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Properties

Label 16.0.24073289246...0624.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 17^{4}$
Root discriminant $25.05$
Ramified primes $2, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3592, -14912, 26080, -28736, 27504, -22432, 13072, -6912, 4188, -1904, 768, -384, 108, -32, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 32*x^13 + 108*x^12 - 384*x^11 + 768*x^10 - 1904*x^9 + 4188*x^8 - 6912*x^7 + 13072*x^6 - 22432*x^5 + 27504*x^4 - 28736*x^3 + 26080*x^2 - 14912*x + 3592)
 
gp: K = bnfinit(x^16 + 16*x^14 - 32*x^13 + 108*x^12 - 384*x^11 + 768*x^10 - 1904*x^9 + 4188*x^8 - 6912*x^7 + 13072*x^6 - 22432*x^5 + 27504*x^4 - 28736*x^3 + 26080*x^2 - 14912*x + 3592, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 32 x^{13} + 108 x^{12} - 384 x^{11} + 768 x^{10} - 1904 x^{9} + 4188 x^{8} - 6912 x^{7} + 13072 x^{6} - 22432 x^{5} + 27504 x^{4} - 28736 x^{3} + 26080 x^{2} - 14912 x + 3592 \)

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:  \(24073289246567116570624=2^{58}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.05$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{68} a^{13} - \frac{7}{68} a^{12} - \frac{5}{68} a^{11} - \frac{5}{34} a^{10} - \frac{7}{34} a^{9} - \frac{3}{17} a^{8} - \frac{3}{34} a^{7} + \frac{1}{34} a^{6} - \frac{3}{17} a^{5} + \frac{7}{17} a^{4} + \frac{7}{17} a^{3} + \frac{3}{17} a^{2} + \frac{5}{17} a - \frac{2}{17}$, $\frac{1}{1292} a^{14} - \frac{1}{323} a^{13} + \frac{2}{323} a^{12} - \frac{25}{1292} a^{11} + \frac{23}{323} a^{10} - \frac{5}{34} a^{9} - \frac{55}{646} a^{8} + \frac{9}{646} a^{7} - \frac{78}{323} a^{6} + \frac{15}{323} a^{5} + \frac{45}{323} a^{4} + \frac{160}{323} a^{3} - \frac{88}{323} a^{2} - \frac{89}{323} a + \frac{96}{323}$, $\frac{1}{77577281778071208436} a^{15} + \frac{5855751890608114}{19394320444517802109} a^{14} - \frac{207095546092725881}{38788640889035604218} a^{13} - \frac{2985117633818389037}{38788640889035604218} a^{12} - \frac{6111493124641897681}{77577281778071208436} a^{11} - \frac{1460524320537065}{5972996749158547} a^{10} + \frac{9036811811200857861}{38788640889035604218} a^{9} + \frac{1569683637216196717}{19394320444517802109} a^{8} + \frac{117742847333659875}{2041507415212400222} a^{7} + \frac{6570893409821779421}{38788640889035604218} a^{6} + \frac{1070599008642917616}{19394320444517802109} a^{5} + \frac{7300711893980079087}{19394320444517802109} a^{4} + \frac{7733274936096850}{60044335741541183} a^{3} - \frac{445146406492426493}{19394320444517802109} a^{2} + \frac{1515620754755217394}{19394320444517802109} a - \frac{5265341465078828215}{19394320444517802109}$

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:  \( \frac{854619493220024581}{77577281778071208436} a^{15} + \frac{472494107532292263}{38788640889035604218} a^{14} + \frac{3671838094942320736}{19394320444517802109} a^{13} - \frac{11141618271468825555}{77577281778071208436} a^{12} + \frac{39711170682474053581}{38788640889035604218} a^{11} - \frac{314108224470853900}{101540944735695299} a^{10} + \frac{193859133589933353715}{38788640889035604218} a^{9} - \frac{297472253639090538992}{19394320444517802109} a^{8} + \frac{561891619779912564228}{19394320444517802109} a^{7} - \frac{1688174679371675552033}{38788640889035604218} a^{6} + \frac{1838439844780301464174}{19394320444517802109} a^{5} - \frac{2727225454557692582988}{19394320444517802109} a^{4} + \frac{2791826873976322372216}{19394320444517802109} a^{3} - \frac{155532611004384363014}{1020753707606200111} a^{2} + \frac{2203030704246453118558}{19394320444517802109} a - \frac{649353253793129842311}{19394320444517802109} \) (order $16$)
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:  \( 233059.010847 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$