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Properties

Label 16.0.59834937239...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{12}\cdot 5^{4}$
Root discriminant $35.36$
Ramified primes $2, 3, 5$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^2:Q_8$ (as 16T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1438, 4592, 4968, 1392, 764, 432, -304, -232, 1026, 256, 56, 120, 128, -48, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 48*x^13 + 128*x^12 + 120*x^11 + 56*x^10 + 256*x^9 + 1026*x^8 - 232*x^7 - 304*x^6 + 432*x^5 + 764*x^4 + 1392*x^3 + 4968*x^2 + 4592*x + 1438)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 48*x^13 + 128*x^12 + 120*x^11 + 56*x^10 + 256*x^9 + 1026*x^8 - 232*x^7 - 304*x^6 + 432*x^5 + 764*x^4 + 1392*x^3 + 4968*x^2 + 4592*x + 1438, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 48 x^{13} + 128 x^{12} + 120 x^{11} + 56 x^{10} + 256 x^{9} + 1026 x^{8} - 232 x^{7} - 304 x^{6} + 432 x^{5} + 764 x^{4} + 1392 x^{3} + 4968 x^{2} + 4592 x + 1438 \)

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:  \(5983493723923509411840000=2^{54}\cdot 3^{12}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.36$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{26028720450766303043119221579313} a^{15} - \frac{8570950898060585819446262573169}{26028720450766303043119221579313} a^{14} - \frac{11504010022200966393704904382325}{26028720450766303043119221579313} a^{13} - \frac{4457585046592797777685807779597}{26028720450766303043119221579313} a^{12} + \frac{10315260382734451249825284072142}{26028720450766303043119221579313} a^{11} - \frac{3767660141805920449028344066246}{26028720450766303043119221579313} a^{10} - \frac{2371726445436838608378183628676}{26028720450766303043119221579313} a^{9} + \frac{6746558009173750061637261821696}{26028720450766303043119221579313} a^{8} + \frac{4392026430514951353035897300}{26028720450766303043119221579313} a^{7} - \frac{8632445318804913322391124041731}{26028720450766303043119221579313} a^{6} - \frac{11649470477774355306587774772098}{26028720450766303043119221579313} a^{5} + \frac{481192628148104127724059168475}{26028720450766303043119221579313} a^{4} - \frac{12183512625374584084971132094620}{26028720450766303043119221579313} a^{3} - \frac{12551545462021860966159962902533}{26028720450766303043119221579313} a^{2} - \frac{6457837223495907705443374546009}{26028720450766303043119221579313} a + \frac{9519442224124770528266766366592}{26028720450766303043119221579313}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 34258.1672484 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:Q_8$ (as 16T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:Q_8$
Character table for $C_2^2:Q_8$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.276480.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.276480.2, 8.0.305764761600.24, 8.0.12230590464.1, 8.8.8493465600.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$