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Properties

Label 16.0.10966149720...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 401^{2}$
Root discriminant $75.42$
Ramified primes $2, 5, 101, 401$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![180416, 2194816, 6567808, -3840832, 3030528, -1888832, 1322496, -773696, 397768, -158208, 57824, -15024, 3868, -596, 112, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 112*x^14 - 596*x^13 + 3868*x^12 - 15024*x^11 + 57824*x^10 - 158208*x^9 + 397768*x^8 - 773696*x^7 + 1322496*x^6 - 1888832*x^5 + 3030528*x^4 - 3840832*x^3 + 6567808*x^2 + 2194816*x + 180416)
 
gp: K = bnfinit(x^16 - 8*x^15 + 112*x^14 - 596*x^13 + 3868*x^12 - 15024*x^11 + 57824*x^10 - 158208*x^9 + 397768*x^8 - 773696*x^7 + 1322496*x^6 - 1888832*x^5 + 3030528*x^4 - 3840832*x^3 + 6567808*x^2 + 2194816*x + 180416, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 112 x^{14} - 596 x^{13} + 3868 x^{12} - 15024 x^{11} + 57824 x^{10} - 158208 x^{9} + 397768 x^{8} - 773696 x^{7} + 1322496 x^{6} - 1888832 x^{5} + 3030528 x^{4} - 3840832 x^{3} + 6567808 x^{2} + 2194816 x + 180416 \)

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:  \(1096614972044148736000000000000=2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 401^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.42$
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:  $2, 5, 101, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{8} a^{10}$, $\frac{1}{16} a^{11}$, $\frac{1}{16} a^{12}$, $\frac{1}{16} a^{13}$, $\frac{1}{32} a^{14}$, $\frac{1}{6557947575444957349394654535965316268120288} a^{15} + \frac{1666113816297292704497885626786862308375}{819743446930619668674331816995664533515036} a^{14} - \frac{13135487030806161720543694716230087692541}{3278973787722478674697327267982658134060144} a^{13} + \frac{54819830418825352559813546855839996341969}{3278973787722478674697327267982658134060144} a^{12} - \frac{40307208644801649929490962604362177988077}{1639486893861239337348663633991329067030072} a^{11} + \frac{67770330078474301931744007191921362225897}{1639486893861239337348663633991329067030072} a^{10} + \frac{62093128826427175198410352033906857749807}{1639486893861239337348663633991329067030072} a^{9} - \frac{9895446571818932825569983734176320237950}{204935861732654917168582954248916133378759} a^{8} + \frac{66491829471296790135832236095529981620357}{819743446930619668674331816995664533515036} a^{7} + \frac{1388697968409316984005010402037476489955}{19993742608063894357910532121845476427196} a^{6} - \frac{917684291775222358238214459025671205663}{9996871304031947178955266060922738213598} a^{5} - \frac{9645616471217107540646592443110752867523}{204935861732654917168582954248916133378759} a^{4} - \frac{16347516777796326607366000153572729375262}{204935861732654917168582954248916133378759} a^{3} - \frac{101903835424394016838445693944126713331641}{204935861732654917168582954248916133378759} a^{2} + \frac{13790211107609465994306568846565459147159}{204935861732654917168582954248916133378759} a - \frac{76461845154623284783381980482548595908645}{204935861732654917168582954248916133378759}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 36090357.4029 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.6464000000.11

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
401Data not computed