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Properties

Label 16.0.78345418253...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{2}\cdot 661^{4}$
Root discriminant $26.97$
Ramified primes $5, 41, 661$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, -131, 530, -967, 2263, -3328, 4616, -4498, 3950, -2539, 1537, -682, 302, -88, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 28*x^14 - 88*x^13 + 302*x^12 - 682*x^11 + 1537*x^10 - 2539*x^9 + 3950*x^8 - 4498*x^7 + 4616*x^6 - 3328*x^5 + 2263*x^4 - 967*x^3 + 530*x^2 - 131*x + 61)
 
gp: K = bnfinit(x^16 - 4*x^15 + 28*x^14 - 88*x^13 + 302*x^12 - 682*x^11 + 1537*x^10 - 2539*x^9 + 3950*x^8 - 4498*x^7 + 4616*x^6 - 3328*x^5 + 2263*x^4 - 967*x^3 + 530*x^2 - 131*x + 61, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 28 x^{14} - 88 x^{13} + 302 x^{12} - 682 x^{11} + 1537 x^{10} - 2539 x^{9} + 3950 x^{8} - 4498 x^{7} + 4616 x^{6} - 3328 x^{5} + 2263 x^{4} - 967 x^{3} + 530 x^{2} - 131 x + 61 \)

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:  \(78345418253203369140625=5^{12}\cdot 41^{2}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.97$
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:  $5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{275} a^{14} - \frac{1}{25} a^{13} + \frac{4}{275} a^{12} + \frac{12}{55} a^{11} - \frac{82}{275} a^{10} - \frac{63}{275} a^{9} - \frac{1}{11} a^{8} - \frac{16}{275} a^{7} - \frac{13}{275} a^{6} + \frac{69}{275} a^{5} + \frac{111}{275} a^{4} - \frac{19}{275} a^{3} + \frac{8}{55} a^{2} + \frac{12}{275} a - \frac{129}{275}$, $\frac{1}{4677850897161475} a^{15} + \frac{6677044622187}{4677850897161475} a^{14} + \frac{462513393511316}{4677850897161475} a^{13} + \frac{192154565283992}{4677850897161475} a^{12} + \frac{485882330339503}{4677850897161475} a^{11} + \frac{192934201793081}{425259172469225} a^{10} - \frac{705203818663189}{4677850897161475} a^{9} + \frac{1460050689928184}{4677850897161475} a^{8} + \frac{483253821271854}{4677850897161475} a^{7} - \frac{324921323908453}{935570179432295} a^{6} + \frac{1838031591367328}{4677850897161475} a^{5} - \frac{257492916920836}{4677850897161475} a^{4} + \frac{156525935342838}{425259172469225} a^{3} + \frac{305174358359247}{4677850897161475} a^{2} + \frac{1185318538036762}{4677850897161475} a + \frac{1730620352359518}{4677850897161475}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{76561602918138}{4677850897161475} a^{15} - \frac{4850607025328}{85051834493845} a^{14} + \frac{2009220578212374}{4677850897161475} a^{13} - \frac{5756098686318798}{4677850897161475} a^{12} + \frac{20351078334816134}{4677850897161475} a^{11} - \frac{8586604710809118}{935570179432295} a^{10} + \frac{98630314767380306}{4677850897161475} a^{9} - \frac{153344833481748663}{4677850897161475} a^{8} + \frac{240932215279915198}{4677850897161475} a^{7} - \frac{257007708296569822}{4677850897161475} a^{6} + \frac{53187620311320226}{935570179432295} a^{5} - \frac{178935928740576599}{4677850897161475} a^{4} + \frac{120664703103162758}{4677850897161475} a^{3} - \frac{44585127393037399}{4677850897161475} a^{2} + \frac{24336367694307539}{4677850897161475} a - \frac{405281674207967}{425259172469225} \) (order $10$)
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:  \( 119348.84075 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.16525.1, 4.4.82625.1, 8.0.6826890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
661Data not computed