↑

Properties

Label 16.0.11576669171...7137.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 43^{8}$
Root discriminant $65.54$
Ramified primes $17, 43$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![180323, 651963, 856157, 506625, 190104, 110519, 68036, 49059, 24634, 3624, 3549, 615, -208, 13, -13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 13*x^14 + 13*x^13 - 208*x^12 + 615*x^11 + 3549*x^10 + 3624*x^9 + 24634*x^8 + 49059*x^7 + 68036*x^6 + 110519*x^5 + 190104*x^4 + 506625*x^3 + 856157*x^2 + 651963*x + 180323)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 13*x^13 - 208*x^12 + 615*x^11 + 3549*x^10 + 3624*x^9 + 24634*x^8 + 49059*x^7 + 68036*x^6 + 110519*x^5 + 190104*x^4 + 506625*x^3 + 856157*x^2 + 651963*x + 180323, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 13 x^{14} + 13 x^{13} - 208 x^{12} + 615 x^{11} + 3549 x^{10} + 3624 x^{9} + 24634 x^{8} + 49059 x^{7} + 68036 x^{6} + 110519 x^{5} + 190104 x^{4} + 506625 x^{3} + 856157 x^{2} + 651963 x + 180323 \)

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:  \(115766691713731939356121017137=17^{13}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.54$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{2}{13} a^{10} + \frac{5}{13} a^{9} - \frac{2}{13} a^{8} - \frac{6}{13} a^{6} + \frac{5}{13} a^{5} - \frac{6}{13} a^{4} + \frac{1}{13} a^{3} - \frac{3}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{13} a^{13} + \frac{2}{13} a^{11} + \frac{5}{13} a^{10} - \frac{2}{13} a^{9} - \frac{6}{13} a^{7} + \frac{5}{13} a^{6} - \frac{6}{13} a^{5} + \frac{1}{13} a^{4} - \frac{3}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{143} a^{14} + \frac{5}{143} a^{13} + \frac{1}{143} a^{12} + \frac{54}{143} a^{11} + \frac{60}{143} a^{10} - \frac{2}{143} a^{9} - \frac{56}{143} a^{8} + \frac{14}{143} a^{7} - \frac{14}{143} a^{6} + \frac{4}{13} a^{5} - \frac{5}{143} a^{4} - \frac{54}{143} a^{3} + \frac{34}{143} a^{2} + \frac{64}{143} a$, $\frac{1}{26775257239240181767698246226962214121} a^{15} - \frac{17804018302739049786033467062295625}{26775257239240181767698246226962214121} a^{14} + \frac{80864253920346668654506860188061665}{2434114294476380160699840566087474011} a^{13} + \frac{146639854293183180531724685421055264}{26775257239240181767698246226962214121} a^{12} - \frac{188586281854588325729405937037144811}{26775257239240181767698246226962214121} a^{11} - \frac{8106799260609961251237235705497852010}{26775257239240181767698246226962214121} a^{10} - \frac{1556685144104249074531001558136693041}{26775257239240181767698246226962214121} a^{9} + \frac{11135612014573218233342575516995290233}{26775257239240181767698246226962214121} a^{8} - \frac{3165091641888175660769294211703880155}{26775257239240181767698246226962214121} a^{7} - \frac{12994860828575389101203266356657040114}{26775257239240181767698246226962214121} a^{6} + \frac{12229776947702704319371553489785019916}{26775257239240181767698246226962214121} a^{5} - \frac{4277855350923048907278144065833254112}{26775257239240181767698246226962214121} a^{4} + \frac{9487786847769851812442269873277040819}{26775257239240181767698246226962214121} a^{3} - \frac{956958580018454001624201737135890421}{2434114294476380160699840566087474011} a^{2} + \frac{860369377641638422094365039872715910}{2059635172249244751361403555920170317} a + \frac{4193913812197255585254496982533214}{14403043162582131128401423467973219}$

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

Class group and class number

$C_{4}$, 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:  $7$
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:  \( 49367944.0923 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.31433.1, 8.0.4854208531457.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$17$17.8.6.4$x^{8} + 136 x^{4} + 7803$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
17.8.7.8$x^{8} + 4131$$8$$1$$7$$C_8$$[\ ]_{8}$
43Data not computed