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Properties

Label 16.8.35025358550...1641.2
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 83^{8}$
Root discriminant $296.16$
Ramified primes $41, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8881, 491775, -1021420, -2755458, 10611616, -10197714, 1824938, 1551360, -632204, 149658, -15460, -9464, 2912, -644, 112, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 112*x^14 - 644*x^13 + 2912*x^12 - 9464*x^11 - 15460*x^10 + 149658*x^9 - 632204*x^8 + 1551360*x^7 + 1824938*x^6 - 10197714*x^5 + 10611616*x^4 - 2755458*x^3 - 1021420*x^2 + 491775*x - 8881)
 
gp: K = bnfinit(x^16 - 8*x^15 + 112*x^14 - 644*x^13 + 2912*x^12 - 9464*x^11 - 15460*x^10 + 149658*x^9 - 632204*x^8 + 1551360*x^7 + 1824938*x^6 - 10197714*x^5 + 10611616*x^4 - 2755458*x^3 - 1021420*x^2 + 491775*x - 8881, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 112 x^{14} - 644 x^{13} + 2912 x^{12} - 9464 x^{11} - 15460 x^{10} + 149658 x^{9} - 632204 x^{8} + 1551360 x^{7} + 1824938 x^{6} - 10197714 x^{5} + 10611616 x^{4} - 2755458 x^{3} - 1021420 x^{2} + 491775 x - 8881 \)

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:  \(3502535855067952407242642672315293971641=41^{15}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.16$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1223877146484019035787943} a^{14} - \frac{7}{1223877146484019035787943} a^{13} - \frac{108846569425706484857012}{1223877146484019035787943} a^{12} - \frac{570797729929780126645780}{1223877146484019035787943} a^{11} - \frac{216249096115067798143631}{1223877146484019035787943} a^{10} - \frac{9807251902441533266734}{1223877146484019035787943} a^{9} - \frac{598422035166631154486415}{1223877146484019035787943} a^{8} - \frac{154611937694350468193401}{1223877146484019035787943} a^{7} + \frac{510601625241957485144223}{1223877146484019035787943} a^{6} + \frac{596188320802408800633048}{1223877146484019035787943} a^{5} - \frac{478150329312138483293866}{1223877146484019035787943} a^{4} - \frac{277807281288673287907962}{1223877146484019035787943} a^{3} - \frac{567873586501835694902950}{1223877146484019035787943} a^{2} - \frac{571978421675779325655400}{1223877146484019035787943} a + \frac{54909660295677440601530}{1223877146484019035787943}$, $\frac{1}{6955293823468680180382880069} a^{15} + \frac{2834}{6955293823468680180382880069} a^{14} + \frac{1112395479584547597046363288}{6955293823468680180382880069} a^{13} + \frac{2893082590880315913051629959}{6955293823468680180382880069} a^{12} + \frac{2579717644053016942261203708}{6955293823468680180382880069} a^{11} + \frac{1909261186735237195735416061}{6955293823468680180382880069} a^{10} + \frac{2152488250344518894608836717}{6955293823468680180382880069} a^{9} + \frac{1533211807164684832187646990}{6955293823468680180382880069} a^{8} + \frac{1869490384904452178843752480}{6955293823468680180382880069} a^{7} + \frac{2001960121551012190199944941}{6955293823468680180382880069} a^{6} + \frac{1842605880941381586481324548}{6955293823468680180382880069} a^{5} + \frac{2926090996983476539967807175}{6955293823468680180382880069} a^{4} + \frac{3212259909275119600385150618}{6955293823468680180382880069} a^{3} + \frac{75049624474555280842424990}{6955293823468680180382880069} a^{2} - \frac{880818481258309812987320526}{6955293823468680180382880069} a + \frac{2453215748850523338922915741}{6955293823468680180382880069}$

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:  $11$
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:  \( 51935584821200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.9242710845966413801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
41Data not computed
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$