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Properties

Label 16.0.23455086603...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\cdot 1049^{2}$
Root discriminant $121.80$
Ramified primes $2, 5, 13, 29, 1049$
Class number $797936$ (GRH)
Class group $[2, 2, 2, 99742]$ (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5393065301, 0, 10745001410, 0, 3928831985, 0, 559128700, 0, 38263662, 0, 1377210, 0, 26620, 0, 260, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 260*x^14 + 26620*x^12 + 1377210*x^10 + 38263662*x^8 + 559128700*x^6 + 3928831985*x^4 + 10745001410*x^2 + 5393065301)
 
gp: K = bnfinit(x^16 + 260*x^14 + 26620*x^12 + 1377210*x^10 + 38263662*x^8 + 559128700*x^6 + 3928831985*x^4 + 10745001410*x^2 + 5393065301, 1)
 

Normalized defining polynomial

\( x^{16} + 260 x^{14} + 26620 x^{12} + 1377210 x^{10} + 38263662 x^{8} + 559128700 x^{6} + 3928831985 x^{4} + 10745001410 x^{2} + 5393065301 \)

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:  \(2345508660327164504585753600000000=2^{16}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.80$
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:  $2, 5, 13, 29, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{12} - \frac{1}{13} a^{8} - \frac{23}{52} a^{6} + \frac{21}{52} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{52} a^{13} - \frac{1}{13} a^{9} - \frac{23}{52} a^{7} + \frac{21}{52} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{139452047357761774384516837394156} a^{14} - \frac{16026976018300146056782185986}{34863011839440443596129209348539} a^{12} + \frac{1533579806560409637655315908191}{34863011839440443596129209348539} a^{10} + \frac{477696145877622125729593061151}{7339581439882198651816675652324} a^{8} - \frac{67625736449197536833822789482431}{139452047357761774384516837394156} a^{6} + \frac{20115727259318474096688034258143}{69726023678880887192258418697078} a^{4} + \frac{388615141793657015174129348316}{2681770141495418738163785334503} a^{2} + \frac{181437513810361391323146707}{2556501564819274297582254847}$, $\frac{1}{139452047357761774384516837394156} a^{15} - \frac{16026976018300146056782185986}{34863011839440443596129209348539} a^{13} + \frac{1533579806560409637655315908191}{34863011839440443596129209348539} a^{11} + \frac{477696145877622125729593061151}{7339581439882198651816675652324} a^{9} - \frac{67625736449197536833822789482431}{139452047357761774384516837394156} a^{7} + \frac{20115727259318474096688034258143}{69726023678880887192258418697078} a^{5} + \frac{388615141793657015174129348316}{2681770141495418738163785334503} a^{3} + \frac{181437513810361391323146707}{2556501564819274297582254847} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{99742}$, which has order $797936$ (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:  \( 10968.6213178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.2576088125.1

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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
1049Data not computed