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Properties

Label 16.12.3007692940...5376.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{64}\cdot 113^{4}$
Root discriminant $52.17$
Ramified primes $2, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2686, -16400, 6616, 55360, -8852, -43664, -6384, 16800, 9130, -4496, -3720, 784, 712, -32, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 - 32*x^13 + 712*x^12 + 784*x^11 - 3720*x^10 - 4496*x^9 + 9130*x^8 + 16800*x^7 - 6384*x^6 - 43664*x^5 - 8852*x^4 + 55360*x^3 + 6616*x^2 - 16400*x - 2686)
 
gp: K = bnfinit(x^16 - 48*x^14 - 32*x^13 + 712*x^12 + 784*x^11 - 3720*x^10 - 4496*x^9 + 9130*x^8 + 16800*x^7 - 6384*x^6 - 43664*x^5 - 8852*x^4 + 55360*x^3 + 6616*x^2 - 16400*x - 2686, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{14} - 32 x^{13} + 712 x^{12} + 784 x^{11} - 3720 x^{10} - 4496 x^{9} + 9130 x^{8} + 16800 x^{7} - 6384 x^{6} - 43664 x^{5} - 8852 x^{4} + 55360 x^{3} + 6616 x^{2} - 16400 x - 2686 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3007692940260731871482085376=2^{64}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.17$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{4}{17} a^{13} + \frac{7}{17} a^{12} - \frac{1}{17} a^{11} - \frac{5}{17} a^{10} - \frac{6}{17} a^{9} + \frac{5}{17} a^{8} - \frac{1}{17} a^{7} + \frac{5}{17} a^{6} + \frac{2}{17} a^{5} + \frac{7}{17} a^{4} - \frac{4}{17} a^{3} + \frac{7}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{55113638642060963320368147778657} a^{15} + \frac{1572673243136322407624423841799}{55113638642060963320368147778657} a^{14} + \frac{19509977824883698099395099666929}{55113638642060963320368147778657} a^{13} - \frac{3363354316434049120749340018871}{7873376948865851902909735396951} a^{12} - \frac{19575030311709887503331694159905}{55113638642060963320368147778657} a^{11} + \frac{24541563544649031098875909070809}{55113638642060963320368147778657} a^{10} - \frac{21897138615653829147863917533003}{55113638642060963320368147778657} a^{9} - \frac{18698950115062100618507688977314}{55113638642060963320368147778657} a^{8} + \frac{11839769860150413066490623696549}{55113638642060963320368147778657} a^{7} + \frac{12219695019711043095438244806138}{55113638642060963320368147778657} a^{6} + \frac{525243424267742654290696204523}{55113638642060963320368147778657} a^{5} + \frac{21328978151162455630774960410905}{55113638642060963320368147778657} a^{4} - \frac{40973258046750877647289056697}{463139820521520700171160905703} a^{3} + \frac{1426602301424387080866043551482}{55113638642060963320368147778657} a^{2} + \frac{1701003180543480828454113564924}{7873376948865851902909735396951} a + \frac{917464218136822622488624588149}{3241978743650644901198126339921}$

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:  $13$
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:  \( 374028203.834 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.242665652224.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
113Data not computed