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Properties

Label 16.16.2283388028...2144.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 3^{4}\cdot 17^{8}\cdot 97^{2}$
Root discriminant $38.45$
Ramified primes $2, 3, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -120, -304, 4164, -8411, 12, 12526, -5048, -7057, 3576, 1788, -1060, -177, 148, -2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 2*x^14 + 148*x^13 - 177*x^12 - 1060*x^11 + 1788*x^10 + 3576*x^9 - 7057*x^8 - 5048*x^7 + 12526*x^6 + 12*x^5 - 8411*x^4 + 4164*x^3 - 304*x^2 - 120*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 - 2*x^14 + 148*x^13 - 177*x^12 - 1060*x^11 + 1788*x^10 + 3576*x^9 - 7057*x^8 - 5048*x^7 + 12526*x^6 + 12*x^5 - 8411*x^4 + 4164*x^3 - 304*x^2 - 120*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 2 x^{14} + 148 x^{13} - 177 x^{12} - 1060 x^{11} + 1788 x^{10} + 3576 x^{9} - 7057 x^{8} - 5048 x^{7} + 12526 x^{6} + 12 x^{5} - 8411 x^{4} + 4164 x^{3} - 304 x^{2} - 120 x + 4 \)

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:  \(22833880280189357097222144=2^{32}\cdot 3^{4}\cdot 17^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.45$
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:  $2, 3, 17, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{12} + \frac{3}{8} a^{4} - \frac{1}{2}$, $\frac{1}{8} a^{13} + \frac{3}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} + \frac{3}{16} a^{6} - \frac{3}{16} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{29708176} a^{15} - \frac{349165}{14854088} a^{14} + \frac{1349979}{29708176} a^{13} - \frac{203331}{3713522} a^{12} + \frac{205091}{14854088} a^{11} + \frac{1701903}{14854088} a^{10} + \frac{1131769}{14854088} a^{9} + \frac{413625}{14854088} a^{8} - \frac{258671}{29708176} a^{7} - \frac{75352}{1856761} a^{6} + \frac{13792083}{29708176} a^{5} - \frac{3011499}{14854088} a^{4} + \frac{784419}{7427044} a^{3} - \frac{245985}{1856761} a^{2} - \frac{456773}{7427044} a + \frac{831343}{3713522}$

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:  \( 107212709.99 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$