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Properties

Label 16.4.11940778378...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{12}\cdot 13^{2}\cdot 29^{7}$
Root discriminant $56.86$
Ramified primes $2, 5, 13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1606

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-359, 100734, -568564, -82004, -2907, -13716, -17136, -38462, -13097, -3026, 216, -512, 53, -82, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 14*x^14 - 82*x^13 + 53*x^12 - 512*x^11 + 216*x^10 - 3026*x^9 - 13097*x^8 - 38462*x^7 - 17136*x^6 - 13716*x^5 - 2907*x^4 - 82004*x^3 - 568564*x^2 + 100734*x - 359)
 
gp: K = bnfinit(x^16 - 2*x^15 + 14*x^14 - 82*x^13 + 53*x^12 - 512*x^11 + 216*x^10 - 3026*x^9 - 13097*x^8 - 38462*x^7 - 17136*x^6 - 13716*x^5 - 2907*x^4 - 82004*x^3 - 568564*x^2 + 100734*x - 359, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 14 x^{14} - 82 x^{13} + 53 x^{12} - 512 x^{11} + 216 x^{10} - 3026 x^{9} - 13097 x^{8} - 38462 x^{7} - 17136 x^{6} - 13716 x^{5} - 2907 x^{4} - 82004 x^{3} - 568564 x^{2} + 100734 x - 359 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11940778378121216000000000000=2^{24}\cdot 5^{12}\cdot 13^{2}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.86$
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:  $2, 5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{258769475526021512243057533762738145072195} a^{15} - \frac{7553544096580583608105090133210032642658}{258769475526021512243057533762738145072195} a^{14} + \frac{3461944100375577524328967531002626809567}{258769475526021512243057533762738145072195} a^{13} + \frac{6339402882048156508809729933286811383252}{258769475526021512243057533762738145072195} a^{12} + \frac{2452503983365443629916555595128734684989}{258769475526021512243057533762738145072195} a^{11} + \frac{25697272401503122950284192750742256337888}{258769475526021512243057533762738145072195} a^{10} + \frac{23322853672284105149906940605320196256584}{258769475526021512243057533762738145072195} a^{9} - \frac{22461334485432776781061424244256367152829}{258769475526021512243057533762738145072195} a^{8} + \frac{101170916176603015583416173491742175751484}{258769475526021512243057533762738145072195} a^{7} + \frac{2293322047943196872100427430681815980416}{13619446080316921697003028092775691845905} a^{6} + \frac{25668706551915772590071597419804153095039}{51753895105204302448611506752547629014439} a^{5} + \frac{23220230744646879935661934773640836525142}{51753895105204302448611506752547629014439} a^{4} + \frac{75251471822296038009122778950219181566792}{258769475526021512243057533762738145072195} a^{3} - \frac{99971442937068067942388816398685958823496}{258769475526021512243057533762738145072195} a^{2} - \frac{65402530601881395235965235952365242389788}{258769475526021512243057533762738145072195} a + \frac{24330126748572712855670125151397319627284}{51753895105204302448611506752547629014439}$

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:  $9$
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:  \( 19560904.4249 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1606:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1606 are not computed
Character table for t16n1606 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.97556000000.1

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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$