234}{883} a^{13} - \frac{286}{883} a^{12} + \frac{160}{883} a^{11} + \frac{46}{883} a^{10} - \frac{337}{883} a^{9} + \frac{314}{883} a^{8} - \frac{6}{883} a^{7} - \frac{22}{883} a^{6} + \frac{8}{883} a^{5} + \frac{181}{883} a^{4} - \frac{267}{883} a^{3} - \frac{143}{883} a^{2} + \frac{117}{883} a - \frac{289}{883}$, $\frac{1}{22296956500607539085182810116491} a^{15} + \frac{445033861990178930425498211}{22296956500607539085182810116491} a^{14} - \frac{1544547739460840643070937107969}{22296956500607539085182810116491} a^{13} + \frac{5280434034832534544138696660649}{22296956500607539085182810116491} a^{12} - \frac{2912085517481590681930199339592}{22296956500607539085182810116491} a^{11} - \frac{2111050407284085872619372754620}{22296956500607539085182810116491} a^{10} - \frac{7281915473110410969019449953161}{22296956500607539085182810116491} a^{9} + \frac{2468833028594095731433376401813}{22296956500607539085182810116491} a^{8} + \frac{10395689246422273152992012477380}{22296956500607539085182810116491} a^{7} + \frac{746995761465498502089647526032}{22296956500607539085182810116491} a^{6} + \frac{9344699878355052064280214815695}{22296956500607539085182810116491} a^{5} - \frac{8076986963633161119765448648000}{22296956500607539085182810116491} a^{4} + \frac{8153417425084359606683605382055}{22296956500607539085182810116491} a^{3} - \frac{7426680465401776395929430314527}{22296956500607539085182810116491} a^{2} + \frac{324774074427902793573582409361}{22296956500607539085182810116491} a + \frac{10292068366952025491536998214264}{22296956500607539085182810116491}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 5141408.95892 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 5141408.95892 \cdot 1}{2\sqrt{57681033264163530732453953}}\approx 0.135050151997$ (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
17Data not computed
$67$67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$