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Properties

Label 16.8.38882231785...3904.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{46}\cdot 17^{8}\cdot 89^{2}$
Root discriminant $53.01$
Ramified primes $2, 17, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7921, 0, -35244, 0, 37682, 0, 7196, 0, -8510, 0, -1388, 0, 130, 0, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 28*x^14 + 130*x^12 - 1388*x^10 - 8510*x^8 + 7196*x^6 + 37682*x^4 - 35244*x^2 + 7921)
 
gp: K = bnfinit(x^16 + 28*x^14 + 130*x^12 - 1388*x^10 - 8510*x^8 + 7196*x^6 + 37682*x^4 - 35244*x^2 + 7921, 1)
 

Normalized defining polynomial

\( x^{16} + 28 x^{14} + 130 x^{12} - 1388 x^{10} - 8510 x^{8} + 7196 x^{6} + 37682 x^{4} - 35244 x^{2} + 7921 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3888223178515238551136763904=2^{46}\cdot 17^{8}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.01$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{8} + \frac{3}{7} a^{6} - \frac{3}{7} a^{4} - \frac{3}{14} a^{2} + \frac{1}{14}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{9} + \frac{3}{7} a^{7} - \frac{3}{7} a^{5} - \frac{3}{14} a^{3} + \frac{1}{14} a$, $\frac{1}{112} a^{12} + \frac{1}{56} a^{10} - \frac{25}{112} a^{8} - \frac{1}{7} a^{6} + \frac{7}{16} a^{4} + \frac{17}{56} a^{2} - \frac{25}{112}$, $\frac{1}{112} a^{13} + \frac{1}{56} a^{11} - \frac{25}{112} a^{9} - \frac{1}{7} a^{7} + \frac{7}{16} a^{5} + \frac{17}{56} a^{3} - \frac{25}{112} a$, $\frac{1}{91982788339792} a^{14} + \frac{57600318781}{13140398334256} a^{12} - \frac{1363002112679}{91982788339792} a^{10} - \frac{20358135768689}{91982788339792} a^{8} + \frac{2567402638071}{13140398334256} a^{6} - \frac{16230623041861}{91982788339792} a^{4} - \frac{1945674702385}{13140398334256} a^{2} + \frac{417617914743}{1033514475728}$, $\frac{1}{183965576679584} a^{15} - \frac{1}{183965576679584} a^{14} + \frac{57600318781}{26280796668512} a^{13} - \frac{57600318781}{26280796668512} a^{12} - \frac{1363002112679}{183965576679584} a^{11} + \frac{1363002112679}{183965576679584} a^{10} - \frac{20358135768689}{183965576679584} a^{9} + \frac{20358135768689}{183965576679584} a^{8} + \frac{2567402638071}{26280796668512} a^{7} - \frac{2567402638071}{26280796668512} a^{6} + \frac{75752165297931}{183965576679584} a^{5} - \frac{75752165297931}{183965576679584} a^{4} - \frac{1945674702385}{26280796668512} a^{3} + \frac{1945674702385}{26280796668512} a^{2} + \frac{417617914743}{2067028951456} a - \frac{417617914743}{2067028951456}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 87058605.7096 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.24.34$x^{8} + 32 x^{4} + 144$$8$$1$$24$$Q_8:C_2$$[2, 3, 4]^{2}$
2.8.22.102$x^{8} + 8 x^{7} + 16 x^{5} + 144$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed