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Properties

Label 16.8.15792174860...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{2}\cdot 5^{12}\cdot 7^{2}\cdot 19^{4}\cdot 103^{4}$
Root discriminant $32.54$
Ramified primes $3, 5, 7, 19, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1046

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![521, 670, -1205, -829, 1236, 335, -739, -273, 481, 206, -257, -77, 82, 14, -14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 14*x^14 + 14*x^13 + 82*x^12 - 77*x^11 - 257*x^10 + 206*x^9 + 481*x^8 - 273*x^7 - 739*x^6 + 335*x^5 + 1236*x^4 - 829*x^3 - 1205*x^2 + 670*x + 521)
 
gp: K = bnfinit(x^16 - x^15 - 14*x^14 + 14*x^13 + 82*x^12 - 77*x^11 - 257*x^10 + 206*x^9 + 481*x^8 - 273*x^7 - 739*x^6 + 335*x^5 + 1236*x^4 - 829*x^3 - 1205*x^2 + 670*x + 521, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 14 x^{14} + 14 x^{13} + 82 x^{12} - 77 x^{11} - 257 x^{10} + 206 x^{9} + 481 x^{8} - 273 x^{7} - 739 x^{6} + 335 x^{5} + 1236 x^{4} - 829 x^{3} - 1205 x^{2} + 670 x + 521 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1579217486082822509765625=3^{2}\cdot 5^{12}\cdot 7^{2}\cdot 19^{4}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.54$
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:  $3, 5, 7, 19, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17879} a^{13} + \frac{3866}{17879} a^{12} + \frac{2970}{17879} a^{11} - \frac{5882}{17879} a^{10} - \frac{8900}{17879} a^{9} + \frac{1223}{17879} a^{8} + \frac{8799}{17879} a^{7} - \frac{8576}{17879} a^{6} - \frac{5547}{17879} a^{5} + \frac{7302}{17879} a^{4} + \frac{5449}{17879} a^{3} - \frac{2145}{17879} a^{2} - \frac{8741}{17879} a - \frac{2893}{17879}$, $\frac{1}{17879} a^{14} + \frac{3858}{17879} a^{12} + \frac{8295}{17879} a^{11} + \frac{6703}{17879} a^{10} - \frac{8452}{17879} a^{9} + \frac{737}{17879} a^{8} - \frac{1773}{17879} a^{7} + \frac{1603}{17879} a^{6} - \frac{2796}{17879} a^{5} + \frac{6858}{17879} a^{4} - \frac{343}{941} a^{3} + \frac{308}{941} a^{2} - \frac{1497}{17879} a - \frac{7916}{17879}$, $\frac{1}{339701} a^{15} + \frac{3}{339701} a^{14} - \frac{2}{339701} a^{13} + \frac{169106}{339701} a^{12} + \frac{152980}{339701} a^{11} + \frac{81363}{339701} a^{10} - \frac{16178}{339701} a^{9} + \frac{160625}{339701} a^{8} - \frac{105030}{339701} a^{7} - \frac{60172}{339701} a^{6} + \frac{80243}{339701} a^{5} - \frac{101633}{339701} a^{4} - \frac{110530}{339701} a^{3} + \frac{53540}{339701} a^{2} - \frac{45578}{339701} a + \frac{94010}{339701}$

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:  $11$
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:  \( 1322820.99782 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1046:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 40 conjugacy class representatives for t16n1046
Character table for t16n1046 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.12.0.1$x^{12} + 3 x^{2} - 2 x + 3$$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}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$103$103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$