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Properties

Label 16.0.22833880280...2144.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{4}\cdot 17^{8}\cdot 97^{2}$
Root discriminant $38.45$
Ramified primes $2, 3, 17, 97$
Class number $64$ (GRH)
Class group $[2, 2, 16]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29584, -52288, 46936, 992, -39679, 43692, -19538, -2096, 9708, -7304, 3430, -1172, 372, -120, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 120*x^13 + 372*x^12 - 1172*x^11 + 3430*x^10 - 7304*x^9 + 9708*x^8 - 2096*x^7 - 19538*x^6 + 43692*x^5 - 39679*x^4 + 992*x^3 + 46936*x^2 - 52288*x + 29584)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 120*x^13 + 372*x^12 - 1172*x^11 + 3430*x^10 - 7304*x^9 + 9708*x^8 - 2096*x^7 - 19538*x^6 + 43692*x^5 - 39679*x^4 + 992*x^3 + 46936*x^2 - 52288*x + 29584, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 120 x^{13} + 372 x^{12} - 1172 x^{11} + 3430 x^{10} - 7304 x^{9} + 9708 x^{8} - 2096 x^{7} - 19538 x^{6} + 43692 x^{5} - 39679 x^{4} + 992 x^{3} + 46936 x^{2} - 52288 x + 29584 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(22833880280189357097222144=2^{32}\cdot 3^{4}\cdot 17^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.45$
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:  $2, 3, 17, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{3}{8} a^{10} - \frac{1}{2} a^{9} + \frac{3}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{13} + \frac{7}{32} a^{11} - \frac{7}{16} a^{10} + \frac{11}{32} a^{9} - \frac{3}{8} a^{8} - \frac{11}{32} a^{7} - \frac{3}{16} a^{6} - \frac{7}{32} a^{5} + \frac{3}{8} a^{4} - \frac{5}{32} a^{3} - \frac{7}{16} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{18846848} a^{14} + \frac{71107}{9423424} a^{13} - \frac{482169}{18846848} a^{12} - \frac{349105}{4711712} a^{11} - \frac{4365289}{18846848} a^{10} + \frac{2083947}{9423424} a^{9} + \frac{6003981}{18846848} a^{8} - \frac{441337}{2355856} a^{7} + \frac{4153077}{18846848} a^{6} - \frac{4033295}{9423424} a^{5} + \frac{2687843}{18846848} a^{4} + \frac{2235005}{4711712} a^{3} - \frac{587095}{1177928} a^{2} - \frac{31823}{1177928} a + \frac{514293}{1177928}$, $\frac{1}{1050260398720958848} a^{15} - \frac{1250383205}{1050260398720958848} a^{14} + \frac{15134251128981361}{1050260398720958848} a^{13} - \frac{53994355989781617}{1050260398720958848} a^{12} + \frac{226602466316939895}{1050260398720958848} a^{11} + \frac{437697087406897553}{1050260398720958848} a^{10} + \frac{188686352297245247}{1050260398720958848} a^{9} + \frac{516001397469792377}{1050260398720958848} a^{8} + \frac{410052121376110441}{1050260398720958848} a^{7} + \frac{357959991614763955}{1050260398720958848} a^{6} + \frac{434862808551182073}{1050260398720958848} a^{5} - \frac{395783641175898653}{1050260398720958848} a^{4} + \frac{6794575636278655}{131282549840119856} a^{3} - \frac{18384117185442889}{131282549840119856} a^{2} + \frac{30642882080653253}{65641274920059928} a + \frac{142178348630953}{1526541277210696}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{16}$, which has order $64$ (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:  $7$
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:  \( 45197.9459907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), 4.4.9248.1 x2, 4.4.4352.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{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}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$