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Properties

Label 16.0.16777216000...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}$
Root discriminant $15.91$
Ramified primes $2, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8:C_2^2$ (as 16T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 28, 32, -32, 328, 634, -1204, 1963, -1812, 1426, -820, 398, -144, 42, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 42*x^14 - 144*x^13 + 398*x^12 - 820*x^11 + 1426*x^10 - 1812*x^9 + 1963*x^8 - 1204*x^7 + 634*x^6 + 328*x^5 - 32*x^4 + 32*x^3 + 28*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 42*x^14 - 144*x^13 + 398*x^12 - 820*x^11 + 1426*x^10 - 1812*x^9 + 1963*x^8 - 1204*x^7 + 634*x^6 + 328*x^5 - 32*x^4 + 32*x^3 + 28*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 42 x^{14} - 144 x^{13} + 398 x^{12} - 820 x^{11} + 1426 x^{10} - 1812 x^{9} + 1963 x^{8} - 1204 x^{7} + 634 x^{6} + 328 x^{5} - 32 x^{4} + 32 x^{3} + 28 x^{2} + 8 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16777216000000000000=2^{36}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.91$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{116857} a^{14} + \frac{53033}{116857} a^{13} - \frac{52718}{116857} a^{12} - \frac{19992}{116857} a^{11} + \frac{49017}{116857} a^{10} + \frac{15361}{116857} a^{9} - \frac{55252}{116857} a^{8} + \frac{27015}{116857} a^{7} - \frac{56837}{116857} a^{6} + \frac{45986}{116857} a^{5} - \frac{53585}{116857} a^{4} + \frac{16623}{116857} a^{3} + \frac{42059}{116857} a^{2} - \frac{27658}{116857} a + \frac{50413}{116857}$, $\frac{1}{65335800413} a^{15} - \frac{249646}{65335800413} a^{14} - \frac{23877824558}{65335800413} a^{13} - \frac{87928738}{734110117} a^{12} - \frac{23435085511}{65335800413} a^{11} - \frac{17972141920}{65335800413} a^{10} - \frac{27703874330}{65335800413} a^{9} + \frac{7852681682}{65335800413} a^{8} + \frac{18971989360}{65335800413} a^{7} + \frac{31246117999}{65335800413} a^{6} - \frac{18989358452}{65335800413} a^{5} + \frac{4436262671}{65335800413} a^{4} - \frac{26786658648}{65335800413} a^{3} - \frac{26361772769}{65335800413} a^{2} - \frac{6454802537}{65335800413} a - \frac{1967063107}{5025830801}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2151902026}{5025830801} a^{15} + \frac{18102444780}{5025830801} a^{14} - \frac{97805667800}{5025830801} a^{13} + \frac{349836514580}{5025830801} a^{12} - \frac{998589508470}{5025830801} a^{11} + \frac{2168027543999}{5025830801} a^{10} - \frac{3937384081090}{5025830801} a^{9} + \frac{5463205208965}{5025830801} a^{8} - \frac{6361942701740}{5025830801} a^{7} + \frac{5037951870445}{5025830801} a^{6} - \frac{3224105485028}{5025830801} a^{5} + \frac{421599938165}{5025830801} a^{4} + \frac{27323013070}{5025830801} a^{3} - \frac{124991472095}{5025830801} a^{2} - \frac{21197006850}{5025830801} a - \frac{5130713880}{5025830801} \) (order $4$)
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:  \( 958.044559076 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.320.1 x2, 4.2.400.1 x2, \(\Q(i, \sqrt{5})\), 8.2.1024000000.1 x2, 8.2.1024000000.2 x2, 8.0.2560000.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$