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Properties

Label 16.0.635134283447265625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{4}$
Root discriminant $12.96$
Ramified primes $5, 101$
Class number $1$
Class group Trivial
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 1, 30, -8, -85, 83, 5, 5, -55, 17, 15, 7, -10, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 - 10*x^13 + 7*x^12 + 15*x^11 + 17*x^10 - 55*x^9 + 5*x^8 + 5*x^7 + 83*x^6 - 85*x^5 - 8*x^4 + 30*x^3 + x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - x^14 - 10*x^13 + 7*x^12 + 15*x^11 + 17*x^10 - 55*x^9 + 5*x^8 + 5*x^7 + 83*x^6 - 85*x^5 - 8*x^4 + 30*x^3 + x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} - 10 x^{13} + 7 x^{12} + 15 x^{11} + 17 x^{10} - 55 x^{9} + 5 x^{8} + 5 x^{7} + 83 x^{6} - 85 x^{5} - 8 x^{4} + 30 x^{3} + x^{2} - 5 x + 1 \)

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:  \(635134283447265625=5^{14}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.96$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} + \frac{2}{11} a^{13} + \frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{3}{11} a^{10} + \frac{1}{11} a^{9} + \frac{2}{11} a^{8} + \frac{2}{11} a^{7} + \frac{5}{11} a^{6} - \frac{4}{11} a^{4} - \frac{5}{11} a^{3} + \frac{1}{11} a^{2} - \frac{2}{11} a - \frac{5}{11}$, $\frac{1}{61370089} a^{15} + \frac{1425420}{61370089} a^{14} - \frac{688365}{5579099} a^{13} + \frac{987694}{61370089} a^{12} - \frac{1272064}{61370089} a^{11} + \frac{4024531}{61370089} a^{10} - \frac{11356822}{61370089} a^{9} - \frac{27980677}{61370089} a^{8} - \frac{18932195}{61370089} a^{7} + \frac{23525560}{61370089} a^{6} + \frac{15494893}{61370089} a^{5} + \frac{3569409}{61370089} a^{4} - \frac{24884565}{61370089} a^{3} - \frac{21542595}{61370089} a^{2} + \frac{25865517}{61370089} a + \frac{17297387}{61370089}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{48592075}{61370089} a^{15} - \frac{47789420}{61370089} a^{14} - \frac{18013769}{61370089} a^{13} + \frac{444126588}{61370089} a^{12} + \frac{85962634}{61370089} a^{11} - \frac{456430154}{61370089} a^{10} - \frac{1176041941}{61370089} a^{9} + \frac{1353251914}{61370089} a^{8} + \frac{507848882}{61370089} a^{7} + \frac{55500389}{5579099} a^{6} - \frac{3069686329}{61370089} a^{5} + \frac{1588185289}{61370089} a^{4} + \frac{757845227}{61370089} a^{3} - \frac{520734228}{61370089} a^{2} - \frac{221376685}{61370089} a + \frac{11950721}{5579099} \) (order $10$)
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:  \( 705.592331631 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.3.4$x^{4} + 808$$4$$1$$3$$C_4$$[\ ]_{4}$