Properties

Label 16.0.6561000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $9.74$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois Group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 18, -28, 9, 42, -98, 131, -134, 114, -85, 56, -32, 15, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 15 x^{14} \) \(\mathstrut -\mathstrut 32 x^{13} \) \(\mathstrut +\mathstrut 56 x^{12} \) \(\mathstrut -\mathstrut 85 x^{11} \) \(\mathstrut +\mathstrut 114 x^{10} \) \(\mathstrut -\mathstrut 134 x^{9} \) \(\mathstrut +\mathstrut 131 x^{8} \) \(\mathstrut -\mathstrut 98 x^{7} \) \(\mathstrut +\mathstrut 42 x^{6} \) \(\mathstrut +\mathstrut 9 x^{5} \) \(\mathstrut -\mathstrut 28 x^{4} \) \(\mathstrut +\mathstrut 18 x^{3} \) \(\mathstrut -\mathstrut x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6561000000000000=2^{12}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.74$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13129} a^{15} - \frac{1370}{13129} a^{14} + \frac{5747}{13129} a^{13} + \frac{6455}{13129} a^{12} - \frac{1460}{13129} a^{11} - \frac{147}{691} a^{10} + \frac{271}{691} a^{9} - \frac{4504}{13129} a^{8} + \frac{3719}{13129} a^{7} + \frac{4390}{13129} a^{6} - \frac{5484}{13129} a^{5} + \frac{2139}{13129} a^{4} - \frac{5125}{13129} a^{3} - \frac{2114}{13129} a^{2} - \frac{2771}{13129} a + \frac{1260}{13129}$

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:  \( -\frac{11709}{13129} a^{15} + \frac{50208}{13129} a^{14} - \frac{136788}{13129} a^{13} + \frac{264638}{13129} a^{12} - \frac{432075}{13129} a^{11} + \frac{32419}{691} a^{10} - \frac{40836}{691} a^{9} + \frac{838399}{13129} a^{8} - \frac{705844}{13129} a^{7} + \frac{378266}{13129} a^{6} + \frac{24475}{13129} a^{5} - \frac{257999}{13129} a^{4} + \frac{206030}{13129} a^{3} - \frac{34726}{13129} a^{2} - \frac{61765}{13129} a + \frac{16785}{13129} \) (order $30$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{458}{13129} a^{15} + \frac{2732}{13129} a^{14} - \frac{6803}{13129} a^{13} + \frac{15494}{13129} a^{12} - \frac{25359}{13129} a^{11} + \frac{2465}{691} a^{10} - \frac{3717}{691} a^{9} + \frac{90324}{13129} a^{8} - \frac{108500}{13129} a^{7} + \frac{106915}{13129} a^{6} - \frac{95936}{13129} a^{5} + \frac{86890}{13129} a^{4} - \frac{62804}{13129} a^{3} + \frac{29592}{13129} a^{2} + \frac{30653}{13129} a - \frac{13725}{13129} \),  \( \frac{9829}{13129} a^{15} - \frac{34763}{13129} a^{14} + \frac{85079}{13129} a^{13} - \frac{150681}{13129} a^{12} + \frac{235979}{13129} a^{11} - \frac{17257}{691} a^{10} + \frac{20584}{691} a^{9} - \frac{405827}{13129} a^{8} + \frac{304882}{13129} a^{7} - \frac{137003}{13129} a^{6} - \frac{33949}{13129} a^{5} + \frac{83476}{13129} a^{4} - \frac{10781}{13129} a^{3} - \frac{21557}{13129} a^{2} + \frac{32774}{13129} a + \frac{3893}{13129} \),  \( \frac{9392}{13129} a^{15} - \frac{40007}{13129} a^{14} + \frac{107537}{13129} a^{13} - \frac{214426}{13129} a^{12} + \frac{361968}{13129} a^{11} - \frac{28337}{691} a^{10} + \frac{36902}{691} a^{9} - \frac{800799}{13129} a^{8} + \frac{754061}{13129} a^{7} - \frac{532469}{13129} a^{6} + \frac{222532}{13129} a^{5} + \frac{28376}{13129} a^{4} - \frac{94989}{13129} a^{3} + \frac{62005}{13129} a^{2} - \frac{3554}{13129} a - \frac{8438}{13129} \),  \( \frac{12204}{13129} a^{15} - \frac{58779}{13129} a^{14} + \frac{171947}{13129} a^{13} - \frac{364792}{13129} a^{12} + \frac{641534}{13129} a^{11} - \frac{51977}{691} a^{10} + \frac{70640}{691} a^{9} - \frac{1610560}{13129} a^{8} + \frac{1627719}{13129} a^{7} - \frac{1316789}{13129} a^{6} + \frac{740130}{13129} a^{5} - \frac{179902}{13129} a^{4} - \frac{90847}{13129} a^{3} + \frac{104261}{13129} a^{2} - \frac{23238}{13129} a + \frac{2981}{13129} \),  \( \frac{2452}{13129} a^{15} - \frac{11345}{13129} a^{14} + \frac{30485}{13129} a^{13} - \frac{58430}{13129} a^{12} + \frac{96200}{13129} a^{11} - \frac{7343}{691} a^{10} + \frac{9424}{691} a^{9} - \frac{199254}{13129} a^{8} + \frac{178139}{13129} a^{7} - \frac{119661}{13129} a^{6} + \frac{36715}{13129} a^{5} + \frac{6357}{13129} a^{4} - \frac{2047}{13129} a^{3} - \frac{23831}{13129} a^{2} + \frac{19459}{13129} a + \frac{4205}{13129} \),  \( \frac{5442}{13129} a^{15} - \frac{24526}{13129} a^{14} + \frac{67541}{13129} a^{13} - \frac{136384}{13129} a^{12} + \frac{234047}{13129} a^{11} - \frac{18453}{691} a^{10} + \frac{24373}{691} a^{9} - \frac{537214}{13129} a^{8} + \frac{519040}{13129} a^{7} - \frac{385141}{13129} a^{6} + \frac{182095}{13129} a^{5} - \frac{18114}{13129} a^{4} - \frac{43641}{13129} a^{3} + \frac{36003}{13129} a^{2} + \frac{5439}{13129} a - \frac{9547}{13129} \),  \( \frac{5448}{13129} a^{15} - \frac{19617}{13129} a^{14} + \frac{49507}{13129} a^{13} - \frac{84525}{13129} a^{12} + \frac{120255}{13129} a^{11} - \frac{7588}{691} a^{10} + \frac{7342}{691} a^{9} - \frac{91594}{13129} a^{8} - \frac{36322}{13129} a^{7} + \frac{192617}{13129} a^{6} - \frac{323453}{13129} a^{5} + \frac{322945}{13129} a^{4} - \frac{166294}{13129} a^{3} + \frac{10190}{13129} a^{2} + \frac{67587}{13129} a - \frac{28245}{13129} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 175.01473438 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.9000.2, 4.0.9000.1, 8.0.81000000.1, \(\Q(\zeta_{15})\), 8.0.3240000.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
5Data not computed