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Properties

Label 16.12.3393001956...1409.3
Degree $16$
Signature $[12, 2]$
Discriminant $17^{14}\cdot 67^{4}$
Root discriminant $34.13$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, 510, -187, -2856, 34, 4063, -612, -1343, 1157, -934, -228, 419, -155, 61, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 + 61*x^13 - 155*x^12 + 419*x^11 - 228*x^10 - 934*x^9 + 1157*x^8 - 1343*x^7 - 612*x^6 + 4063*x^5 + 34*x^4 - 2856*x^3 - 187*x^2 + 510*x + 17)
 
gp: K = bnfinit(x^16 - 4*x^15 - 10*x^14 + 61*x^13 - 155*x^12 + 419*x^11 - 228*x^10 - 934*x^9 + 1157*x^8 - 1343*x^7 - 612*x^6 + 4063*x^5 + 34*x^4 - 2856*x^3 - 187*x^2 + 510*x + 17, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 10 x^{14} + 61 x^{13} - 155 x^{12} + 419 x^{11} - 228 x^{10} - 934 x^{9} + 1157 x^{8} - 1343 x^{7} - 612 x^{6} + 4063 x^{5} + 34 x^{4} - 2856 x^{3} - 187 x^{2} + 510 x + 17 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3393001956715501807791409=17^{14}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.13$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4406900509454021775923} a^{15} + \frac{991087376478263406350}{4406900509454021775923} a^{14} + \frac{1332032632938736014147}{4406900509454021775923} a^{13} + \frac{365848433806554042486}{4406900509454021775923} a^{12} - \frac{1194643952684160351116}{4406900509454021775923} a^{11} + \frac{1850844087655510846333}{4406900509454021775923} a^{10} - \frac{1990802537457081683504}{4406900509454021775923} a^{9} - \frac{240165016397279379753}{4406900509454021775923} a^{8} - \frac{1281850640283621956791}{4406900509454021775923} a^{7} + \frac{1513330992658188261045}{4406900509454021775923} a^{6} + \frac{293054251104229910943}{4406900509454021775923} a^{5} - \frac{1489281207802085077271}{4406900509454021775923} a^{4} + \frac{73018397550949689295}{4406900509454021775923} a^{3} + \frac{1828474867407877597581}{4406900509454021775923} a^{2} + \frac{2146596776069470496029}{4406900509454021775923} a + \frac{677727578056637679332}{4406900509454021775923}$

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:  $13$
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:  \( 3731014.34737 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed