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Properties

Label 16.16.1123010068...7408.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{67}\cdot 17^{4}\cdot 977^{4}$
Root discriminant $206.85$
Ramified primes $2, 17, 977$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1433

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7289004894728, 0, -7669495426592, 0, 846231957776, 0, -40118683872, 0, 1023306828, 0, -15027760, 0, 126364, 0, -560, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 560*x^14 + 126364*x^12 - 15027760*x^10 + 1023306828*x^8 - 40118683872*x^6 + 846231957776*x^4 - 7669495426592*x^2 + 7289004894728)
 
gp: K = bnfinit(x^16 - 560*x^14 + 126364*x^12 - 15027760*x^10 + 1023306828*x^8 - 40118683872*x^6 + 846231957776*x^4 - 7669495426592*x^2 + 7289004894728, 1)
 

Normalized defining polynomial

\( x^{16} - 560 x^{14} + 126364 x^{12} - 15027760 x^{10} + 1023306828 x^{8} - 40118683872 x^{6} + 846231957776 x^{4} - 7669495426592 x^{2} + 7289004894728 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11230100681627560957071852383825297408=2^{67}\cdot 17^{4}\cdot 977^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $206.85$
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:  $2, 17, 977$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{1954} a^{10} + \frac{417}{1954} a^{8} + \frac{331}{1954} a^{6} + \frac{227}{977} a^{4} + \frac{468}{977} a^{2}$, $\frac{1}{3908} a^{11} - \frac{140}{977} a^{9} - \frac{323}{1954} a^{7} - \frac{375}{977} a^{5} + \frac{234}{977} a^{3}$, $\frac{1}{3818116} a^{12} - \frac{140}{954529} a^{10} + \frac{31591}{954529} a^{8} + \frac{61176}{954529} a^{6} + \frac{12935}{954529} a^{4} - \frac{445}{977} a^{2}$, $\frac{1}{3818116} a^{13} + \frac{417}{3818116} a^{11} - \frac{105189}{954529} a^{9} - \frac{193219}{1909058} a^{7} - \frac{353440}{954529} a^{5} - \frac{211}{977} a^{3}$, $\frac{1}{14184146402296145857266300560452} a^{14} + \frac{695853517097288246870921}{14184146402296145857266300560452} a^{12} - \frac{135381389871627155358739164}{3546036600574036464316575140113} a^{10} + \frac{354669170248201093015110442576}{3546036600574036464316575140113} a^{8} + \frac{616231173333510567607867181148}{3546036600574036464316575140113} a^{6} - \frac{1370011136455019906005877633}{3629515456063496892852175169} a^{4} - \frac{1004813310163921691763700}{3714959525141757311005297} a^{2} + \frac{1467511276814820812930}{3802415071792996224161}$, $\frac{1}{14184146402296145857266300560452} a^{15} + \frac{695853517097288246870921}{14184146402296145857266300560452} a^{13} - \frac{135381389871627155358739164}{3546036600574036464316575140113} a^{11} + \frac{354669170248201093015110442576}{3546036600574036464316575140113} a^{9} + \frac{616231173333510567607867181148}{3546036600574036464316575140113} a^{7} - \frac{1370011136455019906005877633}{3629515456063496892852175169} a^{5} - \frac{1004813310163921691763700}{3714959525141757311005297} a^{3} + \frac{1467511276814820812930}{3802415071792996224161} a$

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:  $15$
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:  \( 42076984096600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1433:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 68 conjugacy class representatives for t16n1433 are not computed
Character table for t16n1433 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 R $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
977Data not computed