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Properties

Label 16.10.5064182024...8827.3
Degree $16$
Signature $[10, 3]$
Discriminant $-\,17^{14}\cdot 67^{3}$
Root discriminant $26.24$
Ramified primes $17, 67$
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![-1, 10, 68, -96, -547, -26, 541, -214, -181, 213, 11, -28, -12, 9, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 9*x^13 - 12*x^12 - 28*x^11 + 11*x^10 + 213*x^9 - 181*x^8 - 214*x^7 + 541*x^6 - 26*x^5 - 547*x^4 - 96*x^3 + 68*x^2 + 10*x - 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 9*x^13 - 12*x^12 - 28*x^11 + 11*x^10 + 213*x^9 - 181*x^8 - 214*x^7 + 541*x^6 - 26*x^5 - 547*x^4 - 96*x^3 + 68*x^2 + 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 9 x^{13} - 12 x^{12} - 28 x^{11} + 11 x^{10} + 213 x^{9} - 181 x^{8} - 214 x^{7} + 541 x^{6} - 26 x^{5} - 547 x^{4} - 96 x^{3} + 68 x^{2} + 10 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-50641820249485101608827=-\,17^{14}\cdot 67^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.24$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1072} a^{14} - \frac{3}{536} a^{13} + \frac{495}{1072} a^{12} + \frac{409}{1072} a^{11} + \frac{271}{1072} a^{10} - \frac{271}{1072} a^{9} - \frac{205}{536} a^{8} - \frac{37}{268} a^{7} + \frac{405}{1072} a^{6} + \frac{97}{268} a^{5} + \frac{69}{268} a^{4} + \frac{149}{536} a^{3} + \frac{309}{1072} a^{2} - \frac{107}{268} a + \frac{91}{1072}$, $\frac{1}{2548901603398336} a^{15} + \frac{58350322357}{2548901603398336} a^{14} - \frac{184820171766339}{2548901603398336} a^{13} + \frac{517536633660063}{1274450801699168} a^{12} - \frac{88219693008567}{1274450801699168} a^{11} + \frac{59916700529771}{1274450801699168} a^{10} - \frac{738240171205695}{2548901603398336} a^{9} + \frac{167111892413471}{1274450801699168} a^{8} - \frac{246675844268231}{2548901603398336} a^{7} + \frac{1187356921889819}{2548901603398336} a^{6} + \frac{24848068171397}{159306350212396} a^{5} + \frac{3766261514363}{1274450801699168} a^{4} - \frac{154520755132189}{2548901603398336} a^{3} - \frac{869989745978293}{2548901603398336} a^{2} - \frac{292979274493401}{2548901603398336} a + \frac{947697218320441}{2548901603398336}$

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:  $12$
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:  \( 338820.573524 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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