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Properties

Label 16.8.21122310253...9312.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 13^{2}\cdot 17^{15}\cdot 2857^{4}$
Root discriminant $286.94$
Ramified primes $2, 13, 17, 2857$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191415273175747673, 0, 10786789982952529, 0, 160261082893754, 0, -306906491051, 0, -14600815653, 0, 20490185, 0, 333982, 0, -1139, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1139*x^14 + 333982*x^12 + 20490185*x^10 - 14600815653*x^8 - 306906491051*x^6 + 160261082893754*x^4 + 10786789982952529*x^2 + 191415273175747673)
 
gp: K = bnfinit(x^16 - 1139*x^14 + 333982*x^12 + 20490185*x^10 - 14600815653*x^8 - 306906491051*x^6 + 160261082893754*x^4 + 10786789982952529*x^2 + 191415273175747673, 1)
 

Normalized defining polynomial

\( x^{16} - 1139 x^{14} + 333982 x^{12} + 20490185 x^{10} - 14600815653 x^{8} - 306906491051 x^{6} + 160261082893754 x^{4} + 10786789982952529 x^{2} + 191415273175747673 \)

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:  \(2112231025384252423772630322104776589312=2^{16}\cdot 13^{2}\cdot 17^{15}\cdot 2857^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $286.94$
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:  $2, 13, 17, 2857$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2857} a^{10} - \frac{1139}{2857} a^{8} - \frac{287}{2857} a^{6} - \frac{219}{2857} a^{4} - \frac{16}{2857} a^{2}$, $\frac{1}{2857} a^{11} - \frac{1139}{2857} a^{9} - \frac{287}{2857} a^{7} - \frac{219}{2857} a^{5} - \frac{16}{2857} a^{3}$, $\frac{1}{8162449} a^{12} - \frac{1139}{8162449} a^{10} + \frac{333982}{8162449} a^{8} - \frac{3997162}{8162449} a^{6} + \frac{1805608}{8162449} a^{4} + \frac{557}{2857} a^{2}$, $\frac{1}{8162449} a^{13} - \frac{1139}{8162449} a^{11} + \frac{333982}{8162449} a^{9} - \frac{3997162}{8162449} a^{7} + \frac{1805608}{8162449} a^{5} + \frac{557}{2857} a^{3}$, $\frac{1}{249493744450750962635700169220329066393664876884879} a^{14} - \frac{422789438056122618074268079500173648236025}{249493744450750962635700169220329066393664876884879} a^{12} - \frac{31064932073133485442764764492627669640930013457}{249493744450750962635700169220329066393664876884879} a^{10} - \frac{72844103203641624689931061210847248623943779784620}{249493744450750962635700169220329066393664876884879} a^{8} + \frac{84684249538311660854856364914279513365879029758633}{249493744450750962635700169220329066393664876884879} a^{6} + \frac{22014834329946677163557264137376973047004536702}{87327176916608667355862852369733659920778745847} a^{4} + \frac{12692751541845467628400334039190936751737945}{30566040222824174783291162887551158530199071} a^{2} + \frac{17924812858469224123964309490234623293}{822973000802998701792928647251047589731}$, $\frac{1}{249493744450750962635700169220329066393664876884879} a^{15} - \frac{422789438056122618074268079500173648236025}{249493744450750962635700169220329066393664876884879} a^{13} - \frac{31064932073133485442764764492627669640930013457}{249493744450750962635700169220329066393664876884879} a^{11} - \frac{72844103203641624689931061210847248623943779784620}{249493744450750962635700169220329066393664876884879} a^{9} + \frac{84684249538311660854856364914279513365879029758633}{249493744450750962635700169220329066393664876884879} a^{7} + \frac{22014834329946677163557264137376973047004536702}{87327176916608667355862852369733659920778745847} a^{5} + \frac{12692751541845467628400334039190936751737945}{30566040222824174783291162887551158530199071} a^{3} + \frac{17924812858469224123964309490234623293}{822973000802998701792928647251047589731} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 4898213901570 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ 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 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 $16$ $16$ $16$ $16$ R 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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed
2857Data not computed