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Properties

Label 16.0.51399544780...7056.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -512, 512, -448, 368, -496, 648, -780, 481, -12, -24, 14, -1, 2, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 2*x^13 - x^12 + 14*x^11 - 24*x^10 - 12*x^9 + 481*x^8 - 780*x^7 + 648*x^6 - 496*x^5 + 368*x^4 - 448*x^3 + 512*x^2 - 512*x + 256)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 + 2*x^13 - x^12 + 14*x^11 - 24*x^10 - 12*x^9 + 481*x^8 - 780*x^7 + 648*x^6 - 496*x^5 + 368*x^4 - 448*x^3 + 512*x^2 - 512*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} + 2 x^{13} - x^{12} + 14 x^{11} - 24 x^{10} - 12 x^{9} + 481 x^{8} - 780 x^{7} + 648 x^{6} - 496 x^{5} + 368 x^{4} - 448 x^{3} + 512 x^{2} - 512 x + 256 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{48} a^{12} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{5}{24} a^{9} - \frac{17}{48} a^{8} + \frac{1}{24} a^{7} - \frac{1}{12} a^{6} + \frac{5}{12} a^{5} + \frac{1}{48} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{288} a^{13} - \frac{1}{144} a^{12} + \frac{7}{144} a^{11} + \frac{1}{16} a^{10} + \frac{13}{96} a^{9} + \frac{59}{144} a^{8} + \frac{3}{8} a^{7} - \frac{3}{8} a^{6} + \frac{113}{288} a^{5} + \frac{11}{24} a^{4} - \frac{3}{8} a^{3} + \frac{5}{36} a^{2} - \frac{1}{18} a + \frac{4}{9}$, $\frac{1}{18697536} a^{14} - \frac{1381}{849888} a^{13} - \frac{191}{1038752} a^{12} + \frac{151861}{9348768} a^{11} - \frac{697171}{6232512} a^{10} + \frac{1349477}{9348768} a^{9} - \frac{821671}{2337192} a^{8} + \frac{249815}{1558128} a^{7} + \frac{1355969}{18697536} a^{6} - \frac{274451}{2337192} a^{5} + \frac{45861}{129844} a^{4} + \frac{79117}{292149} a^{3} - \frac{2477}{7491} a^{2} - \frac{20179}{44946} a - \frac{132280}{292149}$, $\frac{1}{1616551567488} a^{15} - \frac{1175}{808275783744} a^{14} + \frac{128987581}{808275783744} a^{13} - \frac{6755256811}{808275783744} a^{12} - \frac{84263252537}{1616551567488} a^{11} + \frac{36219229757}{808275783744} a^{10} - \frac{703169719}{50517236484} a^{9} + \frac{615831751}{1780343136} a^{8} - \frac{94185113839}{1616551567488} a^{7} - \frac{74284612343}{202068945936} a^{6} - \frac{80659167337}{202068945936} a^{5} - \frac{42784311929}{101034472968} a^{4} + \frac{4335657057}{11226052552} a^{3} - \frac{49720809}{215885626} a^{2} + \frac{14619097}{1403256569} a + \frac{5827502099}{12629309121}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1293814223}{124350120576} a^{15} - \frac{162934985}{15543765072} a^{14} + \frac{272723365}{20725020096} a^{13} + \frac{1789136257}{62175060288} a^{12} + \frac{2766616789}{124350120576} a^{11} + \frac{5437625981}{31087530144} a^{10} - \frac{17201065}{215885626} a^{9} - \frac{5174418331}{31087530144} a^{8} + \frac{593915295079}{124350120576} a^{7} - \frac{23653512543}{6908340032} a^{6} + \frac{144659771671}{31087530144} a^{5} - \frac{876967729}{353267388} a^{4} + \frac{20405190757}{7771882536} a^{3} - \frac{1238336515}{431771252} a^{2} + \frac{2654426255}{971485317} a - \frac{3253322179}{971485317} \) (order $12$)
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:  \( 7754.35289362 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{21})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), 4.2.84672.5 x2, 4.2.1728.1 x2, 4.0.21168.1 x2, 4.0.432.1 x2, 8.0.49787136.1, 8.0.7169347584.11, 8.0.2985984.1, 8.4.7169347584.1 x2, 8.0.448084224.8 x2, 8.0.7169347584.3 x2, 8.0.7169347584.5 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$