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Properties

Label 16.4.30076929402...5376.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{64}\cdot 113^{4}$
Root discriminant $52.17$
Ramified primes $2, 113$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4216, 14848, -1536, 12480, 40112, 9536, 9776, 2176, -1876, -1632, -432, -432, 28, -16, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 16*x^13 + 28*x^12 - 432*x^11 - 432*x^10 - 1632*x^9 - 1876*x^8 + 2176*x^7 + 9776*x^6 + 9536*x^5 + 40112*x^4 + 12480*x^3 - 1536*x^2 + 14848*x - 4216)
 
gp: K = bnfinit(x^16 + 16*x^14 - 16*x^13 + 28*x^12 - 432*x^11 - 432*x^10 - 1632*x^9 - 1876*x^8 + 2176*x^7 + 9776*x^6 + 9536*x^5 + 40112*x^4 + 12480*x^3 - 1536*x^2 + 14848*x - 4216, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 16 x^{13} + 28 x^{12} - 432 x^{11} - 432 x^{10} - 1632 x^{9} - 1876 x^{8} + 2176 x^{7} + 9776 x^{6} + 9536 x^{5} + 40112 x^{4} + 12480 x^{3} - 1536 x^{2} + 14848 x - 4216 \)

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:  \(3007692940260731871482085376=2^{64}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.17$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{68} a^{14} - \frac{1}{34} a^{13} + \frac{1}{34} a^{12} - \frac{1}{68} a^{11} - \frac{3}{34} a^{10} + \frac{3}{34} a^{9} + \frac{1}{17} a^{8} - \frac{7}{34} a^{7} - \frac{4}{17} a^{6} + \frac{3}{17} a^{5} - \frac{6}{17} a^{4} - \frac{4}{17} a^{3} - \frac{5}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{156706575452695692733468437931396} a^{15} + \frac{217691664054756074278709115252}{39176643863173923183367109482849} a^{14} - \frac{137662478718202419683996120324}{2304508462539642540198065263697} a^{13} + \frac{17270470556202129047027969255501}{156706575452695692733468437931396} a^{12} - \frac{18498523887303917515027768598121}{156706575452695692733468437931396} a^{11} + \frac{209492922269900577493134651161}{4609016925079285080396130527394} a^{10} - \frac{12614593958744668083186293150931}{78353287726347846366734218965698} a^{9} + \frac{5568470456417918517836205984993}{78353287726347846366734218965698} a^{8} - \frac{2745618998726225332795742758423}{78353287726347846366734218965698} a^{7} + \frac{8961507882653332150175406659292}{39176643863173923183367109482849} a^{6} - \frac{5796214640097964900128862711717}{39176643863173923183367109482849} a^{5} - \frac{7965058102731091463786944394599}{39176643863173923183367109482849} a^{4} - \frac{1460387532589043343586000499362}{39176643863173923183367109482849} a^{3} - \frac{19166708452332817620907637119193}{39176643863173923183367109482849} a^{2} - \frac{13843611372751506939433463066965}{39176643863173923183367109482849} a + \frac{815311548585983516530932522021}{2304508462539642540198065263697}$

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:  \( 29801511.0477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.242665652224.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
113Data not computed