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Properties

Label 16.16.8055289472...8125.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 101^{6}\cdot 181^{5}$
Root discriminant $64.07$
Ramified primes $5, 101, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![263509, 735684, -711570, -1387281, 783788, 984154, -431147, -341579, 129025, 62058, -21340, -5771, 1864, 243, -76, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 76*x^14 + 243*x^13 + 1864*x^12 - 5771*x^11 - 21340*x^10 + 62058*x^9 + 129025*x^8 - 341579*x^7 - 431147*x^6 + 984154*x^5 + 783788*x^4 - 1387281*x^3 - 711570*x^2 + 735684*x + 263509)
 
gp: K = bnfinit(x^16 - 3*x^15 - 76*x^14 + 243*x^13 + 1864*x^12 - 5771*x^11 - 21340*x^10 + 62058*x^9 + 129025*x^8 - 341579*x^7 - 431147*x^6 + 984154*x^5 + 783788*x^4 - 1387281*x^3 - 711570*x^2 + 735684*x + 263509, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 76 x^{14} + 243 x^{13} + 1864 x^{12} - 5771 x^{11} - 21340 x^{10} + 62058 x^{9} + 129025 x^{8} - 341579 x^{7} - 431147 x^{6} + 984154 x^{5} + 783788 x^{4} - 1387281 x^{3} - 711570 x^{2} + 735684 x + 263509 \)

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:  \(80552894728007447787305078125=5^{8}\cdot 101^{6}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.07$
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:  $5, 101, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{219046614576650677384450591860265057} a^{15} - \frac{23504917000919668417033704046895786}{219046614576650677384450591860265057} a^{14} - \frac{78379363265287248530064180616817409}{219046614576650677384450591860265057} a^{13} + \frac{51755900542200810503129653417323793}{219046614576650677384450591860265057} a^{12} + \frac{34393028000977184280733277895600403}{219046614576650677384450591860265057} a^{11} - \frac{6937222837194280803806188672352596}{219046614576650677384450591860265057} a^{10} - \frac{59288452807911942850661756309168412}{219046614576650677384450591860265057} a^{9} - \frac{4525315263611827199110479439010953}{219046614576650677384450591860265057} a^{8} + \frac{87413451824223537089361505183691382}{219046614576650677384450591860265057} a^{7} + \frac{82021864573680436817503508047902888}{219046614576650677384450591860265057} a^{6} - \frac{9431367108215436633333309217595398}{219046614576650677384450591860265057} a^{5} + \frac{59401474848579780474762307851133303}{219046614576650677384450591860265057} a^{4} + \frac{58482583907096549128816967689160531}{219046614576650677384450591860265057} a^{3} - \frac{97343021353032647521870357861949709}{219046614576650677384450591860265057} a^{2} - \frac{59632567059051294106794627544095450}{219046614576650677384450591860265057} a + \frac{55152226336513059662188647167284279}{219046614576650677384450591860265057}$

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:  \( 1702321140.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T994:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n994
Character table for t16n994 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.1153988125.1

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
Arithmetically equvalently 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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101Data not computed
181Data not computed