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Properties

Label 16.8.72504279208...1856.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{66}\cdot 7^{6}\cdot 17^{4}$
Root discriminant $73.50$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14353967, 19567936, 9936896, 2369616, -625800, -426976, 227816, 33296, -89178, -23632, 3736, 2512, 544, -64, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 - 64*x^13 + 544*x^12 + 2512*x^11 + 3736*x^10 - 23632*x^9 - 89178*x^8 + 33296*x^7 + 227816*x^6 - 426976*x^5 - 625800*x^4 + 2369616*x^3 + 9936896*x^2 + 19567936*x + 14353967)
 
gp: K = bnfinit(x^16 - 48*x^14 - 64*x^13 + 544*x^12 + 2512*x^11 + 3736*x^10 - 23632*x^9 - 89178*x^8 + 33296*x^7 + 227816*x^6 - 426976*x^5 - 625800*x^4 + 2369616*x^3 + 9936896*x^2 + 19567936*x + 14353967, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{14} - 64 x^{13} + 544 x^{12} + 2512 x^{11} + 3736 x^{10} - 23632 x^{9} - 89178 x^{8} + 33296 x^{7} + 227816 x^{6} - 426976 x^{5} - 625800 x^{4} + 2369616 x^{3} + 9936896 x^{2} + 19567936 x + 14353967 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(725042792081759922538839801856=2^{66}\cdot 7^{6}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.50$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{39403322054108486826587617787535304152662356764835} a^{15} + \frac{3643368067685982963393905443193824153715790500929}{39403322054108486826587617787535304152662356764835} a^{14} - \frac{8995061441978461046481412513962168383774033273993}{39403322054108486826587617787535304152662356764835} a^{13} - \frac{13206555031333955589054604899384047612782219045528}{39403322054108486826587617787535304152662356764835} a^{12} + \frac{12086680967223646479060999978373693927762615689949}{39403322054108486826587617787535304152662356764835} a^{11} - \frac{216802590256153679665678930884222870846572925543}{39403322054108486826587617787535304152662356764835} a^{10} + \frac{17176952513447856660243846169186075146678172666187}{39403322054108486826587617787535304152662356764835} a^{9} + \frac{6817390637998579509284169945095089690584369790613}{39403322054108486826587617787535304152662356764835} a^{8} + \frac{5001947417834713665992160068433860160521165995268}{39403322054108486826587617787535304152662356764835} a^{7} + \frac{11862331139946639825914937806185726969795435969367}{39403322054108486826587617787535304152662356764835} a^{6} + \frac{100393794200223252155652570241032783145607857673}{39403322054108486826587617787535304152662356764835} a^{5} + \frac{4451226975444927751206699367127182853730011811786}{39403322054108486826587617787535304152662356764835} a^{4} - \frac{16464224803280554200920488311696621525613878339909}{39403322054108486826587617787535304152662356764835} a^{3} + \frac{2916786745526093894960304815272068338017277508576}{7880664410821697365317523557507060830532471352967} a^{2} - \frac{1190077350814508914725750594383988175768994798560}{7880664410821697365317523557507060830532471352967} a + \frac{6717045428162881880322449986389145892542916298608}{39403322054108486826587617787535304152662356764835}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 773772280.323 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14336.1, \(\Q(\zeta_{16})^+\), 4.4.7168.1, 8.8.3288334336.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$