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Properties

Label 16.12.2967385182...8125.1
Degree $16$
Signature $[12, 2]$
Discriminant $5^{12}\cdot 19^{3}\cdot 29^{6}\cdot 31^{3}$
Root discriminant $39.09$
Ramified primes $5, 19, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1477

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14669, -2204, 61506, -23715, -67883, 55818, -309, -14111, 8152, -3025, 709, -176, -35, 86, -21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 21*x^14 + 86*x^13 - 35*x^12 - 176*x^11 + 709*x^10 - 3025*x^9 + 8152*x^8 - 14111*x^7 - 309*x^6 + 55818*x^5 - 67883*x^4 - 23715*x^3 + 61506*x^2 - 2204*x - 14669)
 
gp: K = bnfinit(x^16 - 3*x^15 - 21*x^14 + 86*x^13 - 35*x^12 - 176*x^11 + 709*x^10 - 3025*x^9 + 8152*x^8 - 14111*x^7 - 309*x^6 + 55818*x^5 - 67883*x^4 - 23715*x^3 + 61506*x^2 - 2204*x - 14669, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 21 x^{14} + 86 x^{13} - 35 x^{12} - 176 x^{11} + 709 x^{10} - 3025 x^{9} + 8152 x^{8} - 14111 x^{7} - 309 x^{6} + 55818 x^{5} - 67883 x^{4} - 23715 x^{3} + 61506 x^{2} - 2204 x - 14669 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29673851829099987548828125=5^{12}\cdot 19^{3}\cdot 29^{6}\cdot 31^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.09$
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:  $5, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{4}{13} a^{13} - \frac{5}{13} a^{12} + \frac{3}{13} a^{11} - \frac{6}{13} a^{10} + \frac{6}{13} a^{9} + \frac{4}{13} a^{8} - \frac{1}{13} a^{7} - \frac{2}{13} a^{6} + \frac{5}{13} a^{5} - \frac{3}{13} a^{4} + \frac{6}{13} a^{3} + \frac{3}{13} a^{2} - \frac{2}{13} a + \frac{5}{13}$, $\frac{1}{859544020592093752532733715131247} a^{15} - \frac{25740685726913627445720052109012}{859544020592093752532733715131247} a^{14} + \frac{73617232232321528418058895587687}{859544020592093752532733715131247} a^{13} + \frac{327097891672381765833158850381636}{859544020592093752532733715131247} a^{12} + \frac{403292243506863101179387166356333}{859544020592093752532733715131247} a^{11} - \frac{17096298169454736741370251102509}{859544020592093752532733715131247} a^{10} - \frac{25553872555230461623800608407373}{78140365508372159321157610466477} a^{9} + \frac{3460806467715963029005999031201}{78140365508372159321157610466477} a^{8} - \frac{182126088468918072915336365970669}{859544020592093752532733715131247} a^{7} - \frac{33373230168013834054253442426080}{859544020592093752532733715131247} a^{6} + \frac{102845188508425038598617989301801}{859544020592093752532733715131247} a^{5} + \frac{120194100803740596610588959553167}{859544020592093752532733715131247} a^{4} - \frac{12269137362140670074122697124881}{66118770814776442502517978087019} a^{3} + \frac{57213077298842578807823101377421}{859544020592093752532733715131247} a^{2} - \frac{395770155601175733709805865152730}{859544020592093752532733715131247} a + \frac{397831923600031420194266671620824}{859544020592093752532733715131247}$

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:  $13$
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:  \( 11575529.0042 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1477:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1477 are not computed
Character table for t16n1477 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.309593125.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ R R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$