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Properties

Label 16.2.31938453683...4779.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,41^{15}\cdot 59^{3}$
Root discriminant $69.83$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-48116, -129620, -59961, 108558, 88173, -31766, -18743, 35171, 20706, -2896, -3049, 148, 236, 8, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 + 8*x^13 + 236*x^12 + 148*x^11 - 3049*x^10 - 2896*x^9 + 20706*x^8 + 35171*x^7 - 18743*x^6 - 31766*x^5 + 88173*x^4 + 108558*x^3 - 59961*x^2 - 129620*x - 48116)
 
gp: K = bnfinit(x^16 - 3*x^15 - 15*x^14 + 8*x^13 + 236*x^12 + 148*x^11 - 3049*x^10 - 2896*x^9 + 20706*x^8 + 35171*x^7 - 18743*x^6 - 31766*x^5 + 88173*x^4 + 108558*x^3 - 59961*x^2 - 129620*x - 48116, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 15 x^{14} + 8 x^{13} + 236 x^{12} + 148 x^{11} - 3049 x^{10} - 2896 x^{9} + 20706 x^{8} + 35171 x^{7} - 18743 x^{6} - 31766 x^{5} + 88173 x^{4} + 108558 x^{3} - 59961 x^{2} - 129620 x - 48116 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-319384536834647064600389984779=-\,41^{15}\cdot 59^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.83$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $41, 59$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{3}{8} a^{11} - \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{22629542660236182267515567509408384} a^{15} + \frac{1235602613933452859777485496769415}{22629542660236182267515567509408384} a^{14} + \frac{741266952702770403333805302175879}{22629542660236182267515567509408384} a^{13} + \frac{1151491518994477389635408440774007}{11314771330118091133757783754704192} a^{12} - \frac{286914221127807410769426620068639}{2828692832529522783439445938676048} a^{11} - \frac{2058238475321771398501348577828675}{5657385665059045566878891877352096} a^{10} - \frac{2543380494664442130492911192054449}{22629542660236182267515567509408384} a^{9} - \frac{4280526462367643992528305561222421}{11314771330118091133757783754704192} a^{8} - \frac{2815453162504857703644042106001905}{11314771330118091133757783754704192} a^{7} - \frac{9561619854526343957385077311865905}{22629542660236182267515567509408384} a^{6} - \frac{5444399831095342527389955362941505}{22629542660236182267515567509408384} a^{5} - \frac{178891859438677962186813628106321}{1414346416264761391719722969338024} a^{4} - \frac{1433519233597160561490218114469267}{22629542660236182267515567509408384} a^{3} - \frac{424607478739377269814188519549781}{1414346416264761391719722969338024} a^{2} - \frac{351404625107293631487187140873641}{22629542660236182267515567509408384} a + \frac{637416445064139186055455968209873}{11314771330118091133757783754704192}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 336875783.526 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.11490502158979.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ R

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
41Data not computed
59Data not computed