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Properties

Label 16.0.49087529081...7824.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 17^{8}$
Root discriminant $30.25$
Ramified primes $2, 17$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2\times C_2^2.D_4$ (as 16T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, 0, -980, 0, 1438, 0, -476, 0, 402, 0, -76, 0, -2, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 2*x^12 - 76*x^10 + 402*x^8 - 476*x^6 + 1438*x^4 - 980*x^2 + 2401)
 
gp: K = bnfinit(x^16 - 4*x^14 - 2*x^12 - 76*x^10 + 402*x^8 - 476*x^6 + 1438*x^4 - 980*x^2 + 2401, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 2 x^{12} - 76 x^{10} + 402 x^{8} - 476 x^{6} + 1438 x^{4} - 980 x^{2} + 2401 \)

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:  \(490875290811165073997824=2^{46}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.25$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{6} a$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{4} a^{3} - \frac{5}{12} a^{2} - \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{8} + \frac{1}{3} a^{6} + \frac{1}{12} a^{4} + \frac{1}{3} a^{2} + \frac{1}{4}$, $\frac{1}{84} a^{13} + \frac{1}{28} a^{11} - \frac{1}{12} a^{10} - \frac{1}{42} a^{9} - \frac{1}{12} a^{8} + \frac{3}{7} a^{7} - \frac{11}{84} a^{5} - \frac{1}{3} a^{4} - \frac{1}{4} a^{3} - \frac{5}{12} a^{2} + \frac{5}{42} a + \frac{1}{4}$, $\frac{1}{3144359400} a^{14} - \frac{66272651}{3144359400} a^{12} + \frac{52196299}{628871880} a^{10} + \frac{96119059}{3144359400} a^{8} - \frac{188446957}{1048119800} a^{6} - \frac{215840077}{449194200} a^{4} + \frac{1232342471}{3144359400} a^{2} - \frac{2564533}{64170600}$, $\frac{1}{22010515800} a^{15} - \frac{66272651}{22010515800} a^{13} - \frac{105021671}{4402103160} a^{11} - \frac{1}{12} a^{10} + \frac{119383003}{7336838600} a^{9} - \frac{1}{12} a^{8} + \frac{9915857129}{22010515800} a^{7} - \frac{215840077}{3144359400} a^{5} - \frac{1}{3} a^{4} - \frac{7938705779}{22010515800} a^{3} - \frac{5}{12} a^{2} - \frac{8016123}{21390200} a + \frac{1}{4}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( -\frac{123881}{1048119800} a^{14} - \frac{114869}{1048119800} a^{12} + \frac{2485183}{628871880} a^{10} + \frac{43579663}{3144359400} a^{8} - \frac{69667147}{3144359400} a^{6} - \frac{149185489}{449194200} a^{4} + \frac{71819049}{1048119800} a^{2} + \frac{7596319}{64170600} \) (order $8$)
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:  \( 180834.249785 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2.D_4$ (as 16T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^2.D_4$
Character table for $C_2\times C_2^2.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(i, \sqrt{17})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(i, \sqrt{34})\), 8.8.175156232192.1, 8.0.175156232192.1, 8.0.5473632256.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.22.87$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
2.8.24.14$x^{8} + 12 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 2$$8$$1$$24$$D_4$$[2, 3, 4]$
$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}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$