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Properties

Label 16.0.88284816000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{2}$
Root discriminant $17.64$
Ramified primes $2, 3, 5, 29$
Class number $2$
Class group $[2]$
Galois group $C_2^2.C_2^5.C_2$ (as 16T610)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 42, -140, 313, -482, 534, -498, 437, -304, 173, -112, 49, -16, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 - 16*x^13 + 49*x^12 - 112*x^11 + 173*x^10 - 304*x^9 + 437*x^8 - 498*x^7 + 534*x^6 - 482*x^5 + 313*x^4 - 140*x^3 + 42*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 + 11*x^14 - 16*x^13 + 49*x^12 - 112*x^11 + 173*x^10 - 304*x^9 + 437*x^8 - 498*x^7 + 534*x^6 - 482*x^5 + 313*x^4 - 140*x^3 + 42*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 11 x^{14} - 16 x^{13} + 49 x^{12} - 112 x^{11} + 173 x^{10} - 304 x^{9} + 437 x^{8} - 498 x^{7} + 534 x^{6} - 482 x^{5} + 313 x^{4} - 140 x^{3} + 42 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(88284816000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.64$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{29} a^{13} - \frac{1}{29} a^{12} - \frac{14}{29} a^{11} - \frac{7}{29} a^{10} - \frac{13}{29} a^{9} - \frac{5}{29} a^{8} + \frac{8}{29} a^{7} - \frac{11}{29} a^{6} - \frac{11}{29} a^{5} - \frac{14}{29} a^{4} - \frac{14}{29} a^{3} + \frac{5}{29} a^{2} + \frac{4}{29} a - \frac{14}{29}$, $\frac{1}{29} a^{14} + \frac{14}{29} a^{12} + \frac{8}{29} a^{11} + \frac{9}{29} a^{10} + \frac{11}{29} a^{9} + \frac{3}{29} a^{8} - \frac{3}{29} a^{7} + \frac{7}{29} a^{6} + \frac{4}{29} a^{5} + \frac{1}{29} a^{4} - \frac{9}{29} a^{3} + \frac{9}{29} a^{2} - \frac{10}{29} a - \frac{14}{29}$, $\frac{1}{1789909} a^{15} + \frac{29960}{1789909} a^{14} - \frac{6892}{1789909} a^{13} - \frac{768243}{1789909} a^{12} + \frac{435089}{1789909} a^{11} - \frac{455756}{1789909} a^{10} - \frac{776851}{1789909} a^{9} + \frac{779720}{1789909} a^{8} + \frac{124115}{1789909} a^{7} + \frac{59886}{162719} a^{6} - \frac{616918}{1789909} a^{5} - \frac{52003}{162719} a^{4} - \frac{655507}{1789909} a^{3} + \frac{447177}{1789909} a^{2} + \frac{46326}{162719} a + \frac{260318}{1789909}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 896.342119314 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T610):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, \(\Q(\zeta_{15})^+\), 4.2.18000.1, 8.4.324000000.2

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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
3Data not computed
5Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$