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Properties

Label 16.0.12003260674...9641.9
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{6}$
Root discriminant $31.99$
Ramified primes $13, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2187, -3078, 11718, -2808, 591, -3083, -1526, 849, 17, 342, -43, -7, 9, -10, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^13 + 9*x^12 - 7*x^11 - 43*x^10 + 342*x^9 + 17*x^8 + 849*x^7 - 1526*x^6 - 3083*x^5 + 591*x^4 - 2808*x^3 + 11718*x^2 - 3078*x + 2187)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^13 + 9*x^12 - 7*x^11 - 43*x^10 + 342*x^9 + 17*x^8 + 849*x^7 - 1526*x^6 - 3083*x^5 + 591*x^4 - 2808*x^3 + 11718*x^2 - 3078*x + 2187, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{13} + 9 x^{12} - 7 x^{11} - 43 x^{10} + 342 x^{9} + 17 x^{8} + 849 x^{7} - 1526 x^{6} - 3083 x^{5} + 591 x^{4} - 2808 x^{3} + 11718 x^{2} - 3078 x + 2187 \)

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:  \(1200326067404665657109641=13^{12}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.99$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{3537} a^{13} - \frac{281}{3537} a^{12} - \frac{8}{393} a^{11} + \frac{17}{3537} a^{10} - \frac{19}{131} a^{9} + \frac{1577}{3537} a^{8} + \frac{470}{3537} a^{7} + \frac{85}{393} a^{6} + \frac{71}{3537} a^{5} + \frac{73}{1179} a^{4} - \frac{23}{3537} a^{3} - \frac{1211}{3537} a^{2} - \frac{70}{1179} a - \frac{43}{131}$, $\frac{1}{95499} a^{14} - \frac{5}{95499} a^{13} - \frac{2689}{31833} a^{12} + \frac{15515}{95499} a^{11} - \frac{572}{31833} a^{10} + \frac{3827}{95499} a^{9} - \frac{28804}{95499} a^{8} + \frac{12055}{31833} a^{7} - \frac{4546}{95499} a^{6} - \frac{1255}{31833} a^{5} - \frac{36257}{95499} a^{4} - \frac{40571}{95499} a^{3} - \frac{6944}{31833} a^{2} - \frac{674}{10611} a + \frac{166}{393}$, $\frac{1}{120506490946079559} a^{15} - \frac{258874142552}{120506490946079559} a^{14} + \frac{702107906629}{40168830315359853} a^{13} + \frac{18661024132025273}{120506490946079559} a^{12} - \frac{521328000220453}{40168830315359853} a^{11} + \frac{2219120393302379}{120506490946079559} a^{10} - \frac{10059474301637698}{120506490946079559} a^{9} - \frac{14791556795444071}{40168830315359853} a^{8} - \frac{8973440840636038}{120506490946079559} a^{7} + \frac{2157037174828768}{4463203368373317} a^{6} - \frac{5240606020970414}{120506490946079559} a^{5} + \frac{52106770537737676}{120506490946079559} a^{4} - \frac{613506996166862}{1487734456124439} a^{3} - \frac{1458559322228714}{4463203368373317} a^{2} + \frac{1561705270469636}{4463203368373317} a - \frac{81266395833853}{165303828458271}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
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:  \( 1152643.86726 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

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 t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.294435349.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed