Properties

Label 16.0.3571460308796416.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{10}\cdot 1867553^{2}$
Root discriminant $9.38$
Ramified primes $2, 1867553$
Class number $1$
Class group Trivial
Galois Group 16T1945

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -3, 0, 3, 0, 1, 0, -4, 0, 4, 0, 0, 0, -2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 4*x^10 - 4*x^8 + x^6 + 3*x^4 - 3*x^2 + 1)
gp: K = bnfinit(x^16 - 2*x^14 + 4*x^10 - 4*x^8 + x^6 + 3*x^4 - 3*x^2 + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 4 x^{10} \) \(\mathstrut -\mathstrut 4 x^{8} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(3571460308796416=2^{10}\cdot 1867553^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.38$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 1867553$
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 \(a\))

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

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a^{14} - a^{12} - 2 a^{10} + 4 a^{8} - a^{6} - a^{4} + 4 a^{2} - 2 \),  \( a \),  \( a^{15} - a^{13} - a^{11} + 2 a^{9} + a^{3} + a \),  \( \frac{3}{2} a^{15} - \frac{1}{2} a^{14} - \frac{5}{2} a^{13} + a^{12} - \frac{1}{2} a^{11} + 5 a^{9} - \frac{3}{2} a^{8} - \frac{7}{2} a^{7} + \frac{3}{2} a^{6} + \frac{1}{2} a^{5} - a^{4} + 4 a^{3} - \frac{1}{2} a^{2} - 2 a + 1 \),  \( a^{15} - \frac{3}{2} a^{14} - \frac{3}{2} a^{13} + \frac{5}{2} a^{12} + \frac{1}{2} a^{10} + 2 a^{9} - 5 a^{8} - \frac{3}{2} a^{7} + \frac{7}{2} a^{6} + \frac{3}{2} a^{5} - \frac{1}{2} a^{4} + a^{3} - 4 a^{2} - \frac{1}{2} a + 2 \),  \( \frac{1}{2} a^{15} + \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - a^{12} - \frac{1}{2} a^{11} + a^{9} + \frac{3}{2} a^{8} + \frac{1}{2} a^{7} - \frac{3}{2} a^{6} - \frac{1}{2} a^{5} + a^{4} + \frac{1}{2} a^{2} + a - 1 \),  \( \frac{1}{2} a^{15} - \frac{3}{2} a^{14} + \frac{5}{2} a^{12} - 2 a^{11} + a^{10} + \frac{5}{2} a^{9} - \frac{11}{2} a^{8} + \frac{1}{2} a^{7} + 3 a^{6} - 2 a^{5} + \frac{1}{2} a^{4} + \frac{5}{2} a^{3} - \frac{7}{2} a^{2} - a + \frac{3}{2} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 7.9464198774 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1945:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 5160960
The 100 conjugacy class representatives for t16n1945 are not computed
Character table for t16n1945 is not computed

Intermediate fields

8.0.1867553.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
1867553Data not computed