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Properties

Label 16.0.14986078891...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{6}\cdot 89^{8}$
Root discriminant $182.38$
Ramified primes $2, 5, 29, 89$
Class number $16870528$ (GRH)
Class group $[2, 2, 2, 4, 527204]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1319155617025, 0, 1156113911550, 0, 281375148280, 0, 21323886470, 0, 722041781, 0, 12627854, 0, 117827, 0, 551, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 551*x^14 + 117827*x^12 + 12627854*x^10 + 722041781*x^8 + 21323886470*x^6 + 281375148280*x^4 + 1156113911550*x^2 + 1319155617025)
 
gp: K = bnfinit(x^16 + 551*x^14 + 117827*x^12 + 12627854*x^10 + 722041781*x^8 + 21323886470*x^6 + 281375148280*x^4 + 1156113911550*x^2 + 1319155617025, 1)
 

Normalized defining polynomial

\( x^{16} + 551 x^{14} + 117827 x^{12} + 12627854 x^{10} + 722041781 x^{8} + 21323886470 x^{6} + 281375148280 x^{4} + 1156113911550 x^{2} + 1319155617025 \)

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:  \(1498607889164246756499840640000000000=2^{16}\cdot 5^{10}\cdot 29^{6}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.38$
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:  $2, 5, 29, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{89} a^{8} + \frac{17}{89} a^{6} - \frac{9}{89} a^{4}$, $\frac{1}{89} a^{9} + \frac{17}{89} a^{7} - \frac{9}{89} a^{5}$, $\frac{1}{89} a^{10} - \frac{31}{89} a^{6} - \frac{25}{89} a^{4}$, $\frac{1}{89} a^{11} - \frac{31}{89} a^{7} - \frac{25}{89} a^{5}$, $\frac{1}{1148545} a^{12} + \frac{19}{39605} a^{10} + \frac{58}{39605} a^{8} + \frac{3556}{12905} a^{6} - \frac{5576}{12905} a^{4}$, $\frac{1}{1148545} a^{13} + \frac{19}{39605} a^{11} + \frac{58}{39605} a^{9} + \frac{3556}{12905} a^{7} - \frac{5576}{12905} a^{5}$, $\frac{1}{2413092790911993954285058795067755} a^{14} - \frac{1007044236291798940213618472}{2413092790911993954285058795067755} a^{12} + \frac{350096900220263888963686864666}{83210096238344619113277889485095} a^{10} - \frac{121206492996669660605064574943}{27113402145078583756011896573795} a^{8} - \frac{3970955259576634276859835321}{934944901554433922621099881855} a^{6} - \frac{39233662782477878307190778443}{304644967922231278157437040155} a^{4} + \frac{245306563580828852353401407}{2100999778774008814878876139} a^{2} - \frac{7807071027789639914018195}{23606739087348413650324451}$, $\frac{1}{2413092790911993954285058795067755} a^{15} - \frac{1007044236291798940213618472}{2413092790911993954285058795067755} a^{13} + \frac{350096900220263888963686864666}{83210096238344619113277889485095} a^{11} - \frac{121206492996669660605064574943}{27113402145078583756011896573795} a^{9} - \frac{3970955259576634276859835321}{934944901554433922621099881855} a^{7} - \frac{39233662782477878307190778443}{304644967922231278157437040155} a^{5} + \frac{245306563580828852353401407}{2100999778774008814878876139} a^{3} - \frac{7807071027789639914018195}{23606739087348413650324451} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{527204}$, which has order $16870528$ (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:  $7$
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:  \( 13345.9350553 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.2, 4.4.2225.1, 4.4.725.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$