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Properties

Label 16.0.12365706172...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 41^{4}\cdot 439^{2}\cdot 2411^{2}$
Root discriminant $32.05$
Ramified primes $5, 41, 439, 2411$
Class number $36$ (GRH)
Class group $[36]$ (GRH)
Galois group 16T1648

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10109, -7613, 14051, -11073, 10488, -6916, 5624, -3296, 1851, -910, 498, -196, 54, -20, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 15*x^14 - 20*x^13 + 54*x^12 - 196*x^11 + 498*x^10 - 910*x^9 + 1851*x^8 - 3296*x^7 + 5624*x^6 - 6916*x^5 + 10488*x^4 - 11073*x^3 + 14051*x^2 - 7613*x + 10109)
 
gp: K = bnfinit(x^16 - 6*x^15 + 15*x^14 - 20*x^13 + 54*x^12 - 196*x^11 + 498*x^10 - 910*x^9 + 1851*x^8 - 3296*x^7 + 5624*x^6 - 6916*x^5 + 10488*x^4 - 11073*x^3 + 14051*x^2 - 7613*x + 10109, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 15 x^{14} - 20 x^{13} + 54 x^{12} - 196 x^{11} + 498 x^{10} - 910 x^{9} + 1851 x^{8} - 3296 x^{7} + 5624 x^{6} - 6916 x^{5} + 10488 x^{4} - 11073 x^{3} + 14051 x^{2} - 7613 x + 10109 \)

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:  \(1236570617253236016015625=5^{8}\cdot 41^{4}\cdot 439^{2}\cdot 2411^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.05$
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:  $5, 41, 439, 2411$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{49928318613984978251834169871} a^{15} + \frac{4123211959160217129659686171}{49928318613984978251834169871} a^{14} + \frac{1501400610685768175045192105}{4538938055816816204712197261} a^{13} - \frac{5811496734488562875262444498}{49928318613984978251834169871} a^{12} - \frac{11939103726539963167639427469}{49928318613984978251834169871} a^{11} - \frac{14621086866815407204548913938}{49928318613984978251834169871} a^{10} + \frac{2021122558651136381788350518}{49928318613984978251834169871} a^{9} - \frac{9724200446518437743655320062}{49928318613984978251834169871} a^{8} + \frac{15519052248090227226348177459}{49928318613984978251834169871} a^{7} + \frac{21428412067659260390923528096}{49928318613984978251834169871} a^{6} + \frac{11829030865828952955871882253}{49928318613984978251834169871} a^{5} + \frac{16547949597660654598444709586}{49928318613984978251834169871} a^{4} - \frac{9404340789797428379346384339}{49928318613984978251834169871} a^{3} + \frac{9795074331030984401401258477}{49928318613984978251834169871} a^{2} - \frac{280735659009381094091787699}{49928318613984978251834169871} a + \frac{376800607602205697494518929}{4538938055816816204712197261}$

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

Class group and class number

$C_{36}$, which has order $36$ (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:  \( 4846.77455506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1648:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 65 conjugacy class representatives for t16n1648 are not computed
Character table for t16n1648 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1, 8.8.661518125.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
439Data not computed
2411Data not computed