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Properties

Label 16.0.18055103407...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $77.81$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{16}$ (as 16T1953)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61440, -32768, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32768*x + 61440)
 
gp: K = bnfinit(x^16 - 32768*x + 61440, 1)
 

Normalized defining polynomial

\( x^{16} - 32768 x + 61440 \)

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:  \(1805510340771840000000000000000=2^{38}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.81$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3}$, $\frac{1}{8} a^{4}$, $\frac{1}{16} a^{5} - \frac{1}{2} a$, $\frac{1}{32} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{7}$, $\frac{1}{128} a^{8} - \frac{1}{2}$, $\frac{1}{256} a^{9} - \frac{1}{64} a^{7} - \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{512} a^{10} - \frac{1}{16} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{512} a^{11} - \frac{1}{8} a^{3}$, $\frac{1}{2048} a^{12} - \frac{1}{256} a^{8} + \frac{1}{32} a^{4} - \frac{1}{4}$, $\frac{1}{4096} a^{13} - \frac{1}{1024} a^{11} - \frac{1}{512} a^{9} - \frac{1}{64} a^{7} - \frac{1}{64} a^{5} - \frac{1}{16} a^{3} + \frac{1}{8} a$, $\frac{1}{4096} a^{14} - \frac{1}{64} a^{6} - \frac{1}{16} a^{4} - \frac{1}{2}$, $\frac{1}{8192} a^{15} - \frac{1}{1024} a^{11} + \frac{1}{128} a^{7} - \frac{1}{16} a^{3}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 6244054534.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{16}$ (as 16T1953):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10461394944000
The 123 conjugacy class representatives for $A_{16}$ are not computed
Character table for $A_{16}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $15{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.4.10.4$x^{4} + 6 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.7$x^{4} - 2 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.8.18.77$x^{8} + 8 x^{7} + 36$$8$$1$$18$$C_4\wr C_2$$[2, 2, 3]^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.12.12.10$x^{12} + 9 x^{11} + 36 x^{10} - 72 x^{9} + 18 x^{8} + 45 x^{7} + 99 x^{6} + 54 x^{5} + 81 x^{4} - 81 x^{3} + 81 x^{2} - 81 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.10.1$x^{10} + 10 x^{5} + 75 x^{2} + 25$$5$$2$$10$$C_5^2 : C_8$$[5/4, 5/4]_{4}^{2}$