Properties

Label 16.0.3006123291015625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{4}\cdot 29^{2}$
Root discriminant $9.28$
Ramified primes $5, 11, 29$
Class number $1$
Class group Trivial
Galois Group $C_2^4.C_2^3.C_2$ (as 16T547)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(3006123291015625=5^{12}\cdot 11^{4}\cdot 29^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.28$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 11, 29$
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 \(a\))

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

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:  \( -\frac{4}{3} a^{15} + \frac{8}{3} a^{14} + \frac{4}{3} a^{13} - \frac{31}{3} a^{12} + \frac{29}{3} a^{11} + \frac{20}{3} a^{10} - 22 a^{9} + \frac{40}{3} a^{8} + \frac{35}{3} a^{7} - 23 a^{6} + 8 a^{5} + \frac{41}{3} a^{4} - \frac{49}{3} a^{3} + \frac{10}{3} a^{2} + 6 a - \frac{11}{3} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{10}{3} a^{15} - 4 a^{14} - 7 a^{13} + \frac{58}{3} a^{12} - 6 a^{11} - \frac{62}{3} a^{10} + \frac{92}{3} a^{9} - \frac{13}{3} a^{8} - \frac{80}{3} a^{7} + 25 a^{6} + \frac{11}{3} a^{5} - \frac{71}{3} a^{4} + 15 a^{3} + \frac{11}{3} a^{2} - \frac{17}{3} a + \frac{7}{3} \),  \( a^{15} - 2 a^{14} - \frac{5}{3} a^{13} + \frac{25}{3} a^{12} - \frac{16}{3} a^{11} - \frac{25}{3} a^{10} + \frac{46}{3} a^{9} - 5 a^{8} - 12 a^{7} + 15 a^{6} - a^{5} - \frac{38}{3} a^{4} + \frac{29}{3} a^{3} + \frac{2}{3} a^{2} - \frac{14}{3} a + \frac{4}{3} \),  \( \frac{5}{3} a^{14} - 2 a^{13} - 4 a^{12} + 10 a^{11} - 2 a^{10} - \frac{38}{3} a^{9} + \frac{46}{3} a^{8} + \frac{2}{3} a^{7} - \frac{50}{3} a^{6} + \frac{34}{3} a^{5} + 5 a^{4} - 14 a^{3} + 6 a^{2} + 4 a - \frac{10}{3} \),  \( \frac{7}{3} a^{15} - \frac{7}{3} a^{14} - 6 a^{13} + \frac{38}{3} a^{12} - 17 a^{10} + 17 a^{9} + \frac{13}{3} a^{8} - \frac{61}{3} a^{7} + \frac{32}{3} a^{6} + \frac{26}{3} a^{5} - \frac{46}{3} a^{4} + 5 a^{3} + \frac{19}{3} a^{2} - \frac{8}{3} a - \frac{2}{3} \),  \( \frac{5}{3} a^{15} - 2 a^{14} - 4 a^{13} + \frac{31}{3} a^{12} - 2 a^{11} - 13 a^{10} + \frac{49}{3} a^{9} - 17 a^{7} + \frac{41}{3} a^{6} + \frac{14}{3} a^{5} - \frac{41}{3} a^{4} + 7 a^{3} + \frac{11}{3} a^{2} - \frac{13}{3} a + \frac{4}{3} \),  \( \frac{10}{3} a^{15} - \frac{16}{3} a^{14} - \frac{19}{3} a^{13} + \frac{70}{3} a^{12} - \frac{35}{3} a^{11} - \frac{71}{3} a^{10} + \frac{122}{3} a^{9} - \frac{32}{3} a^{8} - \frac{100}{3} a^{7} + \frac{110}{3} a^{6} + \frac{5}{3} a^{5} - \frac{92}{3} a^{4} + \frac{67}{3} a^{3} + \frac{11}{3} a^{2} - 10 a + 3 \),  \( \frac{2}{3} a^{15} - \frac{5}{3} a^{14} - \frac{2}{3} a^{13} + \frac{20}{3} a^{12} - \frac{19}{3} a^{11} - \frac{16}{3} a^{10} + \frac{46}{3} a^{9} - \frac{25}{3} a^{8} - \frac{29}{3} a^{7} + \frac{52}{3} a^{6} - \frac{14}{3} a^{5} - \frac{34}{3} a^{4} + \frac{38}{3} a^{3} - \frac{2}{3} a^{2} - 6 a + 3 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 35.8754367412 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2^4.C_2^3.C_2$ (as 16T547):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.275.1, 4.2.1375.1, 8.0.2193125.1, 8.4.54828125.1, 8.0.1890625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$