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Properties

Label 16.0.55219815610...0704.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 149^{12}$
Root discriminant $625.72$
Ramified primes $2, 149$
Class number $1909890970$ (GRH)
Class group $[13, 146914690]$ (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![770131876562500, 0, 246442200500000, 0, 21418969775000, 0, 759505090400, 0, 12963208302, 0, 115089984, 0, 533420, 0, 1192, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1192*x^14 + 533420*x^12 + 115089984*x^10 + 12963208302*x^8 + 759505090400*x^6 + 21418969775000*x^4 + 246442200500000*x^2 + 770131876562500)
 
gp: K = bnfinit(x^16 + 1192*x^14 + 533420*x^12 + 115089984*x^10 + 12963208302*x^8 + 759505090400*x^6 + 21418969775000*x^4 + 246442200500000*x^2 + 770131876562500, 1)
 

Normalized defining polynomial

\( x^{16} + 1192 x^{14} + 533420 x^{12} + 115089984 x^{10} + 12963208302 x^{8} + 759505090400 x^{6} + 21418969775000 x^{4} + 246442200500000 x^{2} + 770131876562500 \)

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:  \(552198156105754424298997471653759249446600704=2^{62}\cdot 149^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $625.72$
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:  $2, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4768=2^{5}\cdot 149\)
Dirichlet character group:    $\lbrace$$\chi_{4768}(1,·)$, $\chi_{4768}(3085,·)$, $\chi_{4768}(4365,·)$, $\chi_{4768}(2385,·)$, $\chi_{4768}(789,·)$, $\chi_{4768}(2681,·)$, $\chi_{4768}(3173,·)$, $\chi_{4768}(3873,·)$, $\chi_{4768}(1893,·)$, $\chi_{4768}(1489,·)$, $\chi_{4768}(297,·)$, $\chi_{4768}(1981,·)$, $\chi_{4768}(4277,·)$, $\chi_{4768}(1193,·)$, $\chi_{4768}(3577,·)$, $\chi_{4768}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{149} a^{4}$, $\frac{1}{149} a^{5}$, $\frac{1}{149} a^{6}$, $\frac{1}{149} a^{7}$, $\frac{1}{310814} a^{8} + \frac{2}{1043} a^{6} - \frac{2}{1043} a^{4} + \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{1554070} a^{9} + \frac{9}{5215} a^{7} + \frac{1}{1043} a^{5} + \frac{2}{35} a^{3} + \frac{16}{35} a$, $\frac{1}{7770350} a^{10} - \frac{4}{3885175} a^{8} + \frac{3}{5215} a^{6} + \frac{8}{26075} a^{4} + \frac{26}{175} a^{2} - \frac{1}{7}$, $\frac{1}{38851750} a^{11} - \frac{4}{19425875} a^{9} + \frac{73}{26075} a^{7} - \frac{167}{130375} a^{5} - \frac{324}{875} a^{3} - \frac{3}{7} a$, $\frac{1}{28944553750} a^{12} + \frac{4}{97129375} a^{10} - \frac{17}{19425875} a^{8} - \frac{1158}{651875} a^{6} + \frac{701}{651875} a^{4} - \frac{83}{175} a^{2} - \frac{2}{7}$, $\frac{1}{144722768750} a^{13} + \frac{4}{485646875} a^{11} - \frac{17}{97129375} a^{9} + \frac{3217}{3259375} a^{7} - \frac{8049}{3259375} a^{5} - \frac{83}{875} a^{3} - \frac{16}{35} a$, $\frac{1}{283044309490546504906250} a^{14} - \frac{2416811150729}{141522154745273252453125} a^{12} + \frac{414646344851}{379925247638317456250} a^{10} - \frac{712208444721667}{949813119095793640625} a^{8} - \frac{1723713390595599}{6374584691918078125} a^{6} - \frac{779762252002948}{254983387676723125} a^{4} - \frac{5850821137614}{13690383230965} a^{2} - \frac{104906867277}{391153806599}$, $\frac{1}{1415221547452732524531250} a^{15} - \frac{2416811150729}{707610773726366262265625} a^{13} + \frac{414646344851}{1899626238191587281250} a^{11} - \frac{712208444721667}{4749065595478968203125} a^{9} - \frac{44506160987361224}{31872923459590390625} a^{7} + \frac{2642833555738302}{1274916938383615625} a^{5} - \frac{19541204368579}{68451916154825} a^{3} - \frac{104906867277}{1955769032995} a$

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

Class group and class number

$C_{13}\times C_{146914690}$, which has order $1909890970$ (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:  \( 4026951.331137613 \) (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

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{298}) \), \(\Q(\sqrt{2}, \sqrt{149})\), 4.4.45467648.1, \(\Q(\zeta_{16})^+\), 8.8.2067307014651904.1, 8.0.23498896912530903400448.4, 8.0.23498896912530903400448.3

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
149Data not computed