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Properties

Label 16.4.16182878367...3184.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{34}\cdot 3^{8}\cdot 13^{4}\cdot 709^{2}$
Root discriminant $32.59$
Ramified primes $2, 3, 13, 709$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![913, 1128, -6178, -14536, -16632, -13028, -10746, -6408, -2920, -1568, -526, -128, -44, -16, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 16*x^13 - 44*x^12 - 128*x^11 - 526*x^10 - 1568*x^9 - 2920*x^8 - 6408*x^7 - 10746*x^6 - 13028*x^5 - 16632*x^4 - 14536*x^3 - 6178*x^2 + 1128*x + 913)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 16*x^13 - 44*x^12 - 128*x^11 - 526*x^10 - 1568*x^9 - 2920*x^8 - 6408*x^7 - 10746*x^6 - 13028*x^5 - 16632*x^4 - 14536*x^3 - 6178*x^2 + 1128*x + 913, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 16 x^{13} - 44 x^{12} - 128 x^{11} - 526 x^{10} - 1568 x^{9} - 2920 x^{8} - 6408 x^{7} - 10746 x^{6} - 13028 x^{5} - 16632 x^{4} - 14536 x^{3} - 6178 x^{2} + 1128 x + 913 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1618287836733993526493184=2^{34}\cdot 3^{8}\cdot 13^{4}\cdot 709^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.59$
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:  $2, 3, 13, 709$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{461493088275929035169786546029} a^{15} - \frac{1393223699523284418676221004}{12472786169619703653237474217} a^{14} + \frac{40608914610538622280113251141}{461493088275929035169786546029} a^{13} + \frac{218626883807944980611994113952}{461493088275929035169786546029} a^{12} - \frac{87092460194956486200223626213}{461493088275929035169786546029} a^{11} - \frac{181931695634158591020975572528}{461493088275929035169786546029} a^{10} + \frac{14957267062868109621322031021}{461493088275929035169786546029} a^{9} - \frac{225139955259185855313409032463}{461493088275929035169786546029} a^{8} + \frac{216304149871223039524415637155}{461493088275929035169786546029} a^{7} - \frac{127669638115453357147318570944}{461493088275929035169786546029} a^{6} + \frac{150901588333319685564781047316}{461493088275929035169786546029} a^{5} + \frac{4910318451448741470385301537}{10732397401765791515576431303} a^{4} + \frac{75229991032243296454428538546}{461493088275929035169786546029} a^{3} - \frac{85154233792226250196859776632}{461493088275929035169786546029} a^{2} - \frac{126181119621547979821981831875}{461493088275929035169786546029} a - \frac{56921481335497450024728774615}{461493088275929035169786546029}$

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:  $9$
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:  \( 672412.71446 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 8.4.224280576.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
709Data not computed