Properties

Label 16.0.3207314697265625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 181^{2}\cdot 401$
Root discriminant $9.31$
Ramified primes $5, 181, 401$
Class number $1$
Class group Trivial
Galois Group 16T1771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -4, -2, 11, 3, -15, -10, 22, 5, -10, -11, 11, 1, -1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*x^2 + x + 1)
gp: K = bnfinit(x^16 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut +\mathstrut x^{13} \) \(\mathstrut +\mathstrut 11 x^{12} \) \(\mathstrut -\mathstrut 11 x^{11} \) \(\mathstrut -\mathstrut 10 x^{10} \) \(\mathstrut +\mathstrut 5 x^{9} \) \(\mathstrut +\mathstrut 22 x^{8} \) \(\mathstrut -\mathstrut 10 x^{7} \) \(\mathstrut -\mathstrut 15 x^{6} \) \(\mathstrut +\mathstrut 3 x^{5} \) \(\mathstrut +\mathstrut 11 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut -\mathstrut 4 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3207314697265625=5^{12}\cdot 181^{2}\cdot 401\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.31$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 181, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{131} a^{15} + \frac{63}{131} a^{14} + \frac{33}{131} a^{13} + \frac{50}{131} a^{12} - \frac{14}{131} a^{11} - \frac{4}{131} a^{10} - \frac{8}{131} a^{9} + \frac{9}{131} a^{8} - \frac{48}{131} a^{7} + \frac{14}{131} a^{6} - \frac{22}{131} a^{5} + \frac{14}{131} a^{4} + \frac{4}{131} a^{3} - \frac{4}{131} a^{2} - \frac{2}{131} a + \frac{2}{131}$

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:  \( \frac{109}{131} a^{15} - \frac{207}{131} a^{14} - \frac{202}{131} a^{13} + \frac{79}{131} a^{12} + \frac{1356}{131} a^{11} - \frac{829}{131} a^{10} - \frac{1527}{131} a^{9} - \frac{67}{131} a^{8} + \frac{2628}{131} a^{7} - \frac{308}{131} a^{6} - \frac{1481}{131} a^{5} - \frac{308}{131} a^{4} + \frac{960}{131} a^{3} - \frac{43}{131} a^{2} - \frac{218}{131} a + \frac{87}{131} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{48}{131} a^{15} + \frac{11}{131} a^{14} - \frac{119}{131} a^{13} - \frac{220}{131} a^{12} + \frac{376}{131} a^{11} + \frac{463}{131} a^{10} - \frac{253}{131} a^{9} - \frac{1009}{131} a^{8} + \frac{316}{131} a^{7} + \frac{803}{131} a^{6} - \frac{8}{131} a^{5} - \frac{507}{131} a^{4} + \frac{61}{131} a^{3} + \frac{201}{131} a^{2} - \frac{227}{131} a - \frac{35}{131} \),  \( \frac{11}{131} a^{15} - \frac{93}{131} a^{14} - \frac{30}{131} a^{13} + \frac{157}{131} a^{12} + \frac{370}{131} a^{11} - \frac{437}{131} a^{10} - \frac{612}{131} a^{9} + \frac{230}{131} a^{8} + \frac{782}{131} a^{7} + \frac{154}{131} a^{6} - \frac{635}{131} a^{5} - \frac{239}{131} a^{4} + \frac{175}{131} a^{3} + \frac{218}{131} a^{2} - \frac{22}{131} a - \frac{109}{131} \),  \( \frac{215}{131} a^{15} - \frac{341}{131} a^{14} - \frac{372}{131} a^{13} + \frac{8}{131} a^{12} + \frac{2361}{131} a^{11} - \frac{1122}{131} a^{10} - \frac{2506}{131} a^{9} - \frac{423}{131} a^{8} + \frac{3828}{131} a^{7} + \frac{259}{131} a^{6} - \frac{2372}{131} a^{5} - \frac{658}{131} a^{4} + \frac{1122}{131} a^{3} + \frac{319}{131} a^{2} - \frac{430}{131} a - \frac{94}{131} \),  \( \frac{19}{131} a^{15} + \frac{18}{131} a^{14} - \frac{28}{131} a^{13} - \frac{229}{131} a^{12} + \frac{127}{131} a^{11} + \frac{448}{131} a^{10} + \frac{372}{131} a^{9} - \frac{1270}{131} a^{8} - \frac{388}{131} a^{7} + \frac{921}{131} a^{6} + \frac{1154}{131} a^{5} - \frac{913}{131} a^{4} - \frac{579}{131} a^{3} + \frac{186}{131} a^{2} + \frac{355}{131} a - \frac{93}{131} \),  \( \frac{84}{131} a^{15} - \frac{79}{131} a^{14} - \frac{110}{131} a^{13} - \frac{254}{131} a^{12} + \frac{789}{131} a^{11} - \frac{74}{131} a^{10} - \frac{148}{131} a^{9} - \frac{1209}{131} a^{8} + \frac{1077}{131} a^{7} + \frac{128}{131} a^{6} + \frac{248}{131} a^{5} - \frac{789}{131} a^{4} + \frac{336}{131} a^{3} + \frac{57}{131} a^{2} - \frac{37}{131} a - \frac{94}{131} \),  \( \frac{115}{131} a^{15} - \frac{91}{131} a^{14} - \frac{397}{131} a^{13} - \frac{145}{131} a^{12} + \frac{1403}{131} a^{11} + \frac{457}{131} a^{10} - \frac{2230}{131} a^{9} - \frac{1454}{131} a^{8} + \frac{2602}{131} a^{7} + \frac{1872}{131} a^{6} - \frac{1875}{131} a^{5} - \frac{1665}{131} a^{4} + \frac{984}{131} a^{3} + \frac{850}{131} a^{2} - \frac{492}{131} a - \frac{294}{131} \),  \( \frac{96}{131} a^{15} - \frac{240}{131} a^{14} + \frac{24}{131} a^{13} + \frac{84}{131} a^{12} + \frac{1014}{131} a^{11} - \frac{1563}{131} a^{10} - \frac{244}{131} a^{9} + \frac{602}{131} a^{8} + \frac{1942}{131} a^{7} - \frac{1669}{131} a^{6} - \frac{933}{131} a^{5} + \frac{427}{131} a^{4} + \frac{908}{131} a^{3} - \frac{122}{131} a^{2} - \frac{323}{131} a + \frac{61}{131} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 37.3668375913 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1771:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16384
The 190 conjugacy class representatives for t16n1771 are not computed
Character table for t16n1771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.2828125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$181$181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
401Data not computed