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Properties

Label 16.2.48104202240...0000.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 179$
Root discriminant $22.65$
Ramified primes $2, 3, 5, 179$
Class number $1$
Class group Trivial
Galois group 16T1379

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, 428, -2080, 4266, -4945, 3482, -1096, -458, 786, -422, 82, 38, -22, -8, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 8*x^13 - 22*x^12 + 38*x^11 + 82*x^10 - 422*x^9 + 786*x^8 - 458*x^7 - 1096*x^6 + 3482*x^5 - 4945*x^4 + 4266*x^3 - 2080*x^2 + 428*x + 61)
 
gp: K = bnfinit(x^16 - 6*x^15 + 14*x^14 - 8*x^13 - 22*x^12 + 38*x^11 + 82*x^10 - 422*x^9 + 786*x^8 - 458*x^7 - 1096*x^6 + 3482*x^5 - 4945*x^4 + 4266*x^3 - 2080*x^2 + 428*x + 61, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 14 x^{14} - 8 x^{13} - 22 x^{12} + 38 x^{11} + 82 x^{10} - 422 x^{9} + 786 x^{8} - 458 x^{7} - 1096 x^{6} + 3482 x^{5} - 4945 x^{4} + 4266 x^{3} - 2080 x^{2} + 428 x + 61 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4810420224000000000000=-\,2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 179\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.65$
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:  $2, 3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{937519875226663195841} a^{15} - \frac{373455977211662668141}{937519875226663195841} a^{14} - \frac{251902111575418847985}{937519875226663195841} a^{13} + \frac{278922484659551306564}{937519875226663195841} a^{12} + \frac{9710703250602138885}{937519875226663195841} a^{11} + \frac{313836207979755405390}{937519875226663195841} a^{10} - \frac{214261129199442091}{470878892630167351} a^{9} - \frac{301202926878820014679}{937519875226663195841} a^{8} - \frac{120778248783566196140}{937519875226663195841} a^{7} - \frac{385014754938034210655}{937519875226663195841} a^{6} + \frac{106122539081750095243}{937519875226663195841} a^{5} + \frac{462238372743009768316}{937519875226663195841} a^{4} + \frac{60879188125504916319}{937519875226663195841} a^{3} - \frac{125635450839925611314}{937519875226663195841} a^{2} - \frac{205299285903992057915}{937519875226663195841} a - \frac{284149054012624950001}{937519875226663195841}$

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:  $8$
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:  \( 14836.890631 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1379:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1379 are not computed
Character table for t16n1379 is not computed

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$