Properties

Label 16.0.3914091521024229.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 61\cdot 98893^{2}$
Root discriminant $9.43$
Ramified primes $3, 61, 98893$
Class number $1$
Class group Trivial
Galois Group 16T1905

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(3914091521024229=3^{8}\cdot 61\cdot 98893^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.43$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 61, 98893$
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 \(a\))

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

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:  \( 2 a^{15} - 6 a^{14} + 3 a^{13} + 2 a^{12} + 8 a^{11} - 16 a^{10} + 8 a^{9} - 13 a^{8} + 16 a^{7} - 6 a^{6} + 13 a^{5} - 20 a^{4} + 12 a^{3} - 10 a^{2} + 6 a \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( 2 a^{15} - a^{14} - 4 a^{13} + 8 a^{11} + 2 a^{10} - 3 a^{9} - 4 a^{8} - 4 a^{7} + 3 a^{6} + 6 a^{5} + 2 a^{4} - 4 a^{3} + 2 a^{2} - 3 a + 2 \),  \( 3 a^{15} - 4 a^{14} - 2 a^{13} + a^{12} + 11 a^{11} - 6 a^{10} + 2 a^{9} - 10 a^{8} + 5 a^{7} + 11 a^{5} - 9 a^{4} + 3 a^{3} - 3 a^{2} + a + 2 \),  \( a^{15} - a^{14} - a^{13} - a^{12} + 4 a^{11} + a^{10} + a^{9} - 7 a^{8} - a^{7} + a^{6} + 6 a^{5} - a^{4} - 3 a^{2} - a + 1 \),  \( a^{14} - 2 a^{13} + a^{12} + a^{11} + 2 a^{10} - 6 a^{9} + 5 a^{8} - a^{7} + 4 a^{6} - 4 a^{5} + 3 a^{4} - 6 a^{3} + 5 a^{2} - a + 1 \),  \( a^{15} + a^{14} - 4 a^{13} - a^{12} + 4 a^{11} + 7 a^{10} - 4 a^{9} - a^{8} - 8 a^{7} + 4 a^{6} + 2 a^{5} + 8 a^{4} - 5 a^{3} + 3 a^{2} - 4 a + 1 \),  \( 2 a^{15} - 3 a^{14} - a^{13} + 9 a^{11} - 5 a^{10} + a^{9} - 10 a^{8} + 7 a^{7} - a^{6} + 10 a^{5} - 9 a^{4} + 3 a^{3} - 5 a^{2} + 3 a \),  \( 5 a^{15} - 9 a^{14} + a^{13} + a^{12} + 18 a^{11} - 19 a^{10} + 13 a^{9} - 22 a^{8} + 19 a^{7} - 11 a^{6} + 23 a^{5} - 25 a^{4} + 19 a^{3} - 13 a^{2} + 7 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 25.217952399 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1905:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 294912
The 230 conjugacy class representatives for t16n1905 are not computed
Character table for t16n1905 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 8.0.8010333.1

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 $16$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
98893Data not computed