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Properties

Label 16.0.15010078369140625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 7841^{2}$
Root discriminant $10.26$
Ramified primes $5, 7841$
Class number $1$
Class group Trivial
Galois group 16T1497

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 9, -17, 22, -25, 33, -32, 29, -24, 21, -14, 9, -6, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 3*x^14 - 6*x^13 + 9*x^12 - 14*x^11 + 21*x^10 - 24*x^9 + 29*x^8 - 32*x^7 + 33*x^6 - 25*x^5 + 22*x^4 - 17*x^3 + 9*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 3*x^14 - 6*x^13 + 9*x^12 - 14*x^11 + 21*x^10 - 24*x^9 + 29*x^8 - 32*x^7 + 33*x^6 - 25*x^5 + 22*x^4 - 17*x^3 + 9*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 3 x^{14} - 6 x^{13} + 9 x^{12} - 14 x^{11} + 21 x^{10} - 24 x^{9} + 29 x^{8} - 32 x^{7} + 33 x^{6} - 25 x^{5} + 22 x^{4} - 17 x^{3} + 9 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15010078369140625=5^{12}\cdot 7841^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.26$
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:  $5, 7841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1301} a^{15} + \frac{499}{1301} a^{14} - \frac{289}{1301} a^{13} - \frac{95}{1301} a^{12} + \frac{646}{1301} a^{11} + \frac{338}{1301} a^{10} - \frac{109}{1301} a^{9} + \frac{118}{1301} a^{8} + \frac{484}{1301} a^{7} - \frac{18}{1301} a^{6} + \frac{140}{1301} a^{5} - \frac{279}{1301} a^{4} - \frac{271}{1301} a^{3} - \frac{213}{1301} a^{2} + \frac{191}{1301} a + \frac{523}{1301}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1461}{1301} a^{15} - \frac{822}{1301} a^{14} + \frac{4499}{1301} a^{13} - \frac{7394}{1301} a^{12} + \frac{10989}{1301} a^{11} - \frac{17475}{1301} a^{10} + \frac{25493}{1301} a^{9} - \frac{26655}{1301} a^{8} + \frac{34507}{1301} a^{7} - \frac{34104}{1301} a^{6} + \frac{34109}{1301} a^{5} - \frac{21222}{1301} a^{4} + \frac{22991}{1301} a^{3} - \frac{11963}{1301} a^{2} + \frac{4540}{1301} a - \frac{885}{1301} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 103.338172859 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1497:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4900625.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7841Data not computed