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Properties

Label 16.16.2802086158...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}$
Root discriminant $79.98$
Ramified primes $2, 5, 101, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12845159, -6290462, 34410890, -11366298, -16400650, 9687290, 2128918, -2407142, 83602, 266166, -38738, -13934, 3058, 310, -94, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 94*x^14 + 310*x^13 + 3058*x^12 - 13934*x^11 - 38738*x^10 + 266166*x^9 + 83602*x^8 - 2407142*x^7 + 2128918*x^6 + 9687290*x^5 - 16400650*x^4 - 11366298*x^3 + 34410890*x^2 - 6290462*x - 12845159)
 
gp: K = bnfinit(x^16 - 2*x^15 - 94*x^14 + 310*x^13 + 3058*x^12 - 13934*x^11 - 38738*x^10 + 266166*x^9 + 83602*x^8 - 2407142*x^7 + 2128918*x^6 + 9687290*x^5 - 16400650*x^4 - 11366298*x^3 + 34410890*x^2 - 6290462*x - 12845159, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 94 x^{14} + 310 x^{13} + 3058 x^{12} - 13934 x^{11} - 38738 x^{10} + 266166 x^{9} + 83602 x^{8} - 2407142 x^{7} + 2128918 x^{6} + 9687290 x^{5} - 16400650 x^{4} - 11366298 x^{3} + 34410890 x^{2} - 6290462 x - 12845159 \)

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:  \(2802086158223343616000000000000=2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.98$
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:  $2, 5, 101, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{3}{10} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{31417084469625600470477390} a^{15} - \frac{117339651255887867464413}{15708542234812800235238695} a^{14} + \frac{336123040464012913692293}{31417084469625600470477390} a^{13} + \frac{735689289200125661722043}{31417084469625600470477390} a^{12} - \frac{6758773634766803435597517}{31417084469625600470477390} a^{11} - \frac{198309066949711958083647}{15708542234812800235238695} a^{10} - \frac{1091305266279458863046574}{15708542234812800235238695} a^{9} - \frac{5375063828982933786215051}{31417084469625600470477390} a^{8} + \frac{8937954176256722259885931}{31417084469625600470477390} a^{7} + \frac{488216652359985875015366}{1428049294073890930476245} a^{6} - \frac{19017309454758974264787}{571219717629556372190498} a^{5} - \frac{3748366591443523076557291}{31417084469625600470477390} a^{4} + \frac{324189740292769197654431}{6283416893925120094095478} a^{3} - \frac{4564874670025560229049418}{15708542234812800235238695} a^{2} + \frac{453059995192759958046719}{15708542234812800235238695} a - \frac{3148546785378717013508799}{31417084469625600470477390}$

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:  \( 14205869984.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

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 t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed
641Data not computed