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Properties

Label 16.0.52554919692...6256.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 17^{6}\cdot 89^{2}$
Root discriminant $26.31$
Ramified primes $2, 17, 89$
Class number $2$
Class group $[2]$
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5188, -3112, -572, -7392, 6418, -192, 2080, -2552, 230, 380, -214, 116, -3, -28, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 - 28*x^13 - 3*x^12 + 116*x^11 - 214*x^10 + 380*x^9 + 230*x^8 - 2552*x^7 + 2080*x^6 - 192*x^5 + 6418*x^4 - 7392*x^3 - 572*x^2 - 3112*x + 5188)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 - 28*x^13 - 3*x^12 + 116*x^11 - 214*x^10 + 380*x^9 + 230*x^8 - 2552*x^7 + 2080*x^6 - 192*x^5 + 6418*x^4 - 7392*x^3 - 572*x^2 - 3112*x + 5188, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} - 28 x^{13} - 3 x^{12} + 116 x^{11} - 214 x^{10} + 380 x^{9} + 230 x^{8} - 2552 x^{7} + 2080 x^{6} - 192 x^{5} + 6418 x^{4} - 7392 x^{3} - 572 x^{2} - 3112 x + 5188 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(52554919692301559136256=2^{38}\cdot 17^{6}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{26} a^{13} + \frac{3}{26} a^{12} - \frac{1}{13} a^{11} + \frac{2}{13} a^{10} - \frac{11}{26} a^{9} - \frac{1}{2} a^{8} + \frac{4}{13} a^{7} + \frac{5}{13} a^{6} - \frac{3}{13} a^{5} + \frac{3}{13} a^{4} - \frac{3}{13} a^{3} + \frac{1}{13} a^{2} + \frac{4}{13} a + \frac{4}{13}$, $\frac{1}{52} a^{14} - \frac{11}{52} a^{12} + \frac{5}{26} a^{11} - \frac{5}{26} a^{10} - \frac{3}{26} a^{9} - \frac{9}{26} a^{8} + \frac{3}{13} a^{7} - \frac{5}{26} a^{6} + \frac{6}{13} a^{5} + \frac{1}{26} a^{4} + \frac{5}{13} a^{3} - \frac{6}{13} a^{2} - \frac{4}{13} a - \frac{6}{13}$, $\frac{1}{1078856133818746065467464} a^{15} + \frac{531454219810216111891}{56781901779934003445656} a^{14} - \frac{5211896432823320055513}{1078856133818746065467464} a^{13} - \frac{85659753797665291964809}{1078856133818746065467464} a^{12} - \frac{53087622351721852343803}{539428066909373032733732} a^{11} - \frac{34404300506272170595085}{539428066909373032733732} a^{10} + \frac{2087673992640575371057}{14195475444983500861414} a^{9} - \frac{94160626892792468738}{7097737722491750430707} a^{8} - \frac{7498558726452741209733}{41494466685336387133364} a^{7} - \frac{131806414633487692205661}{539428066909373032733732} a^{6} + \frac{183838063770363895790913}{539428066909373032733732} a^{5} - \frac{5189751953286611728463}{539428066909373032733732} a^{4} + \frac{31278855050192590579977}{134857016727343258183433} a^{3} + \frac{4528479480650764556675}{10373616671334096783341} a^{2} + \frac{899182508134183384531}{20747233342668193566682} a + \frac{134339410160101047403821}{269714033454686516366866}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{3164872700583602375}{904321989789393181448} a^{15} + \frac{426617025850676003}{47595894199441746392} a^{14} - \frac{25748774435143957713}{904321989789393181448} a^{13} + \frac{48985781924817174107}{904321989789393181448} a^{12} + \frac{43771722361302480115}{452160994894696590724} a^{11} - \frac{129681606788970476699}{452160994894696590724} a^{10} + \frac{2005803451071654709}{5949486774930218299} a^{9} - \frac{4512528418528732965}{5949486774930218299} a^{8} - \frac{936238915884145374929}{452160994894696590724} a^{7} + \frac{2814463758292325072291}{452160994894696590724} a^{6} + \frac{832144783025781262169}{452160994894696590724} a^{5} + \frac{628902688467029410685}{452160994894696590724} a^{4} - \frac{2079867121207669237062}{113040248723674147681} a^{3} - \frac{67577857290306446145}{113040248723674147681} a^{2} + \frac{791551948616480110757}{226080497447348295362} a + \frac{2723938291671262742839}{226080497447348295362} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 537572.189549 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), 4.0.1088.2, 4.4.4352.1, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.22.90$x^{8} + 4 x^{7} + 14 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$