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Properties

Label 16.4.64964301804...6416.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{25}\cdot 13^{7}\cdot 17^{8}\cdot 89^{7}$
Root discriminant $266.56$
Ramified primes $2, 13, 17, 89$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2314, 0, -35867, 0, 190905, 0, -409578, 0, 252606, 0, 96630, 0, -3426, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 - 3426*x^12 + 96630*x^10 + 252606*x^8 - 409578*x^6 + 190905*x^4 - 35867*x^2 + 2314)
 
gp: K = bnfinit(x^16 - x^14 - 3426*x^12 + 96630*x^10 + 252606*x^8 - 409578*x^6 + 190905*x^4 - 35867*x^2 + 2314, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} - 3426 x^{12} + 96630 x^{10} + 252606 x^{8} - 409578 x^{6} + 190905 x^{4} - 35867 x^{2} + 2314 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(649643018047387088671190961146262716416=2^{25}\cdot 13^{7}\cdot 17^{8}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $266.56$
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:  $2, 13, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{18} a^{12} + \frac{1}{6} a^{10} + \frac{7}{18} a^{8} + \frac{1}{18} a^{6} + \frac{5}{18} a^{4} - \frac{1}{6} a^{2} + \frac{4}{9}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{7}{36} a^{9} - \frac{7}{36} a^{8} + \frac{1}{36} a^{7} - \frac{1}{36} a^{6} + \frac{5}{36} a^{5} - \frac{5}{36} a^{4} + \frac{5}{12} a^{3} - \frac{5}{12} a^{2} - \frac{5}{18} a + \frac{5}{18}$, $\frac{1}{51225160162023118008} a^{14} + \frac{24939540269998943}{25612580081011559004} a^{12} - \frac{1}{2} a^{11} + \frac{5837675550022999015}{12806290040505779502} a^{10} + \frac{7656301485302884063}{25612580081011559004} a^{8} - \frac{1}{2} a^{7} - \frac{1189918514485765175}{4268763346835259834} a^{6} + \frac{10345489474962678163}{25612580081011559004} a^{4} - \frac{1}{2} a^{3} + \frac{8866132205307247823}{51225160162023118008} a^{2} - \frac{9452245298659365887}{25612580081011559004}$, $\frac{1}{51225160162023118008} a^{15} + \frac{24939540269998943}{25612580081011559004} a^{13} + \frac{5837675550022999015}{12806290040505779502} a^{11} - \frac{1}{2} a^{10} + \frac{7656301485302884063}{25612580081011559004} a^{9} - \frac{1189918514485765175}{4268763346835259834} a^{7} - \frac{1}{2} a^{6} + \frac{10345489474962678163}{25612580081011559004} a^{5} + \frac{8866132205307247823}{51225160162023118008} a^{3} - \frac{1}{2} a^{2} - \frac{9452245298659365887}{25612580081011559004} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $9$
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:  \( 3157912200040 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n1022
Character table for t16n1022 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.334373.2, 8.4.16557918172192384.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.37$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.0.1$x^{8} - x + 62$$1$$8$$0$$C_8$$[\ ]^{8}$