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Properties

Label 16.0.45086848686...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $46.33$
Ramified primes $2, 3, 5$
Class number $80$ (GRH)
Class group $[2, 2, 2, 10]$ (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![6241, 0, 656, 0, 2984, 0, -648, 0, 299, 0, -96, 0, -4, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 4*x^12 - 96*x^10 + 299*x^8 - 648*x^6 + 2984*x^4 + 656*x^2 + 6241)
 
gp: K = bnfinit(x^16 + 8*x^14 - 4*x^12 - 96*x^10 + 299*x^8 - 648*x^6 + 2984*x^4 + 656*x^2 + 6241, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 4 x^{12} - 96 x^{10} + 299 x^{8} - 648 x^{6} + 2984 x^{4} + 656 x^{2} + 6241 \)

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:  \(450868486864896000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.33$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(67,·)$, $\chi_{240}(143,·)$, $\chi_{240}(149,·)$, $\chi_{240}(23,·)$, $\chi_{240}(221,·)$, $\chi_{240}(163,·)$, $\chi_{240}(101,·)$, $\chi_{240}(167,·)$, $\chi_{240}(169,·)$, $\chi_{240}(43,·)$, $\chi_{240}(29,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(121,·)$, $\chi_{240}(187,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{79} a^{11} - \frac{33}{79} a^{9} - \frac{39}{79} a^{7} - \frac{14}{79} a^{5} + \frac{21}{79} a^{3} - \frac{10}{79} a$, $\frac{1}{2607} a^{12} + \frac{757}{2607} a^{10} - \frac{250}{869} a^{8} - \frac{172}{2607} a^{6} - \frac{151}{869} a^{4} - \frac{326}{2607} a^{2} + \frac{16}{33}$, $\frac{1}{2607} a^{13} - \frac{2}{2607} a^{11} + \frac{278}{869} a^{9} + \frac{752}{2607} a^{7} - \frac{85}{869} a^{5} - \frac{623}{2607} a^{3} + \frac{1033}{2607} a$, $\frac{1}{7828067577} a^{14} - \frac{11210}{237214169} a^{12} - \frac{495839515}{7828067577} a^{10} + \frac{300970747}{711642507} a^{8} + \frac{3174229372}{7828067577} a^{6} + \frac{297489638}{711642507} a^{4} + \frac{37143783}{2609355859} a^{2} - \frac{241918}{1254297}$, $\frac{1}{7828067577} a^{15} - \frac{11210}{237214169} a^{13} - \frac{392200}{7828067577} a^{11} + \frac{237913816}{711642507} a^{9} - \frac{492080759}{7828067577} a^{7} - \frac{333079672}{711642507} a^{5} + \frac{895919129}{2609355859} a^{3} + \frac{17263091}{99089463} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}$, which has order $80$ (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:  \( 85299.42553126559 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), 4.0.18432.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.460800.2, 4.4.256000.1, 4.4.256000.2, 4.0.72000.2, 4.0.18000.1, 8.0.212336640000.5, 8.8.65536000000.1, 8.0.82944000000.6

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed