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Properties

Label 16.16.4361204616...3616.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{72}\cdot 31^{4}$
Root discriminant $53.39$
Ramified primes $2, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![317246, 1294816, 1514112, -353952, -1987144, -1213184, 205760, 435712, 76224, -56016, -18528, 3120, 1608, -64, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 64*x^14 - 64*x^13 + 1608*x^12 + 3120*x^11 - 18528*x^10 - 56016*x^9 + 76224*x^8 + 435712*x^7 + 205760*x^6 - 1213184*x^5 - 1987144*x^4 - 353952*x^3 + 1514112*x^2 + 1294816*x + 317246)
 
gp: K = bnfinit(x^16 - 64*x^14 - 64*x^13 + 1608*x^12 + 3120*x^11 - 18528*x^10 - 56016*x^9 + 76224*x^8 + 435712*x^7 + 205760*x^6 - 1213184*x^5 - 1987144*x^4 - 353952*x^3 + 1514112*x^2 + 1294816*x + 317246, 1)
 

Normalized defining polynomial

\( x^{16} - 64 x^{14} - 64 x^{13} + 1608 x^{12} + 3120 x^{11} - 18528 x^{10} - 56016 x^{9} + 76224 x^{8} + 435712 x^{7} + 205760 x^{6} - 1213184 x^{5} - 1987144 x^{4} - 353952 x^{3} + 1514112 x^{2} + 1294816 x + 317246 \)

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:  \(4361204616626257617397743616=2^{72}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.39$
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:  $2, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{589151811828997094365409} a^{15} - \frac{97443625989833228980440}{589151811828997094365409} a^{14} - \frac{213087662293187434891123}{589151811828997094365409} a^{13} - \frac{154446328920901560119358}{589151811828997094365409} a^{12} - \frac{250103331068302736082503}{589151811828997094365409} a^{11} + \frac{146321568912362984831072}{589151811828997094365409} a^{10} - \frac{276192450232358716876617}{589151811828997094365409} a^{9} + \frac{31861874019814541590438}{589151811828997094365409} a^{8} - \frac{104448445306835556620587}{589151811828997094365409} a^{7} + \frac{229392976848002640637604}{589151811828997094365409} a^{6} - \frac{42243866889798753529795}{589151811828997094365409} a^{5} - \frac{236848383741938810336769}{589151811828997094365409} a^{4} + \frac{112780986169484637436460}{589151811828997094365409} a^{3} - \frac{215410789054531992318054}{589151811828997094365409} a^{2} + \frac{192813616164218953989613}{589151811828997094365409} a + \frac{136473368392282932011334}{589151811828997094365409}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 693709557.517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
31Data not computed