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Properties

Label 16.0.65534439047...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 5^{8}\cdot 61^{2}$
Root discriminant $30.80$
Ramified primes $2, 3, 5, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![340, -1320, 2636, -3488, 3858, -3936, 3658, -2616, 1153, 4, -336, 192, -22, -20, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 20*x^13 - 22*x^12 + 192*x^11 - 336*x^10 + 4*x^9 + 1153*x^8 - 2616*x^7 + 3658*x^6 - 3936*x^5 + 3858*x^4 - 3488*x^3 + 2636*x^2 - 1320*x + 340)
 
gp: K = bnfinit(x^16 + 2*x^14 - 20*x^13 - 22*x^12 + 192*x^11 - 336*x^10 + 4*x^9 + 1153*x^8 - 2616*x^7 + 3658*x^6 - 3936*x^5 + 3858*x^4 - 3488*x^3 + 2636*x^2 - 1320*x + 340, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 20 x^{13} - 22 x^{12} + 192 x^{11} - 336 x^{10} + 4 x^{9} + 1153 x^{8} - 2616 x^{7} + 3658 x^{6} - 3936 x^{5} + 3858 x^{4} - 3488 x^{3} + 2636 x^{2} - 1320 x + 340 \)

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:  \(655344390478233600000000=2^{36}\cdot 3^{8}\cdot 5^{8}\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.80$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{1}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{79956114672762004270} a^{15} + \frac{478762271565103827}{79956114672762004270} a^{14} - \frac{4825488446618538217}{39978057336381002135} a^{13} + \frac{269543633324032649}{1378553701254517315} a^{12} - \frac{7528064755871048219}{39978057336381002135} a^{11} - \frac{9349404727211714412}{39978057336381002135} a^{10} - \frac{9439979410872477897}{39978057336381002135} a^{9} - \frac{9035250817295874822}{39978057336381002135} a^{8} - \frac{5570431200044075417}{15991222934552400854} a^{7} - \frac{39060364903243221861}{79956114672762004270} a^{6} - \frac{614788456293315391}{2351650431551823655} a^{5} + \frac{13105083235120638688}{39978057336381002135} a^{4} - \frac{593345067626889390}{7995611467276200427} a^{3} - \frac{17975065026308218529}{39978057336381002135} a^{2} + \frac{1839534391528574028}{7995611467276200427} a - \frac{137946352826270280}{470330086310364731}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{42539483196}{683479237991} a^{15} + \frac{33758596605}{1366958475982} a^{14} + \frac{77575951383}{683479237991} a^{13} - \frac{840778623633}{683479237991} a^{12} - \frac{1320497994982}{683479237991} a^{11} + \frac{7865616628970}{683479237991} a^{10} - \frac{10502231997864}{683479237991} a^{9} - \frac{5854308809947}{683479237991} a^{8} + \frac{48399746623034}{683479237991} a^{7} - \frac{177596309167219}{1366958475982} a^{6} + \frac{107814222830557}{683479237991} a^{5} - \frac{107396425896336}{683479237991} a^{4} + \frac{101706468954364}{683479237991} a^{3} - \frac{90570711419842}{683479237991} a^{2} + \frac{58043819340010}{683479237991} a - \frac{19139600929743}{683479237991} \) (order $4$)
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:  \( 705203.75617 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), 4.0.11520.1, 4.0.1280.1, \(\Q(i, \sqrt{15})\), 8.0.3317760000.11

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.20.75$x^{8} + 8 x^{6} + 16 x^{5} + 80$$8$$1$$20$$C_2^3 : C_4 $$[2, 3, 3]^{2}$
2.8.16.4$x^{8} + 6 x^{6} + 6 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 20$$4$$2$$16$$D_4$$[2, 3]^{2}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed