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Properties

Label 16.16.1364873065...2944.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 449^{2}\cdot 1889^{5}$
Root discriminant $181.32$
Ramified primes $2, 449, 1889$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1186

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-126146235134, -103460794152, 393968857948, 473110428024, 114829145270, -30845655536, -11964273276, 774198304, 448690771, -9347912, -8499648, 54208, 87302, -120, -464, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 464*x^14 - 120*x^13 + 87302*x^12 + 54208*x^11 - 8499648*x^10 - 9347912*x^9 + 448690771*x^8 + 774198304*x^7 - 11964273276*x^6 - 30845655536*x^5 + 114829145270*x^4 + 473110428024*x^3 + 393968857948*x^2 - 103460794152*x - 126146235134)
 
gp: K = bnfinit(x^16 - 464*x^14 - 120*x^13 + 87302*x^12 + 54208*x^11 - 8499648*x^10 - 9347912*x^9 + 448690771*x^8 + 774198304*x^7 - 11964273276*x^6 - 30845655536*x^5 + 114829145270*x^4 + 473110428024*x^3 + 393968857948*x^2 - 103460794152*x - 126146235134, 1)
 

Normalized defining polynomial

\( x^{16} - 464 x^{14} - 120 x^{13} + 87302 x^{12} + 54208 x^{11} - 8499648 x^{10} - 9347912 x^{9} + 448690771 x^{8} + 774198304 x^{7} - 11964273276 x^{6} - 30845655536 x^{5} + 114829145270 x^{4} + 473110428024 x^{3} + 393968857948 x^{2} - 103460794152 x - 126146235134 \)

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:  \(1364873065607290683946275468103122944=2^{48}\cdot 449^{2}\cdot 1889^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $181.32$
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:  $2, 449, 1889$
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}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{15} - \frac{13209542132891098163768552382182302744445600325388398660250718963103419210935}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{14} + \frac{2553836758101200836711971605820532277070097532915012370772176151964280738569}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{13} - \frac{5717808682122466254748338824467894727302653944295288739881583323916392342499}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{12} + \frac{11871518516927979415356915205088738667985294511077284965851315758580177876386}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{11} + \frac{12017037134748624965497596113770226848216828915452893008676913474101283377314}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{10} + \frac{5578248294033835898365972525994204982336507537044518422702093801906820471253}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{9} - \frac{2696686860795062810346243085063567223115390717542206800950287411271645844587}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{8} + \frac{9805315989985724099252340796870551012599177448851283941857750496851571730563}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{7} + \frac{5670015192674314510967166333253812971531171439960753538917292521884985464651}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{6} - \frac{4271392596897805942802520894754881124805913793575415300457877741091740185214}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{5} - \frac{12545573936787429150116779377775106124655930796694173232252784229566606259701}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{4} - \frac{13755305353224974713321041283800850676264882401459370572005860787060138668205}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{3} - \frac{12509181902277537160518942726539837334646871914556707806853893759979020546360}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a^{2} + \frac{744732601945225644514743282327867315828293397730553058321433101456435183483}{27924015905140887822671301071738207736187920148826738710419034684980690435319} a + \frac{7340423118520557734030337727503973183291573907513663920465306914538979138075}{27924015905140887822671301071738207736187920148826738710419034684980690435319}$

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:  \( 22889846170400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
449Data not computed
1889Data not computed