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Properties

Label 16.16.6031518235...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{38}\cdot 3^{4}\cdot 5^{8}\cdot 71^{2}\cdot 89^{4}\cdot 4682551^{2}$
Root discriminant $544.85$
Ramified primes $2, 3, 5, 71, 89, 4682551$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35811848620410831684, 0, -1932548200410874572, 0, 37355497726908528, 0, -336361698262014, 0, 1586607526759, 0, -4064943508, 0, 5554757, 0, -3776, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3776*x^14 + 5554757*x^12 - 4064943508*x^10 + 1586607526759*x^8 - 336361698262014*x^6 + 37355497726908528*x^4 - 1932548200410874572*x^2 + 35811848620410831684)
 
gp: K = bnfinit(x^16 - 3776*x^14 + 5554757*x^12 - 4064943508*x^10 + 1586607526759*x^8 - 336361698262014*x^6 + 37355497726908528*x^4 - 1932548200410874572*x^2 + 35811848620410831684, 1)
 

Normalized defining polynomial

\( x^{16} - 3776 x^{14} + 5554757 x^{12} - 4064943508 x^{10} + 1586607526759 x^{8} - 336361698262014 x^{6} + 37355497726908528 x^{4} - 1932548200410874572 x^{2} + 35811848620410831684 \)

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:  \(60315182355722271054119173521565286400000000=2^{38}\cdot 3^{4}\cdot 5^{8}\cdot 71^{2}\cdot 89^{4}\cdot 4682551^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $544.85$
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:  $2, 3, 5, 71, 89, 4682551$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{2}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{7} - \frac{1}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{342} a^{12} - \frac{1}{57} a^{10} + \frac{5}{342} a^{8} - \frac{1}{9} a^{6} - \frac{97}{342} a^{4} + \frac{16}{57} a^{2} + \frac{2}{19}$, $\frac{1}{342} a^{13} - \frac{1}{57} a^{11} + \frac{5}{342} a^{9} - \frac{1}{9} a^{7} - \frac{97}{342} a^{5} + \frac{16}{57} a^{3} + \frac{2}{19} a$, $\frac{1}{6641158299457770598224464412420954391185715035410160946248228} a^{14} + \frac{3725927737940715493440321095212879009473232173872315537747}{3320579149728885299112232206210477195592857517705080473124114} a^{12} + \frac{88227266838900313514606467470136278464011584562033550260415}{6641158299457770598224464412420954391185715035410160946248228} a^{10} - \frac{123709930447362446329755944409113519133333287031591725548235}{3320579149728885299112232206210477195592857517705080473124114} a^{8} + \frac{669733314629788398250775806813987253899843255594554906444829}{6641158299457770598224464412420954391185715035410160946248228} a^{6} - \frac{95261270734815406671146321062938014701903884953809022342822}{553429858288147549852038701035079532598809586284180078854019} a^{4} + \frac{11051635103200120216464540346923758183443058341775067972255}{122984412952921677744897489118906562799735463618706684189782} a^{2} + \frac{4821739351004453665550421077909640395464860605311}{184960594163613010474234503226178081135290799339371}$, $\frac{1}{6641158299457770598224464412420954391185715035410160946248228} a^{15} + \frac{3725927737940715493440321095212879009473232173872315537747}{3320579149728885299112232206210477195592857517705080473124114} a^{13} + \frac{88227266838900313514606467470136278464011584562033550260415}{6641158299457770598224464412420954391185715035410160946248228} a^{11} - \frac{123709930447362446329755944409113519133333287031591725548235}{3320579149728885299112232206210477195592857517705080473124114} a^{9} + \frac{669733314629788398250775806813987253899843255594554906444829}{6641158299457770598224464412420954391185715035410160946248228} a^{7} - \frac{95261270734815406671146321062938014701903884953809022342822}{553429858288147549852038701035079532598809586284180078854019} a^{5} + \frac{11051635103200120216464540346923758183443058341775067972255}{122984412952921677744897489118906562799735463618706684189782} a^{3} + \frac{4821739351004453665550421077909640395464860605311}{184960594163613010474234503226178081135290799339371} a$

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:  \( 65997855869900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.18.48$x^{8} + 6 x^{6} + 2 x^{4} + 12$$4$$2$$18$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.20.42$x^{8} + 16 x^{7} + 8 x^{6} + 240$$4$$2$$20$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$71$71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
4682551Data not computed