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Properties

Label 16.16.2817928042...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $38.96$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (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![1, 56, 348, -1036, -2020, 4760, 2894, -7132, -1044, 4348, -316, -1108, 193, 116, -26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 26*x^14 + 116*x^13 + 193*x^12 - 1108*x^11 - 316*x^10 + 4348*x^9 - 1044*x^8 - 7132*x^7 + 2894*x^6 + 4760*x^5 - 2020*x^4 - 1036*x^3 + 348*x^2 + 56*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 26*x^14 + 116*x^13 + 193*x^12 - 1108*x^11 - 316*x^10 + 4348*x^9 - 1044*x^8 - 7132*x^7 + 2894*x^6 + 4760*x^5 - 2020*x^4 - 1036*x^3 + 348*x^2 + 56*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 26 x^{14} + 116 x^{13} + 193 x^{12} - 1108 x^{11} - 316 x^{10} + 4348 x^{9} - 1044 x^{8} - 7132 x^{7} + 2894 x^{6} + 4760 x^{5} - 2020 x^{4} - 1036 x^{3} + 348 x^{2} + 56 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28179280429056000000000000=2^{44}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.96$
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$$\chi_{240}(1,·)$, $\chi_{240}(197,·)$, $\chi_{240}(137,·)$, $\chi_{240}(77,·)$, $\chi_{240}(173,·)$, $\chi_{240}(17,·)$, $\chi_{240}(229,·)$, $\chi_{240}(49,·)$, $\chi_{240}(169,·)$, $\chi_{240}(109,·)$, $\chi_{240}(113,·)$, $\chi_{240}(53,·)$, $\chi_{240}(233,·)$, $\chi_{240}(121,·)$, $\chi_{240}(61,·)$, $\chi_{240}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{31} a^{12} - \frac{12}{31} a^{11} - \frac{6}{31} a^{10} - \frac{2}{31} a^{9} - \frac{10}{31} a^{8} + \frac{5}{31} a^{7} - \frac{1}{31} a^{6} + \frac{4}{31} a^{5} - \frac{15}{31} a^{4} + \frac{14}{31} a^{3} + \frac{15}{31} a^{2} - \frac{9}{31} a - \frac{15}{31}$, $\frac{1}{31} a^{13} + \frac{5}{31} a^{11} - \frac{12}{31} a^{10} - \frac{3}{31} a^{9} + \frac{9}{31} a^{8} - \frac{3}{31} a^{7} - \frac{8}{31} a^{6} + \frac{2}{31} a^{5} - \frac{11}{31} a^{4} - \frac{3}{31} a^{3} - \frac{15}{31} a^{2} + \frac{1}{31} a + \frac{6}{31}$, $\frac{1}{7240391} a^{14} - \frac{68793}{7240391} a^{13} - \frac{113240}{7240391} a^{12} + \frac{10253}{7240391} a^{11} - \frac{1004250}{7240391} a^{10} + \frac{1495651}{7240391} a^{9} + \frac{2054692}{7240391} a^{8} - \frac{1737277}{7240391} a^{7} + \frac{89564}{7240391} a^{6} - \frac{1315140}{7240391} a^{5} + \frac{3179338}{7240391} a^{4} - \frac{65961}{233561} a^{3} + \frac{9682}{233561} a^{2} - \frac{2192734}{7240391} a - \frac{3389162}{7240391}$, $\frac{1}{2961319919} a^{15} + \frac{136}{2961319919} a^{14} - \frac{46669154}{2961319919} a^{13} - \frac{9009966}{2961319919} a^{12} - \frac{917991285}{2961319919} a^{11} - \frac{810669260}{2961319919} a^{10} - \frac{949103321}{2961319919} a^{9} + \frac{918470946}{2961319919} a^{8} - \frac{227338434}{2961319919} a^{7} + \frac{570514792}{2961319919} a^{6} + \frac{676347327}{2961319919} a^{5} - \frac{609599323}{2961319919} a^{4} + \frac{1139166482}{2961319919} a^{3} + \frac{86214984}{2961319919} a^{2} + \frac{1352399121}{2961319919} a + \frac{664382723}{2961319919}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 33542030.6703 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, 4.4.2304000.2, 4.4.2304000.1, \(\Q(\zeta_{15})^+\), 4.4.72000.1, 8.8.2621440000.1, 8.8.5308416000000.1, 8.8.5184000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed