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Properties

Label 16.16.3445639081...3536.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{4}\cdot 19^{10}\cdot 37^{8}$
Root discriminant $45.56$
Ramified primes $2, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.Q_8.C_6$ (as 16T732)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-307, -2165, -3895, 4005, 15885, 1733, -20521, -5509, 11778, 2714, -3415, -513, 511, 40, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 37*x^14 + 40*x^13 + 511*x^12 - 513*x^11 - 3415*x^10 + 2714*x^9 + 11778*x^8 - 5509*x^7 - 20521*x^6 + 1733*x^5 + 15885*x^4 + 4005*x^3 - 3895*x^2 - 2165*x - 307)
 
gp: K = bnfinit(x^16 - x^15 - 37*x^14 + 40*x^13 + 511*x^12 - 513*x^11 - 3415*x^10 + 2714*x^9 + 11778*x^8 - 5509*x^7 - 20521*x^6 + 1733*x^5 + 15885*x^4 + 4005*x^3 - 3895*x^2 - 2165*x - 307, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 37 x^{14} + 40 x^{13} + 511 x^{12} - 513 x^{11} - 3415 x^{10} + 2714 x^{9} + 11778 x^{8} - 5509 x^{7} - 20521 x^{6} + 1733 x^{5} + 15885 x^{4} + 4005 x^{3} - 3895 x^{2} - 2165 x - 307 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(344563908178469207780603536=2^{4}\cdot 19^{10}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.56$
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:  $2, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{264735812800456} a^{15} - \frac{12119430571003}{132367906400228} a^{14} - \frac{37787242896299}{264735812800456} a^{13} + \frac{41638033440923}{264735812800456} a^{12} + \frac{7893291655447}{33091976600057} a^{11} + \frac{36764440866595}{264735812800456} a^{10} + \frac{25274100859135}{132367906400228} a^{9} - \frac{5635891391193}{66183953200114} a^{8} - \frac{46593811462829}{132367906400228} a^{7} + \frac{99931103735581}{264735812800456} a^{6} - \frac{44290665660837}{132367906400228} a^{5} - \frac{6836690587189}{264735812800456} a^{4} - \frac{29150227079307}{132367906400228} a^{3} - \frac{57739073092653}{264735812800456} a^{2} + \frac{39558180185437}{132367906400228} a - \frac{31238780103315}{264735812800456}$

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:  $15$
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:  \( 164419529.044 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.Q_8.C_6$ (as 16T732):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 30 conjugacy class representatives for $C_2^3.Q_8.C_6$
Character table for $C_2^3.Q_8.C_6$ is not computed

Intermediate fields

4.4.494209.1, 8.8.244242535681.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.12.8.1$x^{12} - 111 x^{9} + 4107 x^{6} - 50653 x^{3} + 14993288$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$