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Properties

Label 16.8.31236842166...5728.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{15}\cdot 6529^{2}$
Root discriminant $60.38$
Ramified primes $2, 17, 6529$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-546719, -3508966, -4310198, -274843, 56339, -210048, 333964, -116063, -25924, 54370, -12632, -361, 290, -217, 50, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 50*x^14 - 217*x^13 + 290*x^12 - 361*x^11 - 12632*x^10 + 54370*x^9 - 25924*x^8 - 116063*x^7 + 333964*x^6 - 210048*x^5 + 56339*x^4 - 274843*x^3 - 4310198*x^2 - 3508966*x - 546719)
 
gp: K = bnfinit(x^16 - 4*x^15 + 50*x^14 - 217*x^13 + 290*x^12 - 361*x^11 - 12632*x^10 + 54370*x^9 - 25924*x^8 - 116063*x^7 + 333964*x^6 - 210048*x^5 + 56339*x^4 - 274843*x^3 - 4310198*x^2 - 3508966*x - 546719, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 50 x^{14} - 217 x^{13} + 290 x^{12} - 361 x^{11} - 12632 x^{10} + 54370 x^{9} - 25924 x^{8} - 116063 x^{7} + 333964 x^{6} - 210048 x^{5} + 56339 x^{4} - 274843 x^{3} - 4310198 x^{2} - 3508966 x - 546719 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31236842166910780816202985728=2^{8}\cdot 17^{15}\cdot 6529^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.38$
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:  $2, 17, 6529$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2698637572130304654964624700632935413198886321096882217} a^{15} + \frac{824779970493407653641358583892753832299084936464836167}{2698637572130304654964624700632935413198886321096882217} a^{14} + \frac{1042203145939375043606436835503666257573629003080351600}{2698637572130304654964624700632935413198886321096882217} a^{13} - \frac{1120788972043911292857308405833076968872312207619473993}{2698637572130304654964624700632935413198886321096882217} a^{12} - \frac{510132719509016308984729637157067910396774044541240911}{2698637572130304654964624700632935413198886321096882217} a^{11} + \frac{465332783840596791386445465910202934922621512739396889}{2698637572130304654964624700632935413198886321096882217} a^{10} + \frac{759865363387219535412375675232728556684988166117379615}{2698637572130304654964624700632935413198886321096882217} a^{9} + \frac{150025071779661258892075624108955910836231390947201825}{2698637572130304654964624700632935413198886321096882217} a^{8} - \frac{473204557844843740546557901376569439167926213510003944}{2698637572130304654964624700632935413198886321096882217} a^{7} - \frac{137765713451581650977846193848823505885356860821795563}{2698637572130304654964624700632935413198886321096882217} a^{6} - \frac{287448313344076557856707777688166655987819433973975887}{2698637572130304654964624700632935413198886321096882217} a^{5} + \frac{485421883371741237579457321883402039789901601731714502}{2698637572130304654964624700632935413198886321096882217} a^{4} + \frac{1088119578175754961288463035212278146704602691242434500}{2698637572130304654964624700632935413198886321096882217} a^{3} + \frac{1310863818315976637348027906006933613843786654047500156}{2698637572130304654964624700632935413198886321096882217} a^{2} + \frac{1038336753198318893890409921038518179477784603015108273}{2698637572130304654964624700632935413198886321096882217} a + \frac{198955059860220377558762919941070501735862861442318455}{2698637572130304654964624700632935413198886321096882217}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 34110956.0307 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
6529Data not computed