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Properties

Label 16.0.60479166614...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{8}\cdot 7841^{4}$
Root discriminant $35.39$
Ramified primes $2, 5, 7841$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5375956, -2518894, -291719, -327971, 39628, 324405, -3020, -98637, 8573, 15748, -2156, -1596, 307, 95, -22, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 22*x^14 + 95*x^13 + 307*x^12 - 1596*x^11 - 2156*x^10 + 15748*x^9 + 8573*x^8 - 98637*x^7 - 3020*x^6 + 324405*x^5 + 39628*x^4 - 327971*x^3 - 291719*x^2 - 2518894*x + 5375956)
 
gp: K = bnfinit(x^16 - 3*x^15 - 22*x^14 + 95*x^13 + 307*x^12 - 1596*x^11 - 2156*x^10 + 15748*x^9 + 8573*x^8 - 98637*x^7 - 3020*x^6 + 324405*x^5 + 39628*x^4 - 327971*x^3 - 291719*x^2 - 2518894*x + 5375956, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 22 x^{14} + 95 x^{13} + 307 x^{12} - 1596 x^{11} - 2156 x^{10} + 15748 x^{9} + 8573 x^{8} - 98637 x^{7} - 3020 x^{6} + 324405 x^{5} + 39628 x^{4} - 327971 x^{3} - 291719 x^{2} - 2518894 x + 5375956 \)

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:  \(6047916661441537600000000=2^{12}\cdot 5^{8}\cdot 7841^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.39$
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:  $2, 5, 7841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{205630538752617912812279913121763309569763035678} a^{15} - \frac{66591347438088489355195912020874972441361081245}{205630538752617912812279913121763309569763035678} a^{14} + \frac{46003733536253405930898414346286739806615645994}{102815269376308956406139956560881654784881517839} a^{13} + \frac{9988190314396062610577601379103868461978758253}{205630538752617912812279913121763309569763035678} a^{12} - \frac{93294369812739153824024169798987995361788895563}{205630538752617912812279913121763309569763035678} a^{11} + \frac{39331803733902241611607618491563687838595275623}{102815269376308956406139956560881654784881517839} a^{10} - \frac{41622132215557323977639861333429648830723153231}{102815269376308956406139956560881654784881517839} a^{9} - \frac{44473789311866798358267045932069606019800238533}{102815269376308956406139956560881654784881517839} a^{8} + \frac{85397744289873412399606101285505799165514702937}{205630538752617912812279913121763309569763035678} a^{7} + \frac{57984057474417734997388516871880538896744686123}{205630538752617912812279913121763309569763035678} a^{6} + \frac{17073507421294766238937095307895330556434635517}{102815269376308956406139956560881654784881517839} a^{5} - \frac{76277582011449247782055099528205376150597928625}{205630538752617912812279913121763309569763035678} a^{4} - \frac{31019950533967611191440010934712456026717797067}{102815269376308956406139956560881654784881517839} a^{3} - \frac{69222941385436532176627854881889406582271491375}{205630538752617912812279913121763309569763035678} a^{2} + \frac{59294649733763499404232423962991018957876321711}{205630538752617912812279913121763309569763035678} a + \frac{18243551833070226640304449300827583489366315134}{102815269376308956406139956560881654784881517839}$

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:  $7$
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:  \( 1100568.72659 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.0.4900625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.9$x^{12} - 18 x^{10} + 7 x^{8} - 28 x^{6} - x^{4} - 18 x^{2} - 7$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7841Data not computed