Properties

Label 16.0.6741949462890625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 1051^{2}$
Root discriminant $9.76$
Ramified primes $5, 1051$
Class number $1$
Class group Trivial
Galois Group 16T1497

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 9, -10, 17, -20, 17, -20, 15, -10, 13, -5, 2, -5, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + x^14 - 5*x^13 + 2*x^12 - 5*x^11 + 13*x^10 - 10*x^9 + 15*x^8 - 20*x^7 + 17*x^6 - 20*x^5 + 17*x^4 - 10*x^3 + 9*x^2 - 5*x + 1)
gp: K = bnfinit(x^16 + x^14 - 5*x^13 + 2*x^12 - 5*x^11 + 13*x^10 - 10*x^9 + 15*x^8 - 20*x^7 + 17*x^6 - 20*x^5 + 17*x^4 - 10*x^3 + 9*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut +\mathstrut x^{14} \) \(\mathstrut -\mathstrut 5 x^{13} \) \(\mathstrut +\mathstrut 2 x^{12} \) \(\mathstrut -\mathstrut 5 x^{11} \) \(\mathstrut +\mathstrut 13 x^{10} \) \(\mathstrut -\mathstrut 10 x^{9} \) \(\mathstrut +\mathstrut 15 x^{8} \) \(\mathstrut -\mathstrut 20 x^{7} \) \(\mathstrut +\mathstrut 17 x^{6} \) \(\mathstrut -\mathstrut 20 x^{5} \) \(\mathstrut +\mathstrut 17 x^{4} \) \(\mathstrut -\mathstrut 10 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6741949462890625=5^{14}\cdot 1051^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.76$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 1051$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4751} a^{15} - \frac{1736}{4751} a^{14} + \frac{1563}{4751} a^{13} - \frac{552}{4751} a^{12} - \frac{1428}{4751} a^{11} - \frac{1019}{4751} a^{10} + \frac{1625}{4751} a^{9} + \frac{1084}{4751} a^{8} - \frac{413}{4751} a^{7} - \frac{453}{4751} a^{6} - \frac{2241}{4751} a^{5} - \frac{713}{4751} a^{4} - \frac{2226}{4751} a^{3} + \frac{1763}{4751} a^{2} - \frac{915}{4751} a + \frac{1601}{4751}$

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:  \( -\frac{5403}{4751} a^{15} - \frac{3617}{4751} a^{14} - \frac{7113}{4751} a^{13} + \frac{22583}{4751} a^{12} + \frac{4611}{4751} a^{11} + \frac{27754}{4751} a^{10} - \frac{52288}{4751} a^{9} + \frac{20135}{4751} a^{8} - \frac{63294}{4751} a^{7} + \frac{67308}{4751} a^{6} - \frac{49686}{4751} a^{5} + \frac{75294}{4751} a^{4} - \frac{49964}{4751} a^{3} + \frac{28772}{4751} a^{2} - \frac{35303}{4751} a + \frac{15621}{4751} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{3657}{4751} a^{15} + \frac{3535}{4751} a^{14} + \frac{5189}{4751} a^{13} - \frac{13742}{4751} a^{12} - \frac{5598}{4751} a^{11} - \frac{15952}{4751} a^{10} + \frac{32381}{4751} a^{9} - \frac{7648}{4751} a^{8} + \frac{33734}{4751} a^{7} - \frac{36530}{4751} a^{6} + \frac{28644}{4751} a^{5} - \frac{41901}{4751} a^{4} + \frac{26487}{4751} a^{3} - \frac{14069}{4751} a^{2} + \frac{27055}{4751} a - \frac{7877}{4751} \),  \( a \),  \( \frac{3973}{4751} a^{15} + \frac{1324}{4751} a^{14} + \frac{4993}{4751} a^{13} - \frac{17138}{4751} a^{12} + \frac{4001}{4751} a^{11} - \frac{19639}{4751} a^{10} + \frac{42275}{4751} a^{9} - \frac{30931}{4751} a^{8} + \frac{50507}{4751} a^{7} - \frac{60903}{4751} a^{6} + \frac{52142}{4751} a^{5} - \frac{62916}{4751} a^{4} + \frac{49974}{4751} a^{3} - \frac{22330}{4751} a^{2} + \frac{27726}{4751} a - \frac{15069}{4751} \),  \( \frac{1668}{4751} a^{15} - \frac{2289}{4751} a^{14} - \frac{1215}{4751} a^{13} - \frac{13295}{4751} a^{12} + \frac{7849}{4751} a^{11} - \frac{3585}{4751} a^{10} + \frac{35687}{4751} a^{9} - \frac{21023}{4751} a^{8} + \frac{28517}{4751} a^{7} - \frac{52456}{4751} a^{6} + \frac{24804}{4751} a^{5} - \frac{44293}{4751} a^{4} + \frac{35571}{4751} a^{3} - \frac{9687}{4751} a^{2} + \frac{17855}{4751} a - \frac{9096}{4751} \),  \( \frac{2555}{4751} a^{15} + \frac{1954}{4751} a^{14} + \frac{2625}{4751} a^{13} - \frac{8815}{4751} a^{12} - \frac{4523}{4751} a^{11} - \frac{9499}{4751} a^{10} + \frac{13754}{4751} a^{9} - \frac{213}{4751} a^{8} + \frac{18511}{4751} a^{7} - \frac{2922}{4751} a^{6} - \frac{800}{4751} a^{5} - \frac{11584}{4751} a^{4} - \frac{9985}{4751} a^{3} + \frac{5268}{4751} a^{2} - \frac{333}{4751} a + \frac{4695}{4751} \),  \( \frac{8343}{4751} a^{15} + \frac{2351}{4751} a^{14} + \frac{8116}{4751} a^{13} - \frac{39625}{4751} a^{12} + \frac{1704}{4751} a^{11} - \frac{39986}{4751} a^{10} + \frac{93041}{4751} a^{9} - \frac{44851}{4751} a^{8} + \frac{108089}{4751} a^{7} - \frac{121109}{4751} a^{6} + \frac{84040}{4751} a^{5} - \frac{128584}{4751} a^{4} + \frac{80908}{4751} a^{3} - \frac{43146}{4751} a^{2} + \frac{48522}{4751} a - \frac{16922}{4751} \),  \( \frac{3044}{4751} a^{15} - \frac{1272}{4751} a^{14} + \frac{2021}{4751} a^{13} - \frac{17438}{4751} a^{12} + \frac{9835}{4751} a^{11} - \frac{13686}{4751} a^{10} + \frac{48219}{4751} a^{9} - \frac{35506}{4751} a^{8} + \frac{49353}{4751} a^{7} - \frac{77158}{4751} a^{6} + \frac{53093}{4751} a^{5} - \frac{70430}{4751} a^{4} + \frac{60745}{4751} a^{3} - \frac{30564}{4751} a^{2} + \frac{32083}{4751} a - \frac{10584}{4751} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 58.1502195758 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1497:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.2.82109375.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
1051Data not computed