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Properties

Label 16.16.9609771106...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 31^{2}\cdot 61^{2}$
Root discriminant $42.06$
Ramified primes $2, 3, 5, 31, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1224

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -916, 7206, 378, -25808, 8394, 30582, -15392, -13699, 9118, 2062, -2176, 52, 198, -24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 24*x^14 + 198*x^13 + 52*x^12 - 2176*x^11 + 2062*x^10 + 9118*x^9 - 13699*x^8 - 15392*x^7 + 30582*x^6 + 8394*x^5 - 25808*x^4 + 378*x^3 + 7206*x^2 - 916*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 - 24*x^14 + 198*x^13 + 52*x^12 - 2176*x^11 + 2062*x^10 + 9118*x^9 - 13699*x^8 - 15392*x^7 + 30582*x^6 + 8394*x^5 - 25808*x^4 + 378*x^3 + 7206*x^2 - 916*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 24 x^{14} + 198 x^{13} + 52 x^{12} - 2176 x^{11} + 2062 x^{10} + 9118 x^{9} - 13699 x^{8} - 15392 x^{7} + 30582 x^{6} + 8394 x^{5} - 25808 x^{4} + 378 x^{3} + 7206 x^{2} - 916 x + 1 \)

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:  \(96097711067136000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 31^{2}\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.06$
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:  $2, 3, 5, 31, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{9} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{3} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{4} - \frac{1}{2} a^{2} + \frac{1}{10} a$, $\frac{1}{310} a^{14} - \frac{11}{310} a^{13} - \frac{7}{155} a^{12} + \frac{19}{310} a^{11} - \frac{32}{155} a^{10} - \frac{38}{155} a^{9} + \frac{21}{310} a^{8} - \frac{91}{310} a^{7} - \frac{42}{155} a^{6} + \frac{2}{155} a^{5} - \frac{99}{310} a^{4} + \frac{29}{310} a^{3} - \frac{139}{310} a^{2} + \frac{109}{310} a + \frac{51}{310}$, $\frac{1}{9088928792020190} a^{15} + \frac{5322334091507}{9088928792020190} a^{14} + \frac{35115621348055}{908892879202019} a^{13} - \frac{200071568916334}{4544464396010095} a^{12} - \frac{1138980009727347}{9088928792020190} a^{11} + \frac{319121380333267}{1817785758404038} a^{10} + \frac{388723621539229}{4544464396010095} a^{9} - \frac{1380019906898083}{9088928792020190} a^{8} - \frac{564160987051047}{1817785758404038} a^{7} + \frac{2890851605386607}{9088928792020190} a^{6} + \frac{742083653426354}{4544464396010095} a^{5} + \frac{864799059416225}{1817785758404038} a^{4} - \frac{161378435942071}{4544464396010095} a^{3} - \frac{211759291694659}{4544464396010095} a^{2} + \frac{106864690181042}{908892879202019} a + \frac{4096677693451793}{9088928792020190}$

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:  \( 65909534.0034 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1224:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1224 are not computed
Character table for t16n1224 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.8.19764000000.2, 8.8.10044000000.2, 8.8.612684000000.2

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{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}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$