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Properties

Label 16.16.3731902746...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $39.65$
Ramified primes $3, 5, 13$
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![-179, -2083, -1399, 8699, 8419, -12302, -13981, 7313, 10110, -1638, -3437, 24, 535, 18, -38, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 38*x^14 + 18*x^13 + 535*x^12 + 24*x^11 - 3437*x^10 - 1638*x^9 + 10110*x^8 + 7313*x^7 - 13981*x^6 - 12302*x^5 + 8419*x^4 + 8699*x^3 - 1399*x^2 - 2083*x - 179)
 
gp: K = bnfinit(x^16 - x^15 - 38*x^14 + 18*x^13 + 535*x^12 + 24*x^11 - 3437*x^10 - 1638*x^9 + 10110*x^8 + 7313*x^7 - 13981*x^6 - 12302*x^5 + 8419*x^4 + 8699*x^3 - 1399*x^2 - 2083*x - 179, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 38 x^{14} + 18 x^{13} + 535 x^{12} + 24 x^{11} - 3437 x^{10} - 1638 x^{9} + 10110 x^{8} + 7313 x^{7} - 13981 x^{6} - 12302 x^{5} + 8419 x^{4} + 8699 x^{3} - 1399 x^{2} - 2083 x - 179 \)

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:  \(37319027463036582275390625=3^{8}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.65$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(73,·)$, $\chi_{195}(77,·)$, $\chi_{195}(79,·)$, $\chi_{195}(148,·)$, $\chi_{195}(86,·)$, $\chi_{195}(92,·)$, $\chi_{195}(161,·)$, $\chi_{195}(164,·)$, $\chi_{195}(38,·)$, $\chi_{195}(44,·)$, $\chi_{195}(112,·)$, $\chi_{195}(53,·)$, $\chi_{195}(187,·)$, $\chi_{195}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{1138} a^{14} + \frac{57}{1138} a^{13} + \frac{5}{569} a^{12} - \frac{183}{1138} a^{11} + \frac{8}{569} a^{10} + \frac{285}{1138} a^{9} - \frac{343}{1138} a^{8} + \frac{84}{569} a^{7} + \frac{243}{569} a^{6} + \frac{257}{1138} a^{5} - \frac{37}{569} a^{4} - \frac{200}{569} a^{3} + \frac{209}{569} a^{2} + \frac{131}{1138} a - \frac{287}{1138}$, $\frac{1}{580279386211978} a^{15} + \frac{222585444315}{580279386211978} a^{14} - \frac{48720282416745}{290139693105989} a^{13} - \frac{51798932742053}{290139693105989} a^{12} + \frac{63312837742116}{290139693105989} a^{11} - \frac{85502980055963}{580279386211978} a^{10} - \frac{156793660793165}{580279386211978} a^{9} + \frac{113971837506381}{580279386211978} a^{8} - \frac{7660853932027}{580279386211978} a^{7} + \frac{50999472099368}{290139693105989} a^{6} + \frac{75380593513321}{580279386211978} a^{5} + \frac{129510055371110}{290139693105989} a^{4} + \frac{288236115034279}{580279386211978} a^{3} + \frac{245173744463323}{580279386211978} a^{2} - \frac{94057860939875}{290139693105989} a + \frac{99391487170787}{580279386211978}$

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:  \( 36293972.0338 \) (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

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 4.4.274625.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.274625.2, 4.4.190125.1, \(\Q(\zeta_{15})^+\), 4.4.19773.1, 4.4.494325.1, 8.8.75418890625.1, 8.8.36147515625.1, 8.8.244357205625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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
13Data not computed