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Properties

Label 16.0.15045805779...5664.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 17^{4}$
Root discriminant $21.07$
Ramified primes $2, 17$
Class number $2$
Class group $[2]$
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![98, 224, -176, -640, 396, 1776, 840, -872, -558, 440, 296, -136, -84, 40, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 12*x^14 + 40*x^13 - 84*x^12 - 136*x^11 + 296*x^10 + 440*x^9 - 558*x^8 - 872*x^7 + 840*x^6 + 1776*x^5 + 396*x^4 - 640*x^3 - 176*x^2 + 224*x + 98)
 
gp: K = bnfinit(x^16 - 8*x^15 + 12*x^14 + 40*x^13 - 84*x^12 - 136*x^11 + 296*x^10 + 440*x^9 - 558*x^8 - 872*x^7 + 840*x^6 + 1776*x^5 + 396*x^4 - 640*x^3 - 176*x^2 + 224*x + 98, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 12 x^{14} + 40 x^{13} - 84 x^{12} - 136 x^{11} + 296 x^{10} + 440 x^{9} - 558 x^{8} - 872 x^{7} + 840 x^{6} + 1776 x^{5} + 396 x^{4} - 640 x^{3} - 176 x^{2} + 224 x + 98 \)

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:  \(1504580577910444785664=2^{54}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.07$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4381606552811447071} a^{15} + \frac{1673752422143150250}{4381606552811447071} a^{14} + \frac{867089883514788323}{4381606552811447071} a^{13} - \frac{1811513299283159607}{4381606552811447071} a^{12} + \frac{81597790888136788}{625943793258778153} a^{11} - \frac{2038164184274029633}{4381606552811447071} a^{10} + \frac{504509341359863786}{4381606552811447071} a^{9} - \frac{1873367021190781369}{4381606552811447071} a^{8} - \frac{1996564559742881196}{4381606552811447071} a^{7} - \frac{351833829544463235}{4381606552811447071} a^{6} - \frac{20358877468922101}{625943793258778153} a^{5} - \frac{1565340642337187039}{4381606552811447071} a^{4} + \frac{1618727655418314767}{4381606552811447071} a^{3} - \frac{4529997898096767}{55463374086220849} a^{2} - \frac{747247817666089349}{4381606552811447071} a - \frac{232028684128544131}{625943793258778153}$

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:  \( \frac{877142104773888771}{4381606552811447071} a^{15} - \frac{7773991535664563614}{4381606552811447071} a^{14} + \frac{17254143262311956443}{4381606552811447071} a^{13} + \frac{19971660650435002525}{4381606552811447071} a^{12} - \frac{12870593091980603786}{625943793258778153} a^{11} - \frac{42193756838843324397}{4381606552811447071} a^{10} + \frac{293656024126789690826}{4381606552811447071} a^{9} + \frac{136030562277115707832}{4381606552811447071} a^{8} - \frac{600762374311570432453}{4381606552811447071} a^{7} - \frac{256172836583380304885}{4381606552811447071} a^{6} + \frac{135059863488595573002}{625943793258778153} a^{5} + \frac{760261007801853160416}{4381606552811447071} a^{4} - \frac{290562969165764837714}{4381606552811447071} a^{3} - \frac{4149807697716971313}{55463374086220849} a^{2} + \frac{110947048320044832554}{4381606552811447071} a + \frac{14704990659143496875}{625943793258778153} \) (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:  \( 63593.3326838 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

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 $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$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}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$