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Properties

Label 16.0.24313966763...000.20
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 29^{12}$
Root discriminant $334.28$
Ramified primes $2, 5, 29$
Class number $123335680$ (GRH)
Class group $[2, 4, 8, 1927120]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1092493210176, 0, 186876209664, 0, 17044358464, 0, 973354368, 0, 38045364, 0, 1013056, 0, 17916, 0, 192, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 192*x^14 + 17916*x^12 + 1013056*x^10 + 38045364*x^8 + 973354368*x^6 + 17044358464*x^4 + 186876209664*x^2 + 1092493210176)
 
gp: K = bnfinit(x^16 + 192*x^14 + 17916*x^12 + 1013056*x^10 + 38045364*x^8 + 973354368*x^6 + 17044358464*x^4 + 186876209664*x^2 + 1092493210176, 1)
 

Normalized defining polynomial

\( x^{16} + 192 x^{14} + 17916 x^{12} + 1013056 x^{10} + 38045364 x^{8} + 973354368 x^{6} + 17044358464 x^{4} + 186876209664 x^{2} + 1092493210176 \)

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:  \(24313966763341113270967730176000000000000=2^{48}\cdot 5^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $334.28$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2320=2^{4}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{2320}(1,·)$, $\chi_{2320}(1989,·)$, $\chi_{2320}(1607,·)$, $\chi_{2320}(521,·)$, $\chi_{2320}(1549,·)$, $\chi_{2320}(1683,·)$, $\chi_{2320}(987,·)$, $\chi_{2320}(289,·)$, $\chi_{2320}(1507,·)$, $\chi_{2320}(423,·)$, $\chi_{2320}(2089,·)$, $\chi_{2320}(2221,·)$, $\chi_{2320}(2203,·)$, $\chi_{2320}(1781,·)$, $\chi_{2320}(2047,·)$, $\chi_{2320}(2303,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{5} + \frac{1}{3} a$, $\frac{1}{36} a^{6} - \frac{1}{18} a^{4} + \frac{5}{18} a^{2}$, $\frac{1}{36} a^{7} - \frac{1}{18} a^{5} - \frac{1}{18} a^{3} + \frac{1}{3} a$, $\frac{1}{72} a^{8} - \frac{1}{12} a^{4} + \frac{4}{9} a^{2}$, $\frac{1}{1296} a^{9} - \frac{1}{108} a^{7} - \frac{1}{216} a^{5} - \frac{13}{81} a^{3} - \frac{7}{18} a$, $\frac{1}{1296} a^{10} + \frac{1}{216} a^{8} - \frac{1}{216} a^{6} - \frac{25}{324} a^{4} + \frac{7}{18} a^{2}$, $\frac{1}{7776} a^{11} + \frac{1}{3888} a^{9} + \frac{7}{1296} a^{7} - \frac{73}{1944} a^{5} + \frac{59}{972} a^{3} + \frac{17}{54} a$, $\frac{1}{86927904} a^{12} - \frac{827}{10865988} a^{10} - \frac{7331}{14487984} a^{8} + \frac{9463}{10865988} a^{6} + \frac{76505}{2716497} a^{4} - \frac{1280}{301833} a^{2} - \frac{5409}{11179}$, $\frac{1}{86927904} a^{13} + \frac{169}{3219552} a^{11} - \frac{5407}{21731976} a^{9} + \frac{272611}{43463952} a^{7} - \frac{68009}{7243992} a^{5} + \frac{613481}{10865988} a^{3} - \frac{102043}{603666} a$, $\frac{1}{505428941074766688} a^{14} + \frac{518774341}{505428941074766688} a^{12} + \frac{10717793635997}{36102067219626192} a^{10} + \frac{873374983676851}{252714470537383344} a^{8} + \frac{751039652559937}{63178617634345836} a^{6} + \frac{1217805677481805}{15794654408586459} a^{4} - \frac{121445389196716}{250708800136293} a^{2} + \frac{171280262142}{2407354733819}$, $\frac{1}{1630513763907197335488} a^{15} - \frac{3688943370001}{815256881953598667744} a^{13} - \frac{451604941036127}{25476777561049958367} a^{11} + \frac{30451860849157043}{101907110244199833468} a^{9} - \frac{1660083322953376639}{407628440976799333872} a^{7} + \frac{1911994917800048905}{29116317212628523848} a^{5} - \frac{1704761062821226997}{11323012249355537052} a^{3} - \frac{17192849001495197}{209685412025102538} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}\times C_{1927120}$, which has order $123335680$ (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:  \( 394683673.28410536 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{58}) \), \(\Q(\sqrt{10}) \), 4.0.195112000.2, \(\Q(\sqrt{10}, \sqrt{58})\), 4.0.48778000.3, 4.0.49948672.3, 4.0.1248716800.2, 4.4.256000.1, 4.4.215296000.2, 8.0.609099080704000000.4, 8.0.1559293646602240000.5, 8.8.46352367616000000.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
5Data not computed
29Data not computed