Properties

Label 16.0.4186494384765625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{2}\cdot 101^{2}$
Root discriminant $9.47$
Ramified primes $5, 41, 101$
Class number $1$
Class group Trivial
Galois Group 16T1720

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 9, -9, -2, 17, -16, -7, 26, -17, -6, 15, -5, -6, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 5*x^12 + 15*x^11 - 6*x^10 - 17*x^9 + 26*x^8 - 7*x^7 - 16*x^6 + 17*x^5 - 2*x^4 - 9*x^3 + 9*x^2 - 4*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 5*x^12 + 15*x^11 - 6*x^10 - 17*x^9 + 26*x^8 - 7*x^7 - 16*x^6 + 17*x^5 - 2*x^4 - 9*x^3 + 9*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 8 x^{14} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut 5 x^{12} \) \(\mathstrut +\mathstrut 15 x^{11} \) \(\mathstrut -\mathstrut 6 x^{10} \) \(\mathstrut -\mathstrut 17 x^{9} \) \(\mathstrut +\mathstrut 26 x^{8} \) \(\mathstrut -\mathstrut 7 x^{7} \) \(\mathstrut -\mathstrut 16 x^{6} \) \(\mathstrut +\mathstrut 17 x^{5} \) \(\mathstrut -\mathstrut 2 x^{4} \) \(\mathstrut -\mathstrut 9 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(4186494384765625=5^{12}\cdot 41^{2}\cdot 101^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.47$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 41, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{139} a^{15} + \frac{66}{139} a^{14} + \frac{41}{139} a^{13} - \frac{55}{139} a^{12} + \frac{37}{139} a^{11} - \frac{36}{139} a^{10} - \frac{24}{139} a^{9} - \frac{29}{139} a^{8} - \frac{58}{139} a^{7} - \frac{36}{139} a^{6} - \frac{34}{139} a^{5} - \frac{2}{139} a^{3} - \frac{10}{139} a^{2} + \frac{4}{139} a - \frac{2}{139}$

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:  \( \frac{376}{139} a^{15} - \frac{1316}{139} a^{14} + \frac{2211}{139} a^{13} - \frac{664}{139} a^{12} - \frac{3046}{139} a^{11} + \frac{4395}{139} a^{10} + \frac{984}{139} a^{9} - \frac{7429}{139} a^{8} + \frac{5575}{139} a^{7} + \frac{2866}{139} a^{6} - \frac{6529}{139} a^{5} + 14 a^{4} + \frac{2445}{139} a^{3} - \frac{2648}{139} a^{2} + \frac{1087}{139} a - \frac{57}{139} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{62}{139} a^{15} - \frac{78}{139} a^{14} - \frac{99}{139} a^{13} + \frac{621}{139} a^{12} - \frac{625}{139} a^{11} - \frac{425}{139} a^{10} + \frac{1570}{139} a^{9} - \frac{686}{139} a^{8} - \frac{1650}{139} a^{7} + \frac{2077}{139} a^{6} + \frac{116}{139} a^{5} - 13 a^{4} + \frac{988}{139} a^{3} + \frac{353}{139} a^{2} - \frac{725}{139} a + \frac{432}{139} \),  \( \frac{34}{139} a^{15} + \frac{20}{139} a^{14} - \frac{274}{139} a^{13} + \frac{771}{139} a^{12} - \frac{549}{139} a^{11} - \frac{668}{139} a^{10} + \frac{1825}{139} a^{9} - \frac{430}{139} a^{8} - \frac{2250}{139} a^{7} + \frac{2668}{139} a^{6} + \frac{373}{139} a^{5} - 18 a^{4} + \frac{1183}{139} a^{3} + \frac{911}{139} a^{2} - \frac{1115}{139} a + \frac{349}{139} \),  \( \frac{338}{139} a^{15} - \frac{1183}{139} a^{14} + \frac{2043}{139} a^{13} - \frac{798}{139} a^{12} - \frac{2367}{139} a^{11} + \frac{3817}{139} a^{10} + \frac{506}{139} a^{9} - \frac{6049}{139} a^{8} + \frac{5277}{139} a^{7} + \frac{1593}{139} a^{6} - \frac{5237}{139} a^{5} + 17 a^{4} + \frac{1548}{139} a^{3} - \frac{2407}{139} a^{2} + \frac{1352}{139} a - \frac{259}{139} \),  \( \frac{538}{139} a^{15} - \frac{2022}{139} a^{14} + \frac{3710}{139} a^{13} - \frac{1929}{139} a^{12} - \frac{3863}{139} a^{11} + \frac{7459}{139} a^{10} - \frac{541}{139} a^{9} - \frac{10737}{139} a^{8} + \frac{11330}{139} a^{7} + \frac{1343}{139} a^{6} - \frac{10369}{139} a^{5} + 43 a^{4} + \frac{2677}{139} a^{3} - \frac{5102}{139} a^{2} + \frac{2847}{139} a - \frac{520}{139} \),  \( \frac{84}{139} a^{15} - \frac{155}{139} a^{14} + \frac{108}{139} a^{13} + \frac{384}{139} a^{12} - \frac{506}{139} a^{11} - \frac{105}{139} a^{10} + \frac{1181}{139} a^{9} - \frac{629}{139} a^{8} - \frac{841}{139} a^{7} + \frac{1285}{139} a^{6} + \frac{202}{139} a^{5} - 7 a^{4} + \frac{249}{139} a^{3} + \frac{272}{139} a^{2} - \frac{220}{139} a - \frac{29}{139} \),  \( \frac{259}{139} a^{15} - \frac{698}{139} a^{14} + \frac{889}{139} a^{13} + \frac{489}{139} a^{12} - \frac{2093}{139} a^{11} + \frac{1518}{139} a^{10} + \frac{2263}{139} a^{9} - \frac{3897}{139} a^{8} + \frac{685}{139} a^{7} + \frac{3464}{139} a^{6} - \frac{2551}{139} a^{5} - 6 a^{4} + \frac{1845}{139} a^{3} - \frac{783}{139} a^{2} - \frac{76}{139} a + \frac{177}{139} \),  \( \frac{3}{139} a^{15} + \frac{59}{139} a^{14} - \frac{155}{139} a^{13} + \frac{252}{139} a^{12} - \frac{28}{139} a^{11} - \frac{247}{139} a^{10} + \frac{345}{139} a^{9} + \frac{191}{139} a^{8} - \frac{591}{139} a^{7} + \frac{587}{139} a^{6} + \frac{37}{139} a^{5} - 3 a^{4} + \frac{411}{139} a^{3} + \frac{109}{139} a^{2} - \frac{405}{139} a + \frac{272}{139} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 43.5759224128 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1720:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1720 are not computed
Character table for t16n1720 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
41Data not computed
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$