Properties

Label 16.0.6926910400390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{3}$
Root discriminant $9.77$
Ramified primes $5, 61$
Class number $1$
Class group Trivial
Galois Group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 30, -70, 110, -109, 37, 80, -160, 140, -43, -46, 75, -55, 25, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 7 x^{15} \) \(\mathstrut +\mathstrut 25 x^{14} \) \(\mathstrut -\mathstrut 55 x^{13} \) \(\mathstrut +\mathstrut 75 x^{12} \) \(\mathstrut -\mathstrut 46 x^{11} \) \(\mathstrut -\mathstrut 43 x^{10} \) \(\mathstrut +\mathstrut 140 x^{9} \) \(\mathstrut -\mathstrut 160 x^{8} \) \(\mathstrut +\mathstrut 80 x^{7} \) \(\mathstrut +\mathstrut 37 x^{6} \) \(\mathstrut -\mathstrut 109 x^{5} \) \(\mathstrut +\mathstrut 110 x^{4} \) \(\mathstrut -\mathstrut 70 x^{3} \) \(\mathstrut +\mathstrut 30 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6926910400390625=5^{15}\cdot 61^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.77$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1231} a^{15} - \frac{252}{1231} a^{14} + \frac{215}{1231} a^{13} + \frac{203}{1231} a^{12} - \frac{420}{1231} a^{11} - \frac{550}{1231} a^{10} + \frac{528}{1231} a^{9} + \frac{35}{1231} a^{8} - \frac{118}{1231} a^{7} - \frac{554}{1231} a^{6} + \frac{357}{1231} a^{5} - \frac{173}{1231} a^{4} - \frac{590}{1231} a^{3} + \frac{453}{1231} a^{2} - \frac{165}{1231} a - \frac{206}{1231}$

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:  \( \frac{6623}{1231} a^{15} - \frac{44076}{1231} a^{14} + \frac{149860}{1231} a^{13} - \frac{309995}{1231} a^{12} + \frac{383241}{1231} a^{11} - \frac{163844}{1231} a^{10} - \frac{343776}{1231} a^{9} + \frac{796834}{1231} a^{8} - \frac{764280}{1231} a^{7} + \frac{252824}{1231} a^{6} + \frac{327106}{1231} a^{5} - \frac{595522}{1231} a^{4} + \frac{512951}{1231} a^{3} - \frac{281627}{1231} a^{2} + \frac{100044}{1231} a - \frac{17624}{1231} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{11001}{1231} a^{15} - \frac{72669}{1231} a^{14} + \frac{246664}{1231} a^{13} - \frac{509465}{1231} a^{12} + \frac{628564}{1231} a^{11} - \frac{263619}{1231} a^{10} - \frac{577900}{1231} a^{9} + \frac{1326750}{1231} a^{8} - \frac{1258726}{1231} a^{7} + \frac{392816}{1231} a^{6} + \frac{577806}{1231} a^{5} - \frac{998388}{1231} a^{4} + \frac{830167}{1231} a^{3} - \frac{439102}{1231} a^{2} + \frac{147049}{1231} a - \frac{23324}{1231} \),  \( \frac{5015}{1231} a^{15} - \frac{31549}{1231} a^{14} + \frac{102042}{1231} a^{13} - \frac{196952}{1231} a^{12} + \frac{214135}{1231} a^{11} - \frac{32816}{1231} a^{10} - \frac{294170}{1231} a^{9} + \frac{511588}{1231} a^{8} - \frac{377576}{1231} a^{7} + \frac{7443}{1231} a^{6} + \frac{297152}{1231} a^{5} - \frac{350575}{1231} a^{4} + \frac{231902}{1231} a^{3} - \frac{94187}{1231} a^{2} + \frac{20684}{1231} a - \frac{281}{1231} \),  \( \frac{1635}{1231} a^{15} - \frac{8252}{1231} a^{14} + \frac{21617}{1231} a^{13} - \frac{28778}{1231} a^{12} + \frac{6353}{1231} a^{11} + \frac{51082}{1231} a^{10} - \frac{93207}{1231} a^{9} + \frac{65842}{1231} a^{8} + \frac{32343}{1231} a^{7} - \frac{98254}{1231} a^{6} + \frac{90064}{1231} a^{5} - \frac{20652}{1231} a^{4} - \frac{26628}{1231} a^{3} + \frac{37754}{1231} a^{2} - \frac{21113}{1231} a + \frac{6639}{1231} \),  \( \frac{11939}{1231} a^{15} - \frac{76386}{1231} a^{14} + \frac{251374}{1231} a^{13} - \frac{498777}{1231} a^{12} + \frac{575591}{1231} a^{11} - \frac{168943}{1231} a^{10} - \frac{643972}{1231} a^{9} + \frac{1273410}{1231} a^{8} - \frac{1082587}{1231} a^{7} + \frac{226461}{1231} a^{6} + \frac{613539}{1231} a^{5} - \frac{912000}{1231} a^{4} + \frac{716214}{1231} a^{3} - \frac{360099}{1231} a^{2} + \frac{114148}{1231} a - \frac{17130}{1231} \),  \( \frac{2116}{1231} a^{15} - \frac{14981}{1231} a^{14} + \frac{53634}{1231} a^{13} - \frac{117016}{1231} a^{12} + \frac{155168}{1231} a^{11} - \frac{82982}{1231} a^{10} - \frac{114983}{1231} a^{9} + \frac{311643}{1231} a^{8} - \frac{322317}{1231} a^{7} + \frac{123979}{1231} a^{6} + \frac{120216}{1231} a^{5} - \frac{236813}{1231} a^{4} + \frac{211526}{1231} a^{3} - \frac{118577}{1231} a^{2} + \frac{43549}{1231} a - \frac{7508}{1231} \),  \( \frac{3937}{1231} a^{15} - \frac{27020}{1231} a^{14} + \frac{93083}{1231} a^{13} - \frac{194206}{1231} a^{12} + \frac{239738}{1231} a^{11} - \frac{97270}{1231} a^{10} - \frac{229389}{1231} a^{9} + \frac{510788}{1231} a^{8} - \frac{471952}{1231} a^{7} + \frac{127027}{1231} a^{6} + \frac{232366}{1231} a^{5} - \frac{377044}{1231} a^{4} + \frac{302893}{1231} a^{3} - \frac{156595}{1231} a^{2} + \frac{50834}{1231} a - \frac{8410}{1231} \),  \( \frac{2520}{1231} a^{15} - \frac{15847}{1231} a^{14} + \frac{51862}{1231} a^{13} - \frac{101478}{1231} a^{12} + \frac{113512}{1231} a^{11} - \frac{23283}{1231} a^{10} - \frac{146640}{1231} a^{9} + \frac{266695}{1231} a^{8} - \frac{203804}{1231} a^{7} + \frac{13415}{1231} a^{6} + \frac{152423}{1231} a^{5} - \frac{181143}{1231} a^{4} + \frac{125810}{1231} a^{3} - \frac{53741}{1231} a^{2} + \frac{13819}{1231} a - \frac{869}{1231} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 65.2827160145 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1192:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$61$61.2.0.1$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}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$