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Properties

Label 16.16.3578962189...8125.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 13^{6}\cdot 29^{7}\cdot 1049^{2}$
Root discriminant $60.90$
Ramified primes $5, 13, 29, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81089, -267154, -336979, 638886, 497217, -449307, -324977, 126777, 98370, -16935, -15461, 1074, 1311, -26, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 57*x^14 - 26*x^13 + 1311*x^12 + 1074*x^11 - 15461*x^10 - 16935*x^9 + 98370*x^8 + 126777*x^7 - 324977*x^6 - 449307*x^5 + 497217*x^4 + 638886*x^3 - 336979*x^2 - 267154*x + 81089)
 
gp: K = bnfinit(x^16 - 57*x^14 - 26*x^13 + 1311*x^12 + 1074*x^11 - 15461*x^10 - 16935*x^9 + 98370*x^8 + 126777*x^7 - 324977*x^6 - 449307*x^5 + 497217*x^4 + 638886*x^3 - 336979*x^2 - 267154*x + 81089, 1)
 

Normalized defining polynomial

\( x^{16} - 57 x^{14} - 26 x^{13} + 1311 x^{12} + 1074 x^{11} - 15461 x^{10} - 16935 x^{9} + 98370 x^{8} + 126777 x^{7} - 324977 x^{6} - 449307 x^{5} + 497217 x^{4} + 638886 x^{3} - 336979 x^{2} - 267154 x + 81089 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35789621892199165414211328125=5^{8}\cdot 13^{6}\cdot 29^{7}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.90$
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:  $5, 13, 29, 1049$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2945} a^{14} - \frac{994}{2945} a^{13} - \frac{241}{589} a^{12} + \frac{175}{589} a^{11} - \frac{1079}{2945} a^{10} + \frac{224}{589} a^{9} - \frac{223}{589} a^{8} + \frac{7}{19} a^{7} - \frac{12}{589} a^{6} + \frac{262}{2945} a^{5} + \frac{221}{589} a^{4} + \frac{215}{589} a^{3} + \frac{207}{2945} a^{2} - \frac{822}{2945} a - \frac{1074}{2945}$, $\frac{1}{71702413681390329456275} a^{15} + \frac{10500302233787424268}{71702413681390329456275} a^{14} + \frac{7144918541971976508657}{71702413681390329456275} a^{13} - \frac{5627248801359541883432}{14340482736278065891255} a^{12} - \frac{24338056637301187284169}{71702413681390329456275} a^{11} - \frac{29716823529449330960918}{71702413681390329456275} a^{10} + \frac{749971669045583481551}{14340482736278065891255} a^{9} - \frac{6124953682180000081809}{14340482736278065891255} a^{8} - \frac{6115781242272210016708}{14340482736278065891255} a^{7} + \frac{11068695733827778294557}{71702413681390329456275} a^{6} + \frac{4664098511987296079024}{71702413681390329456275} a^{5} + \frac{2706780518954314256876}{14340482736278065891255} a^{4} - \frac{2207755585353778875593}{71702413681390329456275} a^{3} + \frac{30834517041998326990662}{71702413681390329456275} a^{2} + \frac{286938219256329116193}{3773811246388964708225} a - \frac{20938107755434689609153}{71702413681390329456275}$

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:  $15$
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:  \( 1710284974.96 \) (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 $16$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
1049Data not computed