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Properties

Label 16.4.16222633832...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{20}\cdot 5^{12}\cdot 89^{4}\cdot 101$
Root discriminant $32.59$
Ramified primes $2, 5, 89, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1616

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -64, -36, 252, 752, 600, -482, -1226, -591, 276, 294, -176, -27, -2, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 2*x^13 - 27*x^12 - 176*x^11 + 294*x^10 + 276*x^9 - 591*x^8 - 1226*x^7 - 482*x^6 + 600*x^5 + 752*x^4 + 252*x^3 - 36*x^2 - 64*x - 4)
 
gp: K = bnfinit(x^16 - 6*x^14 - 2*x^13 - 27*x^12 - 176*x^11 + 294*x^10 + 276*x^9 - 591*x^8 - 1226*x^7 - 482*x^6 + 600*x^5 + 752*x^4 + 252*x^3 - 36*x^2 - 64*x - 4, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 2 x^{13} - 27 x^{12} - 176 x^{11} + 294 x^{10} + 276 x^{9} - 591 x^{8} - 1226 x^{7} - 482 x^{6} + 600 x^{5} + 752 x^{4} + 252 x^{3} - 36 x^{2} - 64 x - 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1622263383296000000000000=2^{20}\cdot 5^{12}\cdot 89^{4}\cdot 101\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.59$
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:  $2, 5, 89, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{4} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{915175847035935013332} a^{15} + \frac{649491879384017298}{76264653919661251111} a^{14} - \frac{49517878854093358099}{457587923517967506666} a^{13} - \frac{3099905103907413781}{228793961758983753333} a^{12} + \frac{176881299530458693577}{915175847035935013332} a^{11} - \frac{36664050005947535647}{228793961758983753333} a^{10} + \frac{29424060183003842743}{152529307839322502222} a^{9} + \frac{33552353064813168247}{457587923517967506666} a^{8} - \frac{345909215042009029909}{915175847035935013332} a^{7} - \frac{24348126886100015141}{457587923517967506666} a^{6} - \frac{34956366764304215474}{228793961758983753333} a^{5} - \frac{221420099845342662245}{457587923517967506666} a^{4} - \frac{56568478726500184031}{457587923517967506666} a^{3} + \frac{88758581111045544389}{228793961758983753333} a^{2} - \frac{44328931965791319674}{228793961758983753333} a + \frac{70299860635246932170}{228793961758983753333}$

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:  $9$
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:  \( 1405229.48287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1616:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1616 are not computed
Character table for t16n1616 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.178000.1, \(\Q(\zeta_{20})^+\), 4.4.2225.1, 8.8.31684000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
5Data not computed
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$