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Properties

Label 16.0.57460001970...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{4}\cdot 17^{4}$
Root discriminant $14.88$
Ramified primes $2, 3, 5, 17$
Class number $1$
Class group Trivial
Galois group $(C_2\times D_4):C_2^2$ (as 16T115)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -64, 152, -256, 352, -376, 292, -156, 69, -42, 25, 8, -26, 14, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 14*x^13 - 26*x^12 + 8*x^11 + 25*x^10 - 42*x^9 + 69*x^8 - 156*x^7 + 292*x^6 - 376*x^5 + 352*x^4 - 256*x^3 + 152*x^2 - 64*x + 16)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 14*x^13 - 26*x^12 + 8*x^11 + 25*x^10 - 42*x^9 + 69*x^8 - 156*x^7 + 292*x^6 - 376*x^5 + 352*x^4 - 256*x^3 + 152*x^2 - 64*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 14 x^{13} - 26 x^{12} + 8 x^{11} + 25 x^{10} - 42 x^{9} + 69 x^{8} - 156 x^{7} + 292 x^{6} - 376 x^{5} + 352 x^{4} - 256 x^{3} + 152 x^{2} - 64 x + 16 \)

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:  \(5746000197058560000=2^{24}\cdot 3^{8}\cdot 5^{4}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.88$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{56} a^{14} + \frac{1}{14} a^{13} + \frac{1}{14} a^{12} + \frac{5}{28} a^{11} - \frac{5}{28} a^{10} + \frac{5}{14} a^{9} - \frac{19}{56} a^{8} - \frac{5}{28} a^{7} - \frac{1}{56} a^{6} - \frac{1}{14} a^{5} + \frac{5}{28} a^{4} + \frac{5}{14} a^{3} - \frac{3}{14} a^{2} + \frac{1}{7}$, $\frac{1}{548900744} a^{15} - \frac{230259}{39207196} a^{14} - \frac{18596017}{274450372} a^{13} - \frac{18540147}{274450372} a^{12} + \frac{54367911}{274450372} a^{11} - \frac{28243893}{137225186} a^{10} - \frac{164175495}{548900744} a^{9} + \frac{8661523}{68612593} a^{8} + \frac{55595957}{548900744} a^{7} + \frac{9573253}{39207196} a^{6} + \frac{12227589}{137225186} a^{5} - \frac{6790034}{68612593} a^{4} + \frac{44642001}{137225186} a^{3} - \frac{30319185}{68612593} a^{2} - \frac{17751516}{68612593} a - \frac{28690323}{68612593}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{569183}{5601028} a^{15} - \frac{2156837}{5601028} a^{14} + \frac{317861}{2800514} a^{13} + \frac{8472487}{5601028} a^{12} - \frac{3371563}{1400257} a^{11} + \frac{185845}{2800514} a^{10} + \frac{17681753}{5601028} a^{9} - \frac{20554153}{5601028} a^{8} + \frac{30880457}{5601028} a^{7} - \frac{39364751}{2800514} a^{6} + \frac{72644603}{2800514} a^{5} - \frac{168956867}{5601028} a^{4} + \frac{34865335}{1400257} a^{3} - \frac{42347435}{2800514} a^{2} + \frac{11018766}{1400257} a - \frac{2474011}{1400257} \) (order $6$)
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:  \( 1926.34951853 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_2^2$ (as 16T115):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 25 conjugacy class representatives for $(C_2\times D_4):C_2^2$
Character table for $(C_2\times D_4):C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), 4.0.1088.2, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{-3})\), 8.0.95883264.1, 8.0.8294400.1, 8.0.2397081600.7

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$