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Properties

Label 16.8.33693554028...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 19^{4}\cdot 97^{2}\cdot 103^{4}$
Root discriminant $39.40$
Ramified primes $5, 19, 97, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1046

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-569, 12041, 8878, -16285, -7870, 2538, 1405, 1356, 486, 85, -10, -106, -57, -4, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 4*x^13 - 57*x^12 - 106*x^11 - 10*x^10 + 85*x^9 + 486*x^8 + 1356*x^7 + 1405*x^6 + 2538*x^5 - 7870*x^4 - 16285*x^3 + 8878*x^2 + 12041*x - 569)
 
gp: K = bnfinit(x^16 + 2*x^14 - 4*x^13 - 57*x^12 - 106*x^11 - 10*x^10 + 85*x^9 + 486*x^8 + 1356*x^7 + 1405*x^6 + 2538*x^5 - 7870*x^4 - 16285*x^3 + 8878*x^2 + 12041*x - 569, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 4 x^{13} - 57 x^{12} - 106 x^{11} - 10 x^{10} + 85 x^{9} + 486 x^{8} + 1356 x^{7} + 1405 x^{6} + 2538 x^{5} - 7870 x^{4} - 16285 x^{3} + 8878 x^{2} + 12041 x - 569 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33693554028465480712890625=5^{12}\cdot 19^{4}\cdot 97^{2}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.40$
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:  $5, 19, 97, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{38} a^{14} + \frac{5}{38} a^{13} - \frac{3}{38} a^{12} - \frac{17}{38} a^{11} + \frac{5}{38} a^{10} + \frac{11}{38} a^{9} + \frac{9}{38} a^{8} + \frac{9}{38} a^{7} - \frac{5}{38} a^{6} - \frac{3}{38} a^{5} + \frac{1}{38} a^{4} + \frac{11}{38} a^{3} - \frac{17}{38} a^{2} + \frac{1}{38} a + \frac{7}{38}$, $\frac{1}{854519538021792630808461035618} a^{15} - \frac{5235270981074025506202653829}{427259769010896315404230517809} a^{14} - \frac{39264358506721204074496989304}{427259769010896315404230517809} a^{13} + \frac{3088505664241056648455219413}{22487356263731385021275290411} a^{12} - \frac{4370174923289383551731545855}{22487356263731385021275290411} a^{11} + \frac{27088931116070759132708002157}{427259769010896315404230517809} a^{10} - \frac{90703006934357407401554777739}{427259769010896315404230517809} a^{9} + \frac{99690918116117820836367767579}{427259769010896315404230517809} a^{8} - \frac{8751257601743633436374291508}{427259769010896315404230517809} a^{7} - \frac{10101302267779341683591780465}{22487356263731385021275290411} a^{6} - \frac{125958644724966832254018065689}{427259769010896315404230517809} a^{5} - \frac{150486424559027344237841120998}{427259769010896315404230517809} a^{4} + \frac{199381212101835353533514600896}{427259769010896315404230517809} a^{3} - \frac{146563421835557040005525848294}{427259769010896315404230517809} a^{2} - \frac{62750052289369963084874413628}{427259769010896315404230517809} a + \frac{420957596522186756064610511197}{854519538021792630808461035618}$

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:  $11$
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:  \( 7286156.12287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1046:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 40 conjugacy class representatives for t16n1046
Character table for t16n1046 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$