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Properties

Label 16.4.12139888392...1633.2
Degree $16$
Signature $[4, 6]$
Discriminant $61^{8}\cdot 97^{15}$
Root discriminant $569.20$
Ramified primes $61, 97$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (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![-78782439251, 251912468219, -200064755609, 72820441481, -14795099235, 740318833, -338640099, -77115479, 1244977, -4365410, 577204, -101082, 16639, -1045, 223, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 223*x^14 - 1045*x^13 + 16639*x^12 - 101082*x^11 + 577204*x^10 - 4365410*x^9 + 1244977*x^8 - 77115479*x^7 - 338640099*x^6 + 740318833*x^5 - 14795099235*x^4 + 72820441481*x^3 - 200064755609*x^2 + 251912468219*x - 78782439251)
 
gp: K = bnfinit(x^16 - 6*x^15 + 223*x^14 - 1045*x^13 + 16639*x^12 - 101082*x^11 + 577204*x^10 - 4365410*x^9 + 1244977*x^8 - 77115479*x^7 - 338640099*x^6 + 740318833*x^5 - 14795099235*x^4 + 72820441481*x^3 - 200064755609*x^2 + 251912468219*x - 78782439251, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 223 x^{14} - 1045 x^{13} + 16639 x^{12} - 101082 x^{11} + 577204 x^{10} - 4365410 x^{9} + 1244977 x^{8} - 77115479 x^{7} - 338640099 x^{6} + 740318833 x^{5} - 14795099235 x^{4} + 72820441481 x^{3} - 200064755609 x^{2} + 251912468219 x - 78782439251 \)

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:  \(121398883921746872662013204631882371288461633=61^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $569.20$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{206} a^{14} - \frac{15}{206} a^{13} + \frac{25}{206} a^{12} - \frac{43}{103} a^{11} - \frac{79}{206} a^{10} - \frac{45}{206} a^{9} + \frac{27}{206} a^{8} - \frac{51}{206} a^{7} + \frac{33}{206} a^{6} + \frac{3}{206} a^{5} + \frac{5}{103} a^{4} - \frac{9}{206} a^{3} + \frac{16}{103} a^{2} + \frac{81}{206} a + \frac{43}{206}$, $\frac{1}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a^{15} + \frac{372278924929267378161064611564280758351087353321548360029553587670498826322231}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a^{14} - \frac{20671918848140373464545985075753142622932067093069768255738248941308257282512159}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{13} + \frac{1495784500504245270734875847909974481766825488729609698966984649598586718814762}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{12} - \frac{50887357000699908521239784615067178189869091997472340196581032806999001942479163}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{11} + \frac{19868368398222608185072724795767816879556638404723543475143268306864091504122307}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{10} - \frac{19789398987217106621931337966398753406935493157258739953787260943707027188766193}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a^{9} + \frac{45864084813712659880259972720454574284915740535877715261256776559884985196852767}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{8} - \frac{96861361673426438215226665316300606966350837534208512892739721652030858842338791}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a^{7} + \frac{40483951580267238464811940014775089757798148218373374827833746417755155178822539}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{6} + \frac{16517257232506623658472197169621903467837863662197207265311415238416584617816027}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{5} + \frac{22582240965427613361696390025537212966136363426373394175430941280405354024659855}{106641380974530486810022764687122959046368106901663227018034628472635045237064801} a^{4} - \frac{48340287557671611835656647403199960391326696746980362788130330917531918681109271}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a^{3} - \frac{17083221428282243195655514210160978177145600457787159307070355301177007831565073}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a^{2} - \frac{103228781640495612740706421257776206188847365142341287636398573454117182319574421}{213282761949060973620045529374245918092736213803326454036069256945270090474129602} a - \frac{352563506064508179908155460681883851614934622921051085792696587090535538418837}{2111710514347138352673718112616296218739962512904222317188804524210594955189402}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( 140105605131000 \) (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

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.1118720199956720578033.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed