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Properties

Label 16.0.41726830242...7856.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{74}\cdot 47^{2}$
Root discriminant $39.93$
Ramified primes $2, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1502, -4960, 6064, -7680, 11416, -9024, 5936, -3840, 1608, -480, 256, -80, 16, -16, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^13 + 16*x^12 - 80*x^11 + 256*x^10 - 480*x^9 + 1608*x^8 - 3840*x^7 + 5936*x^6 - 9024*x^5 + 11416*x^4 - 7680*x^3 + 6064*x^2 - 4960*x + 1502)
 
gp: K = bnfinit(x^16 - 16*x^13 + 16*x^12 - 80*x^11 + 256*x^10 - 480*x^9 + 1608*x^8 - 3840*x^7 + 5936*x^6 - 9024*x^5 + 11416*x^4 - 7680*x^3 + 6064*x^2 - 4960*x + 1502, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{13} + 16 x^{12} - 80 x^{11} + 256 x^{10} - 480 x^{9} + 1608 x^{8} - 3840 x^{7} + 5936 x^{6} - 9024 x^{5} + 11416 x^{4} - 7680 x^{3} + 6064 x^{2} - 4960 x + 1502 \)

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:  \(41726830242636185108217856=2^{74}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.93$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{799} a^{14} - \frac{380}{799} a^{13} + \frac{302}{799} a^{12} - \frac{347}{799} a^{11} - \frac{20}{799} a^{10} + \frac{116}{799} a^{9} + \frac{88}{799} a^{8} + \frac{149}{799} a^{7} - \frac{375}{799} a^{6} - \frac{240}{799} a^{5} + \frac{38}{799} a^{4} + \frac{110}{799} a^{3} - \frac{199}{799} a^{2} - \frac{193}{799} a - \frac{305}{799}$, $\frac{1}{10090552929838562303494111} a^{15} + \frac{3983102726024609215976}{10090552929838562303494111} a^{14} + \frac{4658782361484686184879218}{10090552929838562303494111} a^{13} + \frac{3371280388461913933157992}{10090552929838562303494111} a^{12} + \frac{3352622716273148372136781}{10090552929838562303494111} a^{11} - \frac{322514580127400247407407}{10090552929838562303494111} a^{10} + \frac{1856438014547487170985827}{10090552929838562303494111} a^{9} + \frac{3219601313414151589146080}{10090552929838562303494111} a^{8} + \frac{2898346107506048215235636}{10090552929838562303494111} a^{7} - \frac{2140850946247649562129644}{10090552929838562303494111} a^{6} + \frac{4207875281179010424870453}{10090552929838562303494111} a^{5} - \frac{2469991876956501610230949}{10090552929838562303494111} a^{4} + \frac{1635928528976544053277099}{10090552929838562303494111} a^{3} - \frac{2296829494800778189200534}{10090552929838562303494111} a^{2} - \frac{404454335754975368098475}{10090552929838562303494111} a - \frac{1457669394129856503793683}{10090552929838562303494111}$

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:  $7$
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:  \( 1871936.37324 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.0.2147483648.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
47Data not computed