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Properties

Label 16.0.43778798779...5536.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 41^{14}$
Root discriminant $300.32$
Ramified primes $2, 3, 41$
Class number $128107648$ (GRH)
Class group $[2, 2, 2, 2, 8006728]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45174951936, 0, 105408221184, 0, 42978947328, 0, 5280868224, 0, 282271716, 0, 7199928, 0, 87822, 0, 492, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 492*x^14 + 87822*x^12 + 7199928*x^10 + 282271716*x^8 + 5280868224*x^6 + 42978947328*x^4 + 105408221184*x^2 + 45174951936)
 
gp: K = bnfinit(x^16 + 492*x^14 + 87822*x^12 + 7199928*x^10 + 282271716*x^8 + 5280868224*x^6 + 42978947328*x^4 + 105408221184*x^2 + 45174951936, 1)
 

Normalized defining polynomial

\( x^{16} + 492 x^{14} + 87822 x^{12} + 7199928 x^{10} + 282271716 x^{8} + 5280868224 x^{6} + 42978947328 x^{4} + 105408221184 x^{2} + 45174951936 \)

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:  \(4377879877971569095410685630463959105536=2^{44}\cdot 3^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $300.32$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1968=2^{4}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1968}(1,·)$, $\chi_{1968}(899,·)$, $\chi_{1968}(73,·)$, $\chi_{1968}(337,·)$, $\chi_{1968}(1667,·)$, $\chi_{1968}(1643,·)$, $\chi_{1968}(409,·)$, $\chi_{1968}(875,·)$, $\chi_{1968}(985,·)$, $\chi_{1968}(1859,·)$, $\chi_{1968}(1883,·)$, $\chi_{1968}(1057,·)$, $\chi_{1968}(1321,·)$, $\chi_{1968}(683,·)$, $\chi_{1968}(1393,·)$, $\chi_{1968}(659,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4}$, $\frac{1}{18} a^{5}$, $\frac{1}{54} a^{6}$, $\frac{1}{54} a^{7}$, $\frac{1}{13284} a^{8}$, $\frac{1}{26568} a^{9} - \frac{1}{36} a^{5} - \frac{1}{2} a$, $\frac{1}{3666384} a^{10} + \frac{2}{76383} a^{8} - \frac{29}{4968} a^{6} + \frac{5}{414} a^{4} - \frac{19}{276} a^{2} + \frac{10}{23}$, $\frac{1}{7332768} a^{11} + \frac{1}{76383} a^{9} + \frac{7}{1104} a^{7} - \frac{1}{46} a^{5} + \frac{73}{552} a^{3} - \frac{13}{46} a$, $\frac{1}{3519728640} a^{12} + \frac{29}{293310720} a^{10} + \frac{799}{195540480} a^{8} + \frac{67}{132480} a^{6} - \frac{5711}{264960} a^{4} + \frac{257}{1840} a^{2} - \frac{79}{920}$, $\frac{1}{7039457280} a^{13} + \frac{29}{586621440} a^{11} + \frac{799}{391080960} a^{9} - \frac{7159}{794880} a^{7} + \frac{1001}{58880} a^{5} + \frac{257}{3680} a^{3} + \frac{841}{1840} a$, $\frac{1}{10690584429588480} a^{14} + \frac{10721}{98986892866560} a^{12} - \frac{10018987}{197973785733120} a^{10} + \frac{317637047}{16497815477760} a^{8} + \frac{1758883729}{804771486720} a^{6} + \frac{74683363}{8383036320} a^{4} + \frac{183761921}{2794345440} a^{2} - \frac{2596773}{11643106}$, $\frac{1}{21381168859176960} a^{15} + \frac{10721}{197973785733120} a^{13} - \frac{10018987}{395947571466240} a^{11} + \frac{317637047}{32995630955520} a^{9} + \frac{1758883729}{1609542973440} a^{7} + \frac{74683363}{16766072640} a^{5} + \frac{183761921}{5588690880} a^{3} + \frac{9046333}{23286212} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8006728}$, which has order $128107648$ (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:  \( 533954.940478744 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{82}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{41})\), 4.4.4410944.2, 4.4.68921.1, 8.8.19456426971136.3, 8.0.66165549026450079744.1, 8.0.66165549026450079744.2

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

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.22.6$x^{8} + 4 x^{6} + 6 x^{4} + 16 x^{3} + 24 x^{2} + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.6$x^{8} + 4 x^{6} + 6 x^{4} + 16 x^{3} + 24 x^{2} + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$