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Properties

Label 16.16.8110145691...1616.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{56}\cdot 103^{4}$
Root discriminant $36.04$
Ramified primes $2, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1027

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 240, 80, -3600, -11424, -12280, 568, 10224, 4642, -2672, -2064, 272, 376, -8, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 8*x^13 + 376*x^12 + 272*x^11 - 2064*x^10 - 2672*x^9 + 4642*x^8 + 10224*x^7 + 568*x^6 - 12280*x^5 - 11424*x^4 - 3600*x^3 + 80*x^2 + 240*x + 31)
 
gp: K = bnfinit(x^16 - 32*x^14 - 8*x^13 + 376*x^12 + 272*x^11 - 2064*x^10 - 2672*x^9 + 4642*x^8 + 10224*x^7 + 568*x^6 - 12280*x^5 - 11424*x^4 - 3600*x^3 + 80*x^2 + 240*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} - 8 x^{13} + 376 x^{12} + 272 x^{11} - 2064 x^{10} - 2672 x^{9} + 4642 x^{8} + 10224 x^{7} + 568 x^{6} - 12280 x^{5} - 11424 x^{4} - 3600 x^{3} + 80 x^{2} + 240 x + 31 \)

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:  \(8110145691709136611311616=2^{56}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.04$
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:  $2, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{96084017609} a^{15} + \frac{44326403450}{96084017609} a^{14} + \frac{15494869842}{96084017609} a^{13} + \frac{767377967}{3099484439} a^{12} - \frac{39155943911}{96084017609} a^{11} - \frac{42227516921}{96084017609} a^{10} + \frac{26044671169}{96084017609} a^{9} - \frac{47621307796}{96084017609} a^{8} - \frac{9302183766}{96084017609} a^{7} - \frac{38278757459}{96084017609} a^{6} + \frac{20315657138}{96084017609} a^{5} - \frac{43155284697}{96084017609} a^{4} - \frac{40565598987}{96084017609} a^{3} + \frac{27076224594}{96084017609} a^{2} + \frac{32678795856}{96084017609} a - \frac{1353413845}{3099484439}$

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:  \( 30670821.583 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1027:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 28 conjugacy class representatives for t16n1027
Character table for t16n1027 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.177989484544.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$103$103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.6.4.1$x^{6} + 1442 x^{3} + 1326125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$