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Properties

Label 16.2.74413923181...6875.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,3^{12}\cdot 5^{12}\cdot 179^{3}$
Root discriminant $20.16$
Ramified primes $3, 5, 179$
Class number $1$
Class group Trivial
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -9, 9, 189, 132, 285, -15, 75, -77, 3, -11, -6, 9, -9, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 9*x^13 + 9*x^12 - 6*x^11 - 11*x^10 + 3*x^9 - 77*x^8 + 75*x^7 - 15*x^6 + 285*x^5 + 132*x^4 + 189*x^3 + 9*x^2 - 9*x - 9)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 9*x^13 + 9*x^12 - 6*x^11 - 11*x^10 + 3*x^9 - 77*x^8 + 75*x^7 - 15*x^6 + 285*x^5 + 132*x^4 + 189*x^3 + 9*x^2 - 9*x - 9, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} - 9 x^{13} + 9 x^{12} - 6 x^{11} - 11 x^{10} + 3 x^{9} - 77 x^{8} + 75 x^{7} - 15 x^{6} + 285 x^{5} + 132 x^{4} + 189 x^{3} + 9 x^{2} - 9 x - 9 \)

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:  \(-744139231811279296875=-\,3^{12}\cdot 5^{12}\cdot 179^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.16$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{10301904861312243} a^{15} - \frac{1619956468349893}{10301904861312243} a^{14} - \frac{310226642537107}{10301904861312243} a^{13} + \frac{659588310349672}{10301904861312243} a^{12} - \frac{4836840136669448}{10301904861312243} a^{11} - \frac{86086893149}{43104204440637} a^{10} - \frac{1847894007743753}{10301904861312243} a^{9} + \frac{432802949878355}{10301904861312243} a^{8} + \frac{2788977383297168}{10301904861312243} a^{7} - \frac{2168244293382497}{10301904861312243} a^{6} - \frac{3257235995873510}{10301904861312243} a^{5} + \frac{3103033745657372}{10301904861312243} a^{4} + \frac{1006486935496674}{3433968287104081} a^{3} + \frac{770394049430346}{3433968287104081} a^{2} + \frac{464581012164971}{3433968287104081} a + \frac{1051981789934202}{3433968287104081}$

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:  \( 7019.90710016 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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
Arithmetically equvalently 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 $16$ R R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{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$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$