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Properties

Label 16.4.36304411013...1216.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{48}\cdot 337^{4}$
Root discriminant $34.28$
Ramified primes $2, 337$
Class number $2$
Class group $[2]$
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-956, -8400, -26072, -39824, -39380, -28512, -15776, -8544, -3082, -1688, -296, -240, 22, -16, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 - 16*x^13 + 22*x^12 - 240*x^11 - 296*x^10 - 1688*x^9 - 3082*x^8 - 8544*x^7 - 15776*x^6 - 28512*x^5 - 39380*x^4 - 39824*x^3 - 26072*x^2 - 8400*x - 956)
 
gp: K = bnfinit(x^16 + 12*x^14 - 16*x^13 + 22*x^12 - 240*x^11 - 296*x^10 - 1688*x^9 - 3082*x^8 - 8544*x^7 - 15776*x^6 - 28512*x^5 - 39380*x^4 - 39824*x^3 - 26072*x^2 - 8400*x - 956, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} - 16 x^{13} + 22 x^{12} - 240 x^{11} - 296 x^{10} - 1688 x^{9} - 3082 x^{8} - 8544 x^{7} - 15776 x^{6} - 28512 x^{5} - 39380 x^{4} - 39824 x^{3} - 26072 x^{2} - 8400 x - 956 \)

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:  \(3630441101393431380361216=2^{48}\cdot 337^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.28$
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:  $2, 337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{254} a^{14} + \frac{18}{127} a^{13} + \frac{1}{127} a^{12} + \frac{15}{127} a^{11} + \frac{7}{127} a^{10} - \frac{27}{127} a^{9} + \frac{25}{127} a^{8} + \frac{12}{127} a^{7} + \frac{23}{127} a^{6} + \frac{61}{127} a^{5} - \frac{43}{127} a^{4} + \frac{34}{127} a^{3} - \frac{27}{127} a^{2} - \frac{10}{127} a + \frac{22}{127}$, $\frac{1}{20503162765341329219933762} a^{15} - \frac{23361660878471491844777}{20503162765341329219933762} a^{14} + \frac{469323317764081070820042}{10251581382670664609966881} a^{13} - \frac{347689384908932369279465}{20503162765341329219933762} a^{12} + \frac{4609865869219833190809705}{20503162765341329219933762} a^{11} + \frac{1140583799490008227102681}{10251581382670664609966881} a^{10} + \frac{330182904884712408357760}{10251581382670664609966881} a^{9} - \frac{937515634175790898515524}{10251581382670664609966881} a^{8} + \frac{4344415371899420407224336}{10251581382670664609966881} a^{7} - \frac{1881035479076101483908937}{10251581382670664609966881} a^{6} + \frac{1735566324469798652189589}{10251581382670664609966881} a^{5} - \frac{835038678483961003410672}{10251581382670664609966881} a^{4} - \frac{1584534871231319197574566}{10251581382670664609966881} a^{3} + \frac{4711895369497651842724623}{10251581382670664609966881} a^{2} + \frac{4307542691594126383576780}{10251581382670664609966881} a - \frac{3025629925126063262850316}{10251581382670664609966881}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 481586.437128 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

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 t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.1413480448.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
337Data not computed