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Properties

Label 16.0.96468058006...3184.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 257^{14}$
Root discriminant $864.02$
Ramified primes $2, 257$
Class number $14079590400$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 4, 40, 687480]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![69257396224, 0, 147171966976, 0, 71556958208, 0, 12850493440, 0, 862829184, 0, 23931840, 0, 255458, 0, 1028, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224)
 
gp: K = bnfinit(x^16 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224, 1)
 

Normalized defining polynomial

\( x^{16} + 1028 x^{14} + 255458 x^{12} + 23931840 x^{10} + 862829184 x^{8} + 12850493440 x^{6} + 71556958208 x^{4} + 147171966976 x^{2} + 69257396224 \)

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:  \(96468058006688010602395260507818496203564253184=2^{44}\cdot 257^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $864.02$
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:  $2, 257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4112=2^{4}\cdot 257\)
Dirichlet character group:    $\lbrace$$\chi_{4112}(1,·)$, $\chi_{4112}(1795,·)$, $\chi_{4112}(513,·)$, $\chi_{4112}(3851,·)$, $\chi_{4112}(2057,·)$, $\chi_{4112}(2763,·)$, $\chi_{4112}(273,·)$, $\chi_{4112}(1803,·)$, $\chi_{4112}(2329,·)$, $\chi_{4112}(2891,·)$, $\chi_{4112}(241,·)$, $\chi_{4112}(3859,·)$, $\chi_{4112}(707,·)$, $\chi_{4112}(2569,·)$, $\chi_{4112}(2297,·)$, $\chi_{4112}(835,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{7} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16448} a^{8} - \frac{1}{16} a^{6} + \frac{1}{32} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{65792} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4473856} a^{10} - \frac{23}{1118464} a^{8} - \frac{287}{8704} a^{6} + \frac{9}{136} a^{4} + \frac{133}{272} a^{2} + \frac{2}{17}$, $\frac{1}{17895424} a^{11} - \frac{23}{4473856} a^{9} + \frac{801}{34816} a^{7} + \frac{9}{544} a^{5} - \frac{343}{1088} a^{3} + \frac{19}{68} a$, $\frac{1}{4853525315584} a^{12} - \frac{60899}{1213381328896} a^{10} - \frac{47042495}{2426762657792} a^{8} + \frac{3372913}{590166016} a^{6} + \frac{9283361}{295083008} a^{4} + \frac{1645255}{4610672} a^{2} + \frac{120883}{576334}$, $\frac{1}{9707050631168} a^{13} - \frac{60899}{2426762657792} a^{11} + \frac{104001}{18885312512} a^{9} + \frac{21815601}{1180332032} a^{7} + \frac{13894033}{590166016} a^{5} + \frac{3950591}{9221344} a^{3} - \frac{41821}{288167} a$, $\frac{1}{22345630552948736} a^{14} - \frac{5}{121443644309504} a^{12} - \frac{981442167}{11172815276474368} a^{10} + \frac{22964557313}{2793203819118592} a^{8} + \frac{60354586683}{1358562168832} a^{6} + \frac{64419170253}{339640542208} a^{4} - \frac{1574765255}{5306883472} a^{2} - \frac{257620721}{663360434}$, $\frac{1}{44691261105897472} a^{15} - \frac{5}{242887288619008} a^{13} + \frac{267236297}{22345630552948736} a^{11} - \frac{5755047359}{5586407638237184} a^{9} - \frac{46953718817}{2717124337664} a^{7} + \frac{75657276429}{679281084416} a^{5} + \frac{8732425499}{21227533888} a^{3} - \frac{137569934}{331680217} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{40}\times C_{687480}$, which has order $14079590400$ (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:  \( 112709711.05474126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{514}) \), \(\Q(\sqrt{2}, \sqrt{257})\), 4.4.1086373952.2, 4.4.16974593.1, 8.8.1180208363584098304.1, 8.0.310593074627699982467072.6, 8.0.310593074627699982467072.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
257Data not computed