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Properties

Label 16.0.16627389542...3824.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 17^{2}\cdot 103^{4}\cdot 3671^{4}$
Root discriminant $282.68$
Ramified primes $2, 17, 103, 3671$
Class number $58788$ (GRH)
Class group $[3, 19596]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28900000000, 0, -6596000000, 0, 980020000, 0, -69044400, 0, 2946513, 0, -118900, 0, 4502, 0, -100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 100*x^14 + 4502*x^12 - 118900*x^10 + 2946513*x^8 - 69044400*x^6 + 980020000*x^4 - 6596000000*x^2 + 28900000000)
 
gp: K = bnfinit(x^16 - 100*x^14 + 4502*x^12 - 118900*x^10 + 2946513*x^8 - 69044400*x^6 + 980020000*x^4 - 6596000000*x^2 + 28900000000, 1)
 

Normalized defining polynomial

\( x^{16} - 100 x^{14} + 4502 x^{12} - 118900 x^{10} + 2946513 x^{8} - 69044400 x^{6} + 980020000 x^{4} - 6596000000 x^{2} + 28900000000 \)

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:  \(1662738954297838893504019372193112653824=2^{48}\cdot 17^{2}\cdot 103^{4}\cdot 3671^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.68$
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:  $2, 17, 103, 3671$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} - \frac{1}{8} a^{5} - \frac{7}{128} a^{4} + \frac{1}{8} a^{3} + \frac{1}{16} a^{2} - \frac{1}{8}$, $\frac{1}{640} a^{9} + \frac{3}{64} a^{7} - \frac{23}{640} a^{5} + \frac{3}{16} a^{3} - \frac{17}{40} a$, $\frac{1}{134400} a^{10} - \frac{1}{1280} a^{9} + \frac{1}{672} a^{8} + \frac{5}{128} a^{7} - \frac{5323}{134400} a^{6} - \frac{137}{1280} a^{5} - \frac{319}{2688} a^{4} - \frac{1}{32} a^{3} + \frac{517}{2100} a^{2} - \frac{23}{80} a - \frac{13}{168}$, $\frac{1}{672000} a^{11} - \frac{13}{26880} a^{9} + \frac{4127}{672000} a^{7} + \frac{1861}{26880} a^{5} + \frac{3611}{84000} a^{3} - \frac{173}{1680} a$, $\frac{1}{27074880000} a^{12} - \frac{1}{360998400} a^{10} - \frac{38459873}{27074880000} a^{8} - \frac{19483843}{360998400} a^{6} - \frac{1}{8} a^{5} + \frac{75445121}{966960000} a^{4} + \frac{1}{8} a^{3} - \frac{1487293}{67687200} a^{2} + \frac{11}{39816}$, $\frac{1}{270748800000} a^{13} - \frac{1}{54149760000} a^{12} - \frac{1}{3609984000} a^{11} + \frac{1}{721996800} a^{10} - \frac{38459873}{270748800000} a^{9} + \frac{38459873}{54149760000} a^{8} - \frac{199983043}{3609984000} a^{7} - \frac{25640957}{721996800} a^{6} + \frac{1163275121}{9669600000} a^{5} + \frac{45424879}{1933920000} a^{4} + \frac{57739007}{676872000} a^{3} + \frac{1487293}{135374400} a^{2} - \frac{19897}{398160} a + \frac{39805}{79632}$, $\frac{1}{190877904000000} a^{14} - \frac{37}{3817558080000} a^{12} - \frac{130451687}{47719476000000} a^{10} + \frac{2008136927}{3817558080000} a^{8} - \frac{1}{16} a^{7} - \frac{10097384857237}{190877904000000} a^{6} - \frac{1}{8} a^{5} + \frac{186646405891}{1908779040000} a^{4} - \frac{1}{16} a^{3} - \frac{536854019}{6362596800} a^{2} + \frac{1}{4} a - \frac{4084765}{11228112}$, $\frac{1}{1908779040000000} a^{15} - \frac{37}{38175580800000} a^{13} - \frac{1}{54149760000} a^{12} - \frac{130451687}{477194760000000} a^{11} + \frac{1}{721996800} a^{10} + \frac{2008136927}{38175580800000} a^{9} + \frac{38459873}{54149760000} a^{8} + \frac{13762353142763}{1908779040000000} a^{7} - \frac{25640957}{721996800} a^{6} + \frac{1141035925891}{19087790400000} a^{5} - \frac{196315121}{1933920000} a^{4} - \frac{1332178619}{63625968000} a^{3} + \frac{18409093}{135374400} a^{2} + \frac{1529291}{112281120} a - \frac{11}{79632}$

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

Class group and class number

$C_{3}\times C_{19596}$, which has order $58788$ (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:  \( -1 \) (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:  \( 479437680.102 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.0.24199232.1, 4.0.774375424.1, 8.0.599657297295179776.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3671Data not computed