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Properties

Label 16.4.17347984308...9629.1
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 101^{3}$
Root discriminant $28.34$
Ramified primes $17, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1351

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-103, -637, -579, -904, -826, -45, -550, 437, -344, 308, -139, 77, -28, 2, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 2*x^13 - 28*x^12 + 77*x^11 - 139*x^10 + 308*x^9 - 344*x^8 + 437*x^7 - 550*x^6 - 45*x^5 - 826*x^4 - 904*x^3 - 579*x^2 - 637*x - 103)
 
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 + 2*x^13 - 28*x^12 + 77*x^11 - 139*x^10 + 308*x^9 - 344*x^8 + 437*x^7 - 550*x^6 - 45*x^5 - 826*x^4 - 904*x^3 - 579*x^2 - 637*x - 103, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 2 x^{14} + 2 x^{13} - 28 x^{12} + 77 x^{11} - 139 x^{10} + 308 x^{9} - 344 x^{8} + 437 x^{7} - 550 x^{6} - 45 x^{5} - 826 x^{4} - 904 x^{3} - 579 x^{2} - 637 x - 103 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(173479843081977336549629=17^{14}\cdot 101^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.34$
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:  $17, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{14} - \frac{3}{26} a^{13} - \frac{3}{13} a^{12} - \frac{1}{2} a^{11} - \frac{3}{13} a^{10} + \frac{6}{13} a^{9} - \frac{9}{26} a^{7} - \frac{3}{13} a^{6} - \frac{11}{26} a^{5} + \frac{5}{26} a^{4} + \frac{2}{13} a^{3} + \frac{5}{26} a^{2} - \frac{4}{13}$, $\frac{1}{8824865413834117414} a^{15} - \frac{27061867817522630}{4412432706917058707} a^{14} - \frac{838749214803501207}{4412432706917058707} a^{13} - \frac{355839704043454816}{4412432706917058707} a^{12} + \frac{1751847921559096745}{8824865413834117414} a^{11} + \frac{730741612690324501}{8824865413834117414} a^{10} + \frac{993190134960543149}{4412432706917058707} a^{9} + \frac{439070225291130957}{8824865413834117414} a^{8} + \frac{3260989799892475623}{8824865413834117414} a^{7} - \frac{586518638058481520}{4412432706917058707} a^{6} - \frac{1362985069893398942}{4412432706917058707} a^{5} + \frac{680873553330038495}{4412432706917058707} a^{4} + \frac{2165471852797463156}{4412432706917058707} a^{3} + \frac{87203063363822957}{8824865413834117414} a^{2} + \frac{1012013708823098493}{8824865413834117414} a + \frac{26548668157043743}{4412432706917058707}$

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:  $9$
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:  \( 160658.508957 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1351:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 56 conjugacy class representatives for t16n1351 are not computed
Character table for t16n1351 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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
17Data not computed
101Data not computed