Properties

Label 16.0.7105708251953125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 181\cdot 401^{2}$
Root discriminant $9.79$
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, -7, 24, -50, 68, -60, 31, -3, -12, 17, -14, 5, 3, -5, 4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 5*x^13 + 3*x^12 + 5*x^11 - 14*x^10 + 17*x^9 - 12*x^8 - 3*x^7 + 31*x^6 - 60*x^5 + 68*x^4 - 50*x^3 + 24*x^2 - 7*x + 1)
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 5*x^13 + 3*x^12 + 5*x^11 - 14*x^10 + 17*x^9 - 12*x^8 - 3*x^7 + 31*x^6 - 60*x^5 + 68*x^4 - 50*x^3 + 24*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{15} \) \(\mathstrut +\mathstrut 4 x^{14} \) \(\mathstrut -\mathstrut 5 x^{13} \) \(\mathstrut +\mathstrut 3 x^{12} \) \(\mathstrut +\mathstrut 5 x^{11} \) \(\mathstrut -\mathstrut 14 x^{10} \) \(\mathstrut +\mathstrut 17 x^{9} \) \(\mathstrut -\mathstrut 12 x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut 31 x^{6} \) \(\mathstrut -\mathstrut 60 x^{5} \) \(\mathstrut +\mathstrut 68 x^{4} \) \(\mathstrut -\mathstrut 50 x^{3} \) \(\mathstrut +\mathstrut 24 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(7105708251953125=5^{12}\cdot 181\cdot 401^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.79$
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 \(a\))

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

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:  \( 32 a^{15} - \frac{77}{2} a^{14} + 88 a^{13} - 81 a^{12} + 8 a^{11} + 184 a^{10} - \frac{599}{2} a^{9} + 252 a^{8} - \frac{219}{2} a^{7} - \frac{477}{2} a^{6} + 820 a^{5} - 1193 a^{4} + 1005 a^{3} - \frac{1013}{2} a^{2} + 144 a - 16 \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a^{15} - 2 a^{14} + 4 a^{13} - 5 a^{12} + 3 a^{11} + 5 a^{10} - 14 a^{9} + 17 a^{8} - 12 a^{7} - 3 a^{6} + 31 a^{5} - 60 a^{4} + 68 a^{3} - 50 a^{2} + 24 a - 7 \),  \( 15 a^{15} - \frac{81}{2} a^{14} + 65 a^{13} - \frac{189}{2} a^{12} + 51 a^{11} + 91 a^{10} - \frac{543}{2} a^{9} + 307 a^{8} - 191 a^{7} - \frac{133}{2} a^{6} + \frac{1127}{2} a^{5} - \frac{2217}{2} a^{4} + 1214 a^{3} - \frac{1591}{2} a^{2} + 299 a - \frac{105}{2} \),  \( \frac{7}{2} a^{15} - 9 a^{14} + \frac{21}{2} a^{13} - \frac{39}{2} a^{12} + 2 a^{11} + \frac{47}{2} a^{10} - 57 a^{9} + 46 a^{8} - 23 a^{7} - \frac{53}{2} a^{6} + 127 a^{5} - \frac{429}{2} a^{4} + \frac{379}{2} a^{3} - 97 a^{2} + \frac{53}{2} a - \frac{7}{2} \),  \( \frac{21}{2} a^{15} + \frac{11}{2} a^{14} + \frac{23}{2} a^{13} + 19 a^{12} - 32 a^{11} + \frac{113}{2} a^{10} + \frac{11}{2} a^{9} - 61 a^{8} + 72 a^{7} - 113 a^{6} + \frac{251}{2} a^{5} + 38 a^{4} - \frac{481}{2} a^{3} + \frac{525}{2} a^{2} - \frac{267}{2} a + 31 \),  \( \frac{19}{2} a^{14} - 8 a^{13} + \frac{43}{2} a^{12} - 16 a^{11} - 7 a^{10} + \frac{105}{2} a^{9} - 68 a^{8} + 42 a^{7} - \frac{21}{2} a^{6} - \frac{153}{2} a^{5} + \frac{431}{2} a^{4} - 262 a^{3} + \frac{349}{2} a^{2} - 61 a + \frac{19}{2} \),  \( 33 a^{15} - \frac{109}{2} a^{14} + \frac{209}{2} a^{13} - \frac{237}{2} a^{12} + 36 a^{11} + 198 a^{10} - \frac{783}{2} a^{9} + \frac{749}{2} a^{8} - 188 a^{7} - 222 a^{6} + 966 a^{5} - \frac{3151}{2} a^{4} + 1478 a^{3} - \frac{1667}{2} a^{2} + \frac{539}{2} a - \frac{77}{2} \),  \( \frac{73}{2} a^{15} - 58 a^{14} + \frac{219}{2} a^{13} - \frac{251}{2} a^{12} + 27 a^{11} + \frac{435}{2} a^{10} - 417 a^{9} + 377 a^{8} - 184 a^{7} - \frac{511}{2} a^{6} + 1048 a^{5} - \frac{3317}{2} a^{4} + \frac{3003}{2} a^{3} - 819 a^{2} + \frac{513}{2} a - \frac{73}{2} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 60.6835925321 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.6265625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$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}$
401Data not computed