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Properties

Label 16.8.77727803230...2256.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{58}\cdot 3^{4}\cdot 577^{2}$
Root discriminant $35.95$
Ramified primes $2, 3, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1385

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-248, -128, 4608, 2816, 1328, -1504, -2176, -1696, -784, 176, 432, 272, 40, -32, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 32*x^13 + 40*x^12 + 272*x^11 + 432*x^10 + 176*x^9 - 784*x^8 - 1696*x^7 - 2176*x^6 - 1504*x^5 + 1328*x^4 + 2816*x^3 + 4608*x^2 - 128*x - 248)
 
gp: K = bnfinit(x^16 - 16*x^14 - 32*x^13 + 40*x^12 + 272*x^11 + 432*x^10 + 176*x^9 - 784*x^8 - 1696*x^7 - 2176*x^6 - 1504*x^5 + 1328*x^4 + 2816*x^3 + 4608*x^2 - 128*x - 248, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 32 x^{13} + 40 x^{12} + 272 x^{11} + 432 x^{10} + 176 x^{9} - 784 x^{8} - 1696 x^{7} - 2176 x^{6} - 1504 x^{5} + 1328 x^{4} + 2816 x^{3} + 4608 x^{2} - 128 x - 248 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7772780323046872376672256=2^{58}\cdot 3^{4}\cdot 577^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.95$
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:  $2, 3, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{84} a^{14} - \frac{1}{28} a^{13} + \frac{1}{84} a^{12} + \frac{1}{21} a^{11} - \frac{1}{14} a^{10} - \frac{1}{6} a^{9} + \frac{1}{14} a^{8} + \frac{1}{21} a^{7} + \frac{2}{21} a^{6} - \frac{2}{21} a^{5} + \frac{1}{7} a^{4} - \frac{2}{21} a^{3} + \frac{5}{21} a^{2} + \frac{1}{21} a - \frac{8}{21}$, $\frac{1}{9446949993409711738284} a^{15} - \frac{2011526246866324024}{2361737498352427934571} a^{14} + \frac{1942336832240562010}{76185080592013804341} a^{13} - \frac{1690556054161397771}{1574491665568285289714} a^{12} - \frac{267210478992630063563}{4723474996704855869142} a^{11} - \frac{986909254002781966993}{4723474996704855869142} a^{10} + \frac{485479983981733090670}{2361737498352427934571} a^{9} - \frac{14492381624225550985}{4723474996704855869142} a^{8} + \frac{299475385292858926199}{4723474996704855869142} a^{7} - \frac{49549963802252672398}{337391071193203990653} a^{6} - \frac{1098740904891797217595}{2361737498352427934571} a^{5} + \frac{191668158272624058670}{2361737498352427934571} a^{4} + \frac{1367274605880892894}{2361737498352427934571} a^{3} + \frac{825045968251996677899}{2361737498352427934571} a^{2} + \frac{35421269062347286070}{112463690397734663551} a + \frac{14013781699082840957}{76185080592013804341}$

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:  $11$
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:  \( 6662652.60444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1385:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1385 are not computed
Character table for t16n1385 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.38721814528.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
577Data not computed