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Properties

Label 16.12.1170881448...2457.1
Degree $16$
Signature $[12, 2]$
Discriminant $43^{2}\cdot 97^{15}$
Root discriminant $116.62$
Ramified primes $43, 97$
Class number $2$ (GRH)
Class group $[2]$ (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![-13969, 61588, 328416, -167467, -367938, 520392, -244894, -199352, 291294, -136300, 33906, -5200, -277, 416, -58, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 58*x^14 + 416*x^13 - 277*x^12 - 5200*x^11 + 33906*x^10 - 136300*x^9 + 291294*x^8 - 199352*x^7 - 244894*x^6 + 520392*x^5 - 367938*x^4 - 167467*x^3 + 328416*x^2 + 61588*x - 13969)
 
gp: K = bnfinit(x^16 - 5*x^15 - 58*x^14 + 416*x^13 - 277*x^12 - 5200*x^11 + 33906*x^10 - 136300*x^9 + 291294*x^8 - 199352*x^7 - 244894*x^6 + 520392*x^5 - 367938*x^4 - 167467*x^3 + 328416*x^2 + 61588*x - 13969, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 58 x^{14} + 416 x^{13} - 277 x^{12} - 5200 x^{11} + 33906 x^{10} - 136300 x^{9} + 291294 x^{8} - 199352 x^{7} - 244894 x^{6} + 520392 x^{5} - 367938 x^{4} - 167467 x^{3} + 328416 x^{2} + 61588 x - 13969 \)

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:  \(1170881448713923574794017240042457=43^{2}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.62$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{97321826053571554890899658476669146465673} a^{15} - \frac{10511714211290174021196294005618362198004}{97321826053571554890899658476669146465673} a^{14} - \frac{25330425093373308042841497016520617378631}{97321826053571554890899658476669146465673} a^{13} + \frac{5131722508157928012913708605371171071263}{97321826053571554890899658476669146465673} a^{12} - \frac{21376930732845786422655641913120394725019}{97321826053571554890899658476669146465673} a^{11} - \frac{6194442456867522211557944472194914077355}{97321826053571554890899658476669146465673} a^{10} + \frac{35135792322663403009629291409598149947485}{97321826053571554890899658476669146465673} a^{9} - \frac{37469914403249194192113300519237613873264}{97321826053571554890899658476669146465673} a^{8} + \frac{28131088404454820211323400201760160210141}{97321826053571554890899658476669146465673} a^{7} + \frac{10142780047058387273573760385208719610882}{97321826053571554890899658476669146465673} a^{6} - \frac{47573578091022577990210162356744240942349}{97321826053571554890899658476669146465673} a^{5} + \frac{12236312625534911022543267519919173922154}{97321826053571554890899658476669146465673} a^{4} + \frac{44089052282751216319960509041338460529381}{97321826053571554890899658476669146465673} a^{3} + \frac{23885669122025627850349693318749577377349}{97321826053571554890899658476669146465673} a^{2} + \frac{8667028292851212408287579219778976426627}{97321826053571554890899658476669146465673} a - \frac{26216515392107906848769657809551748466501}{97321826053571554890899658476669146465673}$

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:  $13$
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:  \( 31705384392.4 \) (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

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed