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Properties

Label 16.0.20928938182...1488.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 13^{4}\cdot 17^{15}$
Root discriminant $38.24$
Ramified primes $2, 13, 17$
Class number $200$ (GRH)
Class group $[2, 2, 50]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38659, 36221, 63157, 14047, 29261, 4447, 12724, -1408, 6221, -1694, 1800, -481, 302, -61, 26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 26*x^14 - 61*x^13 + 302*x^12 - 481*x^11 + 1800*x^10 - 1694*x^9 + 6221*x^8 - 1408*x^7 + 12724*x^6 + 4447*x^5 + 29261*x^4 + 14047*x^3 + 63157*x^2 + 36221*x + 38659)
 
gp: K = bnfinit(x^16 - 3*x^15 + 26*x^14 - 61*x^13 + 302*x^12 - 481*x^11 + 1800*x^10 - 1694*x^9 + 6221*x^8 - 1408*x^7 + 12724*x^6 + 4447*x^5 + 29261*x^4 + 14047*x^3 + 63157*x^2 + 36221*x + 38659, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 26 x^{14} - 61 x^{13} + 302 x^{12} - 481 x^{11} + 1800 x^{10} - 1694 x^{9} + 6221 x^{8} - 1408 x^{7} + 12724 x^{6} + 4447 x^{5} + 29261 x^{4} + 14047 x^{3} + 63157 x^{2} + 36221 x + 38659 \)

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:  \(20928938182187993309151488=2^{8}\cdot 13^{4}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.24$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{214707790924328667108242609799974101} a^{15} + \frac{75400069108543266125190828056387955}{214707790924328667108242609799974101} a^{14} - \frac{13144986065699827824899712058159620}{214707790924328667108242609799974101} a^{13} + \frac{66796799251570995484248220910163919}{214707790924328667108242609799974101} a^{12} + \frac{78212170340871826315801616027167013}{214707790924328667108242609799974101} a^{11} + \frac{50132173732741653469053841112551989}{214707790924328667108242609799974101} a^{10} + \frac{98519230305286593096941664693393160}{214707790924328667108242609799974101} a^{9} + \frac{94297321530362689183345178986734122}{214707790924328667108242609799974101} a^{8} + \frac{28561293920079046057248350676202868}{214707790924328667108242609799974101} a^{7} - \frac{29522122292032831719465532699180615}{214707790924328667108242609799974101} a^{6} + \frac{57846959451099931659558431749937447}{214707790924328667108242609799974101} a^{5} - \frac{10775357290939403731811744204094021}{214707790924328667108242609799974101} a^{4} - \frac{22997979474704512809573781523946127}{214707790924328667108242609799974101} a^{3} + \frac{104025610770156931413549997341478576}{214707790924328667108242609799974101} a^{2} + \frac{89639845629708211913454147562383593}{214707790924328667108242609799974101} a - \frac{73911116025651699633154933751108674}{214707790924328667108242609799974101}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{50}$, which has order $200$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Sibling fields

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 $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17Data not computed