Properties

Label 16.0.4393378612917909.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 61\cdot 104773^{2}$
Root discriminant $9.50$
Ramified primes $3, 61, 104773$
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, 3, 5, 6, 6, 6, 4, 0, -1, 0, 4, -6, 6, -6, 5, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)
gp: K = bnfinit(x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*x + 1, 1)

Normalized defining polynomial

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

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:  \(4393378612917909=3^{8}\cdot 61\cdot 104773^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.50$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 61, 104773$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{779} a^{14} - \frac{172}{779} a^{13} + \frac{251}{779} a^{12} + \frac{248}{779} a^{11} - \frac{368}{779} a^{10} + \frac{6}{41} a^{9} - \frac{155}{779} a^{8} - \frac{177}{779} a^{7} + \frac{155}{779} a^{6} + \frac{6}{41} a^{5} + \frac{368}{779} a^{4} + \frac{248}{779} a^{3} - \frac{251}{779} a^{2} - \frac{172}{779} a - \frac{1}{779}$, $\frac{1}{779} a^{15} + \frac{269}{779} a^{13} - \frac{204}{779} a^{12} + \frac{222}{779} a^{11} - \frac{83}{779} a^{10} - \frac{22}{779} a^{9} - \frac{351}{779} a^{8} + \frac{92}{779} a^{7} + \frac{288}{779} a^{6} - \frac{278}{779} a^{5} - \frac{334}{779} a^{4} + \frac{339}{779} a^{3} + \frac{280}{779} a^{2} + \frac{17}{779} a - \frac{172}{779}$

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:  \( -\frac{9}{41} a^{14} + \frac{31}{41} a^{13} - \frac{45}{41} a^{12} + \frac{23}{41} a^{11} + \frac{32}{41} a^{10} - \frac{83}{41} a^{9} + \frac{124}{41} a^{8} - \frac{170}{41} a^{7} + \frac{163}{41} a^{6} - \frac{83}{41} a^{5} - \frac{32}{41} a^{4} - \frac{18}{41} a^{3} + \frac{4}{41} a^{2} - \frac{10}{41} a + \frac{9}{41} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{9}{41} a^{15} - \frac{31}{41} a^{14} + \frac{45}{41} a^{13} - \frac{23}{41} a^{12} - \frac{32}{41} a^{11} + \frac{83}{41} a^{10} - \frac{124}{41} a^{9} + \frac{170}{41} a^{8} - \frac{163}{41} a^{7} + \frac{83}{41} a^{6} + \frac{32}{41} a^{5} + \frac{18}{41} a^{4} - \frac{4}{41} a^{3} + \frac{10}{41} a^{2} - \frac{9}{41} a \),  \( \frac{320}{779} a^{15} - \frac{967}{779} a^{14} + \frac{1566}{779} a^{13} - \frac{1850}{779} a^{12} + \frac{1825}{779} a^{11} - \frac{1779}{779} a^{10} + \frac{1130}{779} a^{9} + \frac{173}{779} a^{8} - \frac{383}{779} a^{7} - \frac{79}{779} a^{6} + \frac{1005}{779} a^{5} + \frac{2327}{779} a^{4} + \frac{1094}{779} a^{3} + \frac{2021}{779} a^{2} + \frac{384}{779} a + \frac{457}{779} \),  \( \frac{127}{779} a^{15} - \frac{305}{779} a^{14} + \frac{154}{779} a^{13} + \frac{365}{779} a^{12} - \frac{706}{779} a^{11} + \frac{429}{779} a^{10} - \frac{172}{779} a^{9} + \frac{19}{41} a^{8} + \frac{233}{779} a^{7} - \frac{1351}{779} a^{6} + \frac{1592}{779} a^{5} + \frac{1142}{779} a^{4} + \frac{131}{779} a^{3} - \frac{840}{779} a^{2} + \frac{868}{779} a + \frac{273}{779} \),  \( \frac{411}{779} a^{15} - \frac{1359}{779} a^{14} + \frac{2326}{779} a^{13} - \frac{2735}{779} a^{12} + \frac{2711}{779} a^{11} - \frac{2959}{779} a^{10} + \frac{2738}{779} a^{9} - \frac{1389}{779} a^{8} + \frac{1031}{779} a^{7} - \frac{1913}{779} a^{6} + \frac{3466}{779} a^{5} + \frac{34}{19} a^{4} + \frac{942}{779} a^{3} + \frac{2032}{779} a^{2} + \frac{803}{779} a + \frac{777}{779} \),  \( \frac{419}{779} a^{15} - \frac{1049}{779} a^{14} + \frac{1014}{779} a^{13} + \frac{217}{779} a^{12} - \frac{1986}{779} a^{11} + \frac{3042}{779} a^{10} - \frac{4164}{779} a^{9} + \frac{5399}{779} a^{8} - \frac{4026}{779} a^{7} + \frac{922}{779} a^{6} + \frac{2306}{779} a^{5} + \frac{2963}{779} a^{4} + \frac{1855}{779} a^{3} + \frac{2025}{779} a^{2} + \frac{2149}{779} a + \frac{1428}{779} \),  \( \frac{197}{779} a^{15} - \frac{381}{779} a^{14} + \frac{117}{779} a^{13} + \frac{506}{779} a^{12} - \frac{898}{779} a^{11} + \frac{775}{779} a^{10} - \frac{1028}{779} a^{9} + \frac{1593}{779} a^{8} - \frac{908}{779} a^{7} + \frac{18}{779} a^{6} - \frac{46}{779} a^{5} + \frac{3545}{779} a^{4} + \frac{339}{779} a^{3} + \frac{1223}{779} a^{2} + \frac{1108}{779} a + \frac{773}{779} \),  \( \frac{335}{779} a^{15} - \frac{878}{779} a^{14} + \frac{1199}{779} a^{13} - \frac{1267}{779} a^{12} + \frac{80}{41} a^{11} - \frac{2279}{779} a^{10} + \frac{2377}{779} a^{9} - \frac{1749}{779} a^{8} + \frac{2382}{779} a^{7} - \frac{2997}{779} a^{6} + \frac{3086}{779} a^{5} + \frac{2025}{779} a^{4} + \frac{2544}{779} a^{3} + \frac{1799}{779} a^{2} + \frac{1690}{779} a + \frac{904}{779} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 27.0045913409 \)
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

\(\Q(\sqrt{-3}) \), 8.0.8486613.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
104773Data not computed