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Properties

Label 16.16.2802086158...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}$
Root discriminant $79.98$
Ramified primes $2, 5, 101, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125561, -1016658, 385812, 7052948, 3280572, -8196670, -6580482, 544174, 1336235, 54126, -116802, -6530, 5262, 212, -118, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 118*x^14 + 212*x^13 + 5262*x^12 - 6530*x^11 - 116802*x^10 + 54126*x^9 + 1336235*x^8 + 544174*x^7 - 6580482*x^6 - 8196670*x^5 + 3280572*x^4 + 7052948*x^3 + 385812*x^2 - 1016658*x + 125561)
 
gp: K = bnfinit(x^16 - 2*x^15 - 118*x^14 + 212*x^13 + 5262*x^12 - 6530*x^11 - 116802*x^10 + 54126*x^9 + 1336235*x^8 + 544174*x^7 - 6580482*x^6 - 8196670*x^5 + 3280572*x^4 + 7052948*x^3 + 385812*x^2 - 1016658*x + 125561, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 118 x^{14} + 212 x^{13} + 5262 x^{12} - 6530 x^{11} - 116802 x^{10} + 54126 x^{9} + 1336235 x^{8} + 544174 x^{7} - 6580482 x^{6} - 8196670 x^{5} + 3280572 x^{4} + 7052948 x^{3} + 385812 x^{2} - 1016658 x + 125561 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2802086158223343616000000000000=2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.98$
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:  $2, 5, 101, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3768366081270501696273261319774905211849801} a^{15} + \frac{1085152339180981575951446120635286383380651}{3768366081270501696273261319774905211849801} a^{14} + \frac{46283600175297531641626930443833504299015}{3768366081270501696273261319774905211849801} a^{13} - \frac{1861721654034670249326033900167538557229424}{3768366081270501696273261319774905211849801} a^{12} - \frac{833061069952804589034760024665873650439703}{3768366081270501696273261319774905211849801} a^{11} - \frac{1842672602606158396788955877202543520775211}{3768366081270501696273261319774905211849801} a^{10} - \frac{142524474368814769815335000240286038455767}{3768366081270501696273261319774905211849801} a^{9} - \frac{584353921388914428284169811822736114233508}{3768366081270501696273261319774905211849801} a^{8} - \frac{18988472533862226526514060052710111802007}{3768366081270501696273261319774905211849801} a^{7} - \frac{1257932551953804994984151378742850516651459}{3768366081270501696273261319774905211849801} a^{6} + \frac{1636843221790014752583217056484143106306169}{3768366081270501696273261319774905211849801} a^{5} + \frac{1438366105970297553374591267832742904907464}{3768366081270501696273261319774905211849801} a^{4} - \frac{994536768203053611748257630811712933964526}{3768366081270501696273261319774905211849801} a^{3} + \frac{259248519175797912393893234937340705092324}{3768366081270501696273261319774905211849801} a^{2} + \frac{1631636935414606931865939116305280685092218}{3768366081270501696273261319774905211849801} a + \frac{856235672625412024222986934728895329896777}{3768366081270501696273261319774905211849801}$

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:  $15$
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:  \( 12047605031.0 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
101Data not computed
641Data not computed