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Properties

Label 16.16.1068426054...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{10}\cdot 19^{4}\cdot 71^{6}$
Root discriminant $56.47$
Ramified primes $2, 5, 19, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71, 284, -1886, -10046, -4989, 24792, 9856, -27098, -733, 11138, -1394, -2002, 391, 156, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 156*x^13 + 391*x^12 - 2002*x^11 - 1394*x^10 + 11138*x^9 - 733*x^8 - 27098*x^7 + 9856*x^6 + 24792*x^5 - 4989*x^4 - 10046*x^3 - 1886*x^2 + 284*x + 71)
 
gp: K = bnfinit(x^16 - 4*x^15 - 36*x^14 + 156*x^13 + 391*x^12 - 2002*x^11 - 1394*x^10 + 11138*x^9 - 733*x^8 - 27098*x^7 + 9856*x^6 + 24792*x^5 - 4989*x^4 - 10046*x^3 - 1886*x^2 + 284*x + 71, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 36 x^{14} + 156 x^{13} + 391 x^{12} - 2002 x^{11} - 1394 x^{10} + 11138 x^{9} - 733 x^{8} - 27098 x^{7} + 9856 x^{6} + 24792 x^{5} - 4989 x^{4} - 10046 x^{3} - 1886 x^{2} + 284 x + 71 \)

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:  \(10684260544555930240000000000=2^{16}\cdot 5^{10}\cdot 19^{4}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.47$
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:  $2, 5, 19, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{11} - \frac{1}{25} a^{10} - \frac{8}{25} a^{7} + \frac{9}{25} a^{6} + \frac{8}{25} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{6}{25} a^{2} - \frac{7}{25} a - \frac{4}{25}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{10} + \frac{2}{25} a^{8} + \frac{1}{5} a^{7} + \frac{4}{25} a^{5} + \frac{1}{5} a^{4} - \frac{1}{25} a^{3} + \frac{1}{5} a^{2} - \frac{7}{25}$, $\frac{1}{475} a^{14} - \frac{8}{475} a^{13} - \frac{8}{475} a^{12} + \frac{11}{475} a^{11} + \frac{37}{475} a^{10} - \frac{28}{475} a^{9} + \frac{14}{475} a^{8} - \frac{206}{475} a^{7} + \frac{92}{475} a^{6} + \frac{224}{475} a^{5} - \frac{186}{475} a^{4} - \frac{117}{475} a^{3} + \frac{23}{475} a^{2} - \frac{151}{475} a + \frac{108}{475}$, $\frac{1}{2886146301575} a^{15} + \frac{1480960307}{2886146301575} a^{14} - \frac{44534576507}{2886146301575} a^{13} - \frac{25020409926}{2886146301575} a^{12} + \frac{113195359368}{2886146301575} a^{11} + \frac{59209788441}{2886146301575} a^{10} + \frac{174619200619}{2886146301575} a^{9} + \frac{195777792431}{2886146301575} a^{8} + \frac{1435932288173}{2886146301575} a^{7} + \frac{588726231101}{2886146301575} a^{6} - \frac{1220874792888}{2886146301575} a^{5} + \frac{928330624408}{2886146301575} a^{4} + \frac{53220992233}{151902436925} a^{3} - \frac{30622015096}{151902436925} a^{2} - \frac{690626716993}{2886146301575} a + \frac{1111008226621}{2886146301575}$

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:  \( 797746033.308 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.33725.1, 8.8.5440247200000.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
71Data not computed