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Properties

Label 16.8.86030523224...8544.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{64}\cdot 7^{8}\cdot 809$
Root discriminant $64.33$
Ramified primes $2, 7, 809$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3025, -10144, -6200, 24336, 15604, -9440, 1384, 12992, 2104, -5248, -1688, 912, 404, -64, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 64*x^13 + 404*x^12 + 912*x^11 - 1688*x^10 - 5248*x^9 + 2104*x^8 + 12992*x^7 + 1384*x^6 - 9440*x^5 + 15604*x^4 + 24336*x^3 - 6200*x^2 - 10144*x + 3025)
 
gp: K = bnfinit(x^16 - 40*x^14 - 64*x^13 + 404*x^12 + 912*x^11 - 1688*x^10 - 5248*x^9 + 2104*x^8 + 12992*x^7 + 1384*x^6 - 9440*x^5 + 15604*x^4 + 24336*x^3 - 6200*x^2 - 10144*x + 3025, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 64 x^{13} + 404 x^{12} + 912 x^{11} - 1688 x^{10} - 5248 x^{9} + 2104 x^{8} + 12992 x^{7} + 1384 x^{6} - 9440 x^{5} + 15604 x^{4} + 24336 x^{3} - 6200 x^{2} - 10144 x + 3025 \)

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:  \(86030523224437701564163948544=2^{64}\cdot 7^{8}\cdot 809\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.33$
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:  $2, 7, 809$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{29125} a^{14} + \frac{633}{29125} a^{13} + \frac{10683}{29125} a^{12} - \frac{8828}{29125} a^{11} + \frac{8377}{29125} a^{10} - \frac{10599}{29125} a^{9} - \frac{9912}{29125} a^{8} - \frac{2657}{5825} a^{7} - \frac{684}{29125} a^{6} - \frac{2134}{5825} a^{5} + \frac{14268}{29125} a^{4} - \frac{12451}{29125} a^{3} + \frac{2983}{29125} a^{2} - \frac{884}{29125} a + \frac{26}{1165}$, $\frac{1}{1032693317159536233125} a^{15} + \frac{6225674809247749}{1032693317159536233125} a^{14} - \frac{86852180459348656764}{1032693317159536233125} a^{13} - \frac{8827702878731412839}{41307732686381449325} a^{12} + \frac{188740792631026405329}{1032693317159536233125} a^{11} - \frac{87599426889451144117}{1032693317159536233125} a^{10} + \frac{413838167159256202604}{1032693317159536233125} a^{9} - \frac{102450530259903820577}{1032693317159536233125} a^{8} - \frac{25220504690933203119}{1032693317159536233125} a^{7} + \frac{455562923011877469861}{1032693317159536233125} a^{6} - \frac{465372715989193415952}{1032693317159536233125} a^{5} + \frac{358712168631251758012}{1032693317159536233125} a^{4} - \frac{269317280104702833333}{1032693317159536233125} a^{3} - \frac{232694153515316553106}{1032693317159536233125} a^{2} + \frac{36657445951198956231}{1032693317159536233125} a + \frac{18379316514348453321}{41307732686381449325}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 508263462.019 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1638:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1638 are not computed
Character table for t16n1638 is not computed

Intermediate fields

\(\Q(\sqrt{14}) \), 4.4.100352.2, 8.8.5156108238848.2

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ $16$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
809Data not computed