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Properties

Label 16.16.1024092040...8816.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{56}\cdot 17^{10}\cdot 89^{3}$
Root discriminant $154.22$
Ramified primes $2, 17, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![411536, 0, -1646144, 0, 2495872, 0, -1842528, 0, 689800, 0, -120144, 0, 7296, 0, -152, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 152*x^14 + 7296*x^12 - 120144*x^10 + 689800*x^8 - 1842528*x^6 + 2495872*x^4 - 1646144*x^2 + 411536)
 
gp: K = bnfinit(x^16 - 152*x^14 + 7296*x^12 - 120144*x^10 + 689800*x^8 - 1842528*x^6 + 2495872*x^4 - 1646144*x^2 + 411536, 1)
 

Normalized defining polynomial

\( x^{16} - 152 x^{14} + 7296 x^{12} - 120144 x^{10} + 689800 x^{8} - 1842528 x^{6} + 2495872 x^{4} - 1646144 x^{2} + 411536 \)

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:  \(102409204095580621592359633279778816=2^{56}\cdot 17^{10}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $154.22$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{16} a^{9} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{80} a^{10} + \frac{1}{10} a^{6} + \frac{9}{20} a^{2} - \frac{2}{5}$, $\frac{1}{80} a^{11} + \frac{1}{10} a^{7} + \frac{9}{20} a^{3} - \frac{2}{5} a$, $\frac{1}{2720} a^{12} - \frac{1}{170} a^{10} - \frac{41}{1360} a^{8} - \frac{41}{340} a^{6} - \frac{151}{680} a^{4} + \frac{1}{10} a^{2} + \frac{7}{20}$, $\frac{1}{5440} a^{13} + \frac{9}{2720} a^{11} - \frac{41}{2720} a^{9} - \frac{7}{680} a^{7} - \frac{151}{1360} a^{5} - \frac{9}{40} a^{3} + \frac{19}{40} a$, $\frac{1}{11161193670720} a^{14} + \frac{312457}{465049736280} a^{12} + \frac{738360359}{206688771680} a^{10} - \frac{25515490243}{930099472560} a^{8} - \frac{39491387623}{558059683536} a^{6} - \frac{16200782989}{697574604420} a^{4} + \frac{16091907317}{82067600520} a^{2} - \frac{19481418983}{41033800260}$, $\frac{1}{11161193670720} a^{15} + \frac{312457}{465049736280} a^{13} + \frac{738360359}{206688771680} a^{11} - \frac{25515490243}{930099472560} a^{9} - \frac{39491387623}{558059683536} a^{7} - \frac{16200782989}{697574604420} a^{5} + \frac{16091907317}{82067600520} a^{3} - \frac{19481418983}{41033800260} a$

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:  \( 15067369753300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1472 are not computed
Character table for t16n1472 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.7794452332544.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.28.84$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 20 x^{4} + 8 x^{2} + 14$$8$$1$$28$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 9/2]^{2}$
2.8.28.83$x^{8} + 24 x^{6} + 30 x^{4} + 8 x^{2} + 15$$8$$1$$28$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 9/2]^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89Data not computed