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Properties

Label 16.8.46706312975...8064.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{62}\cdot 17^{6}\cdot 127^{6}$
Root discriminant $261.12$
Ramified primes $2, 17, 127$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1528687073, -3117641336, -1527001640, 880219560, 1048954592, 214400888, -94591600, -42616648, -345306, 1888200, 54016, -73240, -5160, 1160, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 + 1160*x^13 - 5160*x^12 - 73240*x^11 + 54016*x^10 + 1888200*x^9 - 345306*x^8 - 42616648*x^7 - 94591600*x^6 + 214400888*x^5 + 1048954592*x^4 + 880219560*x^3 - 1527001640*x^2 - 3117641336*x - 1528687073)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 + 1160*x^13 - 5160*x^12 - 73240*x^11 + 54016*x^10 + 1888200*x^9 - 345306*x^8 - 42616648*x^7 - 94591600*x^6 + 214400888*x^5 + 1048954592*x^4 + 880219560*x^3 - 1527001640*x^2 - 3117641336*x - 1528687073, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} + 1160 x^{13} - 5160 x^{12} - 73240 x^{11} + 54016 x^{10} + 1888200 x^{9} - 345306 x^{8} - 42616648 x^{7} - 94591600 x^{6} + 214400888 x^{5} + 1048954592 x^{4} + 880219560 x^{3} - 1527001640 x^{2} - 3117641336 x - 1528687073 \)

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:  \(467063129754478147962797661051173208064=2^{62}\cdot 17^{6}\cdot 127^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.12$
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:  $2, 17, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{5}{17} a^{13} - \frac{7}{17} a^{12} + \frac{7}{17} a^{11} - \frac{2}{17} a^{10} + \frac{4}{17} a^{9} - \frac{2}{17} a^{8} - \frac{5}{17} a^{7} - \frac{4}{17} a^{6} + \frac{1}{17} a^{5} - \frac{1}{17} a^{4} - \frac{2}{17} a^{3} - \frac{3}{17} a^{2} + \frac{7}{17} a$, $\frac{1}{1113861745463290980157802078053932910607915498245067251001065585369} a^{15} + \frac{6757942887312534575762436110773448283923552513457350294409993546}{1113861745463290980157802078053932910607915498245067251001065585369} a^{14} - \frac{366658037834768799763426878542478948400977536762342095987933159269}{1113861745463290980157802078053932910607915498245067251001065585369} a^{13} - \frac{34970502988572585655399421185014206334184739323521910003740549025}{1113861745463290980157802078053932910607915498245067251001065585369} a^{12} - \frac{223967674992205311095920731236241310430194232867139597163433027936}{1113861745463290980157802078053932910607915498245067251001065585369} a^{11} - \frac{171499051687181031677900940870959006420152327782444472927824536482}{1113861745463290980157802078053932910607915498245067251001065585369} a^{10} - \frac{512113475838496305146709401187050304905322411023408531626455143716}{1113861745463290980157802078053932910607915498245067251001065585369} a^{9} + \frac{46514320643326363865994706171636108959172349464002295734729174318}{1113861745463290980157802078053932910607915498245067251001065585369} a^{8} - \frac{214516338403869051127400363832379950580011565655750275398536797934}{1113861745463290980157802078053932910607915498245067251001065585369} a^{7} + \frac{265102929153781964109951645255077936656925230752437454062041228601}{1113861745463290980157802078053932910607915498245067251001065585369} a^{6} + \frac{20767185531491814487242863270863493272828588238634453496785022933}{1113861745463290980157802078053932910607915498245067251001065585369} a^{5} - \frac{98993652700598999185635800784872703066471798592470588324441726674}{1113861745463290980157802078053932910607915498245067251001065585369} a^{4} + \frac{5606185514086957645630054546548521106824726825296994665094109484}{65521279144899469421047181061996053565171499896768661823592093257} a^{3} - \frac{273489374913835606549284338551757118819030539263352008550971422073}{1113861745463290980157802078053932910607915498245067251001065585369} a^{2} + \frac{106921643417706001151018175173398825284625649908807163758629132505}{1113861745463290980157802078053932910607915498245067251001065585369} a + \frac{26238882301463293737009441527261094065666293379842114812342008737}{65521279144899469421047181061996053565171499896768661823592093257}$

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:  $11$
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:  \( 28893134877000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.4421632.2, 4.4.2210816.1, 8.8.312813272694784.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$