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Properties

Label 16.0.11476349427...184.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 41^{14}$
Root discriminant $654.99$
Ramified primes $2, 3, 41$
Class number $4264960000$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 700, 23800]$ (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![180699807744, 0, 2469564039168, 0, 6663932840448, 0, 343883864448, 0, 6974744274, 0, 70440624, 0, 373428, 0, 984, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 984*x^14 + 373428*x^12 + 70440624*x^10 + 6974744274*x^8 + 343883864448*x^6 + 6663932840448*x^4 + 2469564039168*x^2 + 180699807744)
 
gp: K = bnfinit(x^16 + 984*x^14 + 373428*x^12 + 70440624*x^10 + 6974744274*x^8 + 343883864448*x^6 + 6663932840448*x^4 + 2469564039168*x^2 + 180699807744, 1)
 

Normalized defining polynomial

\( x^{16} + 984 x^{14} + 373428 x^{12} + 70440624 x^{10} + 6974744274 x^{8} + 343883864448 x^{6} + 6663932840448 x^{4} + 2469564039168 x^{2} + 180699807744 \)

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:  \(1147634942730979008947338773912344095761629184=2^{62}\cdot 3^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $654.99$
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:  \(3936=2^{5}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(2627,·)$, $\chi_{3936}(73,·)$, $\chi_{3936}(659,·)$, $\chi_{3936}(875,·)$, $\chi_{3936}(3289,·)$, $\chi_{3936}(2651,·)$, $\chi_{3936}(3361,·)$, $\chi_{3936}(2843,·)$, $\chi_{3936}(1969,·)$, $\chi_{3936}(1321,·)$, $\chi_{3936}(683,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(2867,·)$, $\chi_{3936}(2041,·)$, $\chi_{3936}(899,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{27} a^{4} + \frac{1}{9} a^{2} - \frac{1}{3}$, $\frac{1}{27} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{81} a^{6} + \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{7} + \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{179334} a^{8} + \frac{2}{729} a^{6} - \frac{1}{243} a^{4} + \frac{7}{81} a^{2} - \frac{8}{27}$, $\frac{1}{358668} a^{9} + \frac{1}{729} a^{7} + \frac{4}{243} a^{5} + \frac{8}{81} a^{3} - \frac{17}{54} a$, $\frac{1}{4304016} a^{10} - \frac{7}{2916} a^{6} + \frac{41}{648} a^{2} + \frac{2}{27}$, $\frac{1}{8608032} a^{11} + \frac{29}{5832} a^{7} - \frac{1}{54} a^{5} - \frac{175}{1296} a^{3} - \frac{7}{54} a$, $\frac{1}{9606563712} a^{12} - \frac{1}{133424496} a^{10} + \frac{167}{88949664} a^{8} + \frac{137}{542376} a^{6} - \frac{1541}{482112} a^{4} - \frac{55}{558} a^{2} + \frac{779}{2511}$, $\frac{1}{38426254848} a^{13} - \frac{1}{533697984} a^{11} - \frac{329}{355798656} a^{9} - \frac{8047}{2169504} a^{7} - \frac{17413}{1928448} a^{5} - \frac{371}{20088} a^{3} + \frac{2017}{5022} a$, $\frac{1}{179182708209002990592} a^{14} - \frac{7715}{30349374696646848} a^{12} + \frac{3522083231}{40465832928862464} a^{10} + \frac{12404045291}{10116458232215616} a^{8} + \frac{32905949220155}{8992407317524992} a^{6} - \frac{110885116937}{9138625322688} a^{4} + \frac{165154374397}{1142328165336} a^{2} - \frac{1312704035}{5288556321}$, $\frac{1}{716730832836011962368} a^{15} - \frac{7715}{121397498786587392} a^{13} - \frac{5879794673}{161863331715449856} a^{11} - \frac{44007222133}{40465832928862464} a^{9} - \frac{81195241022789}{35969629270099968} a^{7} + \frac{265189999223}{36554501290752} a^{5} + \frac{187466887565}{2284656330672} a^{3} - \frac{248659162}{587617369} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{700}\times C_{23800}$, which has order $4264960000$ (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:  \( 11418873.067779342 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{41})\), 4.4.141150208.2, 4.4.141150208.1, 8.8.19923381218443264.6, 8.0.33876761101542440828928.1, 8.0.33876761101542440828928.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.3$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{2} + 18$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.3$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{2} + 18$$8$$1$$31$$C_8$$[3, 4, 5]$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$41$41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$