Properties

Label 16.0.2779733938787757.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 7^{2}\cdot 37^{2}\cdot 151^{2}\cdot 277$
Root discriminant $9.23$
Ramified primes $3, 7, 37, 151, 277$
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, -8, 31, -77, 138, -191, 213, -196, 153, -107, 74, -53, 39, -26, 14, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*x^2 - 8*x + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 14 x^{14} \) \(\mathstrut -\mathstrut 26 x^{13} \) \(\mathstrut +\mathstrut 39 x^{12} \) \(\mathstrut -\mathstrut 53 x^{11} \) \(\mathstrut +\mathstrut 74 x^{10} \) \(\mathstrut -\mathstrut 107 x^{9} \) \(\mathstrut +\mathstrut 153 x^{8} \) \(\mathstrut -\mathstrut 196 x^{7} \) \(\mathstrut +\mathstrut 213 x^{6} \) \(\mathstrut -\mathstrut 191 x^{5} \) \(\mathstrut +\mathstrut 138 x^{4} \) \(\mathstrut -\mathstrut 77 x^{3} \) \(\mathstrut +\mathstrut 31 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2779733938787757=3^{8}\cdot 7^{2}\cdot 37^{2}\cdot 151^{2}\cdot 277\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.23$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 7, 37, 151, 277$
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 \(a\))

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

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:  \( -19 a^{15} + 73 a^{14} - 172 a^{13} + 259 a^{12} - 358 a^{11} + 470 a^{10} - 694 a^{9} + 1009 a^{8} - 1411 a^{7} + 1614 a^{6} - 1515 a^{5} + 1126 a^{4} - 629 a^{3} + 235 a^{2} - 49 a + 4 \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{61}{7} a^{15} - \frac{190}{7} a^{14} + \frac{362}{7} a^{13} - \frac{365}{7} a^{12} + \frac{403}{7} a^{11} - \frac{488}{7} a^{10} + \frac{878}{7} a^{9} - \frac{1299}{7} a^{8} + \frac{1667}{7} a^{7} - \frac{1160}{7} a^{6} + \frac{96}{7} a^{5} + 135 a^{4} - \frac{1473}{7} a^{3} + \frac{1240}{7} a^{2} - \frac{599}{7} a + \frac{134}{7} \),  \( \frac{36}{7} a^{15} - \frac{115}{7} a^{14} + \frac{218}{7} a^{13} - \frac{213}{7} a^{12} + \frac{215}{7} a^{11} - \frac{253}{7} a^{10} + \frac{478}{7} a^{9} - \frac{712}{7} a^{8} + \frac{906}{7} a^{7} - \frac{554}{7} a^{6} - \frac{160}{7} a^{5} + 122 a^{4} - \frac{1136}{7} a^{3} + \frac{948}{7} a^{2} - \frac{467}{7} a + \frac{108}{7} \),  \( \frac{26}{7} a^{15} - \frac{99}{7} a^{14} + \frac{201}{7} a^{13} - \frac{239}{7} a^{12} + \frac{242}{7} a^{11} - \frac{306}{7} a^{10} + \frac{493}{7} a^{9} - \frac{781}{7} a^{8} + \frac{1016}{7} a^{7} - \frac{887}{7} a^{6} + \frac{271}{7} a^{5} + 53 a^{4} - \frac{773}{7} a^{3} + \frac{750}{7} a^{2} - \frac{410}{7} a + \frac{106}{7} \),  \( 3 a^{15} + 4 a^{14} - 31 a^{13} + 94 a^{12} - 142 a^{11} + 199 a^{10} - 248 a^{9} + 373 a^{8} - 550 a^{7} + 823 a^{6} - 975 a^{5} + 942 a^{4} - 712 a^{3} + 398 a^{2} - 142 a + 25 \),  \( 15 a^{15} - 72 a^{14} + 189 a^{13} - 327 a^{12} + 461 a^{11} - 616 a^{10} + 868 a^{9} - 1275 a^{8} + 1808 a^{7} - 2241 a^{6} + 2277 a^{5} - 1877 a^{4} + 1203 a^{3} - 556 a^{2} + 160 a - 21 \),  \( \frac{68}{7} a^{15} - \frac{225}{7} a^{14} + \frac{453}{7} a^{13} - \frac{519}{7} a^{12} + \frac{613}{7} a^{11} - \frac{768}{7} a^{10} + \frac{1277}{7} a^{9} - \frac{1887}{7} a^{8} + \frac{2500}{7} a^{7} - \frac{2182}{7} a^{6} + \frac{1104}{7} a^{5} + 21 a^{4} - \frac{976}{7} a^{3} + \frac{1030}{7} a^{2} - \frac{550}{7} a + \frac{120}{7} \),  \( \frac{124}{7} a^{15} - \frac{568}{7} a^{14} + \frac{1482}{7} a^{13} - \frac{2528}{7} a^{12} + \frac{3588}{7} a^{11} - \frac{4758}{7} a^{10} + \frac{6765}{7} a^{9} - \frac{9881}{7} a^{8} + \frac{14043}{7} a^{7} - \frac{17246}{7} a^{6} + \frac{17533}{7} a^{5} - 2050 a^{4} + \frac{9195}{7} a^{3} - \frac{4241}{7} a^{2} + \frac{1235}{7} a - \frac{167}{7} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 20.5837782538 \)
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

\(\Q(\sqrt{-3}) \), 8.0.3167829.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
277Data not computed