Properties

Label 16.0.6472705322265625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 19^{2}\cdot 271^{2}$
Root discriminant $9.73$
Ramified primes $5, 19, 271$
Class number $1$
Class group Trivial
Galois Group 16T1497

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 4, 3, 8, 8, 6, 9, 8, 8, 4, 4, 3, 1, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 1)
gp: K = bnfinit(x^16 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut x^{15} \) \(\mathstrut +\mathstrut x^{14} \) \(\mathstrut +\mathstrut x^{13} \) \(\mathstrut +\mathstrut 3 x^{12} \) \(\mathstrut +\mathstrut 4 x^{11} \) \(\mathstrut +\mathstrut 4 x^{10} \) \(\mathstrut +\mathstrut 8 x^{9} \) \(\mathstrut +\mathstrut 8 x^{8} \) \(\mathstrut +\mathstrut 9 x^{7} \) \(\mathstrut +\mathstrut 6 x^{6} \) \(\mathstrut +\mathstrut 8 x^{5} \) \(\mathstrut +\mathstrut 8 x^{4} \) \(\mathstrut +\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut +\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6472705322265625=5^{12}\cdot 19^{2}\cdot 271^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.73$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 19, 271$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{5} - \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} - \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{5} + \frac{1}{5}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{2}{5} a^{15} - \frac{1}{5} a^{14} + a^{12} + \frac{4}{5} a^{11} + \frac{13}{5} a^{10} + \frac{7}{5} a^{9} + 4 a^{8} + 4 a^{7} + \frac{22}{5} a^{6} + \frac{17}{5} a^{5} + \frac{12}{5} a^{4} + 4 a^{3} + a^{2} + \frac{7}{5} a + \frac{6}{5} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{3}{5} a^{13} - \frac{2}{5} a^{12} + a^{11} - \frac{1}{5} a^{10} + \frac{4}{5} a^{9} + \frac{4}{5} a^{8} + \frac{4}{5} a^{7} + a^{6} - \frac{4}{5} a^{5} + \frac{4}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} \),  \( \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{4}{5} a^{10} - \frac{3}{5} a^{9} - \frac{1}{5} a^{8} - \frac{4}{5} a^{7} + \frac{1}{5} a^{6} - a^{5} - \frac{7}{5} a^{4} - \frac{4}{5} a^{3} - \frac{3}{5} a^{2} - \frac{3}{5} a - \frac{6}{5} \),  \( \frac{1}{5} a^{15} - \frac{3}{5} a^{14} + \frac{3}{5} a^{13} - a^{9} + a^{8} - a^{7} + \frac{2}{5} a^{6} - \frac{7}{5} a^{5} + a^{4} + \frac{4}{5} a^{3} - \frac{7}{5} a^{2} + \frac{6}{5} a - \frac{1}{5} \),  \( \frac{2}{5} a^{14} - \frac{4}{5} a^{13} + \frac{4}{5} a^{12} + \frac{4}{5} a^{10} + \frac{2}{5} a^{9} + \frac{8}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{6}{5} a^{4} + \frac{7}{5} a^{3} + \frac{1}{5} a^{2} - \frac{4}{5} a + \frac{2}{5} \),  \( \frac{2}{5} a^{15} - \frac{3}{5} a^{14} + \frac{3}{5} a^{13} + \frac{1}{5} a^{12} + a^{11} + a^{10} + a^{9} + \frac{11}{5} a^{8} + 2 a^{7} + \frac{8}{5} a^{6} + \frac{8}{5} a^{5} + \frac{14}{5} a^{4} + \frac{12}{5} a^{3} + a^{2} + a + \frac{6}{5} \),  \( \frac{4}{5} a^{15} - a^{14} + \frac{3}{5} a^{13} + \frac{4}{5} a^{12} + 2 a^{11} + \frac{11}{5} a^{10} + a^{9} + \frac{17}{5} a^{8} + \frac{13}{5} a^{7} + \frac{13}{5} a^{6} - \frac{3}{5} a^{5} + \frac{7}{5} a^{4} + \frac{13}{5} a^{3} - \frac{6}{5} a^{2} + \frac{2}{5} a \),  \( \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{4}{5} a^{13} + a^{12} + \frac{3}{5} a^{11} + \frac{8}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{9}{5} a^{7} + a^{6} + \frac{3}{5} a^{5} - \frac{6}{5} a^{4} + \frac{9}{5} a^{3} + \frac{4}{5} a^{2} - \frac{6}{5} a + \frac{3}{5} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 56.3298547938 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1497:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.4.80453125.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
271Data not computed