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Properties

Label 16.16.1587505208...1984.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 41^{6}\cdot 1249^{5}$
Root discriminant $211.37$
Ramified primes $2, 41, 1249$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3275329739569, 0, -3283196824612, 0, 551265806762, 0, -35507044188, 0, 1070625863, 0, -16735612, 0, 139762, 0, -592, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 592*x^14 + 139762*x^12 - 16735612*x^10 + 1070625863*x^8 - 35507044188*x^6 + 551265806762*x^4 - 3283196824612*x^2 + 3275329739569)
 
gp: K = bnfinit(x^16 - 592*x^14 + 139762*x^12 - 16735612*x^10 + 1070625863*x^8 - 35507044188*x^6 + 551265806762*x^4 - 3283196824612*x^2 + 3275329739569, 1)
 

Normalized defining polynomial

\( x^{16} - 592 x^{14} + 139762 x^{12} - 16735612 x^{10} + 1070625863 x^{8} - 35507044188 x^{6} + 551265806762 x^{4} - 3283196824612 x^{2} + 3275329739569 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15875052080993898958961521972146601984=2^{40}\cdot 41^{6}\cdot 1249^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $211.37$
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:  $2, 41, 1249$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{204836} a^{12} - \frac{148}{51209} a^{10} - \frac{13865}{204836} a^{8} + \frac{15235}{51209} a^{6} - \frac{51909}{204836} a^{4} - \frac{671}{2498} a^{2} - \frac{1}{2}$, $\frac{1}{204836} a^{13} - \frac{148}{51209} a^{11} - \frac{13865}{204836} a^{9} + \frac{15235}{51209} a^{7} - \frac{51909}{204836} a^{5} - \frac{671}{2498} a^{3} - \frac{1}{2} a$, $\frac{1}{5123421006462342173170815627488418988} a^{14} + \frac{2711003856575835830523810270755}{2561710503231171086585407813744209494} a^{12} + \frac{330145111643876293457857433450148889}{5123421006462342173170815627488418988} a^{10} - \frac{79855652183779612566580352959326085}{1280855251615585543292703906872104747} a^{8} + \frac{2149854799882323533676193325407882501}{5123421006462342173170815627488418988} a^{6} + \frac{718118501587441366055562852891393743}{2561710503231171086585407813744209494} a^{4} - \frac{3960714679204972090783317527705}{25012307438449990105112458881683} a^{2} - \frac{2848277038811441190932066059}{40051733288150504571837404134}$, $\frac{1}{5123421006462342173170815627488418988} a^{15} + \frac{2711003856575835830523810270755}{2561710503231171086585407813744209494} a^{13} + \frac{330145111643876293457857433450148889}{5123421006462342173170815627488418988} a^{11} - \frac{79855652183779612566580352959326085}{1280855251615585543292703906872104747} a^{9} + \frac{2149854799882323533676193325407882501}{5123421006462342173170815627488418988} a^{7} + \frac{718118501587441366055562852891393743}{2561710503231171086585407813744209494} a^{5} - \frac{3960714679204972090783317527705}{25012307438449990105112458881683} a^{3} - \frac{2848277038811441190932066059}{40051733288150504571837404134} a$

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:  $15$
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:  \( 19611756044400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1262
Character table for t16n1262 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.8599834624.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.20.7$x^{8} + 72 x^{4} + 144$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.20.7$x^{8} + 72 x^{4} + 144$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
1249Data not computed