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Properties

Label 16.16.4281965657...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 5^{8}\cdot 761^{5}$
Root discriminant $71.12$
Ramified primes $2, 5, 761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1163

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![214631, -1743672, 1288520, 6320648, 1688476, -3434480, -1418582, 749840, 375204, -76548, -46258, 3552, 2863, -60, -86, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 86*x^14 - 60*x^13 + 2863*x^12 + 3552*x^11 - 46258*x^10 - 76548*x^9 + 375204*x^8 + 749840*x^7 - 1418582*x^6 - 3434480*x^5 + 1688476*x^4 + 6320648*x^3 + 1288520*x^2 - 1743672*x + 214631)
 
gp: K = bnfinit(x^16 - 86*x^14 - 60*x^13 + 2863*x^12 + 3552*x^11 - 46258*x^10 - 76548*x^9 + 375204*x^8 + 749840*x^7 - 1418582*x^6 - 3434480*x^5 + 1688476*x^4 + 6320648*x^3 + 1288520*x^2 - 1743672*x + 214631, 1)
 

Normalized defining polynomial

\( x^{16} - 86 x^{14} - 60 x^{13} + 2863 x^{12} + 3552 x^{11} - 46258 x^{10} - 76548 x^{9} + 375204 x^{8} + 749840 x^{7} - 1418582 x^{6} - 3434480 x^{5} + 1688476 x^{4} + 6320648 x^{3} + 1288520 x^{2} - 1743672 x + 214631 \)

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:  \(428196565723310037401600000000=2^{32}\cdot 5^{8}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.12$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2642953} a^{14} - \frac{1318125}{2642953} a^{13} + \frac{399060}{2642953} a^{12} + \frac{22270}{114911} a^{11} + \frac{954370}{2642953} a^{10} + \frac{520402}{2642953} a^{9} - \frac{667498}{2642953} a^{8} - \frac{884145}{2642953} a^{7} - \frac{1066568}{2642953} a^{6} + \frac{1190042}{2642953} a^{5} + \frac{870740}{2642953} a^{4} + \frac{43411}{114911} a^{3} + \frac{387838}{2642953} a^{2} - \frac{1174762}{2642953} a - \frac{401750}{2642953}$, $\frac{1}{113084623095490191159314040894483} a^{15} + \frac{682689729039214423640402}{113084623095490191159314040894483} a^{14} - \frac{4461902395807874345654414277473}{10280420281408199196301276444953} a^{13} - \frac{12497884309800376478088862105610}{113084623095490191159314040894483} a^{12} + \frac{14381264129748730270520403036005}{37694874365163397053104680298161} a^{11} + \frac{247063335808220550504621031456}{1638907581094060741439333926007} a^{10} + \frac{34125532137878695227225562770617}{113084623095490191159314040894483} a^{9} + \frac{4768972197269735942269278640016}{10280420281408199196301276444953} a^{8} + \frac{6698961894828523504389200264087}{113084623095490191159314040894483} a^{7} + \frac{14438482831073325494698436418370}{37694874365163397053104680298161} a^{6} + \frac{53758503976680855142805790383035}{113084623095490191159314040894483} a^{5} + \frac{10944268608850850705221682954018}{37694874365163397053104680298161} a^{4} - \frac{16498605800767394042339473749848}{113084623095490191159314040894483} a^{3} + \frac{7637306919086326348665346220005}{113084623095490191159314040894483} a^{2} + \frac{4430435126944529912151332690234}{10280420281408199196301276444953} a + \frac{9151843191103678321955902605428}{113084623095490191159314040894483}$

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:  \( 4161585918.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1163:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 52 conjugacy class representatives for t16n1163 are not computed
Character table for t16n1163 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
761Data not computed