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Properties

Label 16.8.67445192788...6304.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{58}\cdot 3^{4}\cdot 7^{6}\cdot 1567^{2}$
Root discriminant $84.49$
Ramified primes $2, 3, 7, 1567$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11812348, 40496544, 4498080, -25222144, 14305216, -10653152, 2223088, 1362160, -1039068, 425872, -103232, 16720, -1824, 32, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 + 32*x^13 - 1824*x^12 + 16720*x^11 - 103232*x^10 + 425872*x^9 - 1039068*x^8 + 1362160*x^7 + 2223088*x^6 - 10653152*x^5 + 14305216*x^4 - 25222144*x^3 + 4498080*x^2 + 40496544*x + 11812348)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 + 32*x^13 - 1824*x^12 + 16720*x^11 - 103232*x^10 + 425872*x^9 - 1039068*x^8 + 1362160*x^7 + 2223088*x^6 - 10653152*x^5 + 14305216*x^4 - 25222144*x^3 + 4498080*x^2 + 40496544*x + 11812348, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} + 32 x^{13} - 1824 x^{12} + 16720 x^{11} - 103232 x^{10} + 425872 x^{9} - 1039068 x^{8} + 1362160 x^{7} + 2223088 x^{6} - 10653152 x^{5} + 14305216 x^{4} - 25222144 x^{3} + 4498080 x^{2} + 40496544 x + 11812348 \)

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:  \(6744519278804597787590135906304=2^{58}\cdot 3^{4}\cdot 7^{6}\cdot 1567^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.49$
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:  $2, 3, 7, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{68} a^{12} + \frac{2}{17} a^{11} - \frac{1}{34} a^{10} + \frac{2}{17} a^{9} - \frac{7}{34} a^{8} - \frac{7}{17} a^{7} + \frac{4}{17} a^{6} + \frac{5}{17} a^{5} + \frac{15}{34} a^{4} + \frac{4}{17} a^{3} + \frac{3}{17} a^{2} + \frac{2}{17} a$, $\frac{1}{68} a^{13} + \frac{1}{34} a^{11} + \frac{7}{68} a^{10} + \frac{7}{68} a^{9} + \frac{4}{17} a^{8} + \frac{1}{34} a^{7} - \frac{3}{34} a^{6} - \frac{7}{17} a^{5} - \frac{5}{17} a^{4} + \frac{5}{17} a^{3} + \frac{7}{34} a^{2} - \frac{15}{34} a$, $\frac{1}{2652} a^{14} + \frac{5}{884} a^{13} + \frac{1}{2652} a^{12} - \frac{5}{2652} a^{11} + \frac{89}{884} a^{10} - \frac{71}{1326} a^{9} - \frac{157}{663} a^{8} - \frac{49}{102} a^{7} + \frac{94}{663} a^{6} + \frac{161}{1326} a^{5} + \frac{83}{1326} a^{4} + \frac{659}{1326} a^{3} + \frac{47}{102} a^{2} + \frac{200}{663} a - \frac{2}{39}$, $\frac{1}{722944899763920895425013406689362655995413979450875292} a^{15} + \frac{21235963567649468887563680746306845305847398160653}{361472449881960447712506703344681327997706989725437646} a^{14} + \frac{3873774303851518106858254838361066685414342960704607}{722944899763920895425013406689362655995413979450875292} a^{13} - \frac{1016568984685483808158018050860575251292279398300809}{361472449881960447712506703344681327997706989725437646} a^{12} + \frac{37162786806696301260455836534604504921884499411333275}{722944899763920895425013406689362655995413979450875292} a^{11} + \frac{18312007991200564980425359520133710030418346934009875}{361472449881960447712506703344681327997706989725437646} a^{10} + \frac{57464373809401145490356947027410338230859103911409913}{722944899763920895425013406689362655995413979450875292} a^{9} - \frac{26098373875785672126541173321710254703425429328726573}{120490816627320149237502234448227109332568996575145882} a^{8} + \frac{140385566290585678883182718515790092953253831860938971}{361472449881960447712506703344681327997706989725437646} a^{7} + \frac{2259088558466180416026489846013353070273195068044527}{10631542643587071991544314804255333176403146756630519} a^{6} - \frac{83034392211674897426346051598515747839184964637704954}{180736224940980223856253351672340663998853494862718823} a^{5} + \frac{51431486342051108727390201125742305674584845165087741}{180736224940980223856253351672340663998853494862718823} a^{4} - \frac{79551209482437413856123502299523498832221428437641919}{361472449881960447712506703344681327997706989725437646} a^{3} - \frac{8624815048625153208950761149356736432596886062271336}{60245408313660074618751117224113554666284498287572941} a^{2} + \frac{67999441071509011537843975613041847535649960103232737}{361472449881960447712506703344681327997706989725437646} a + \frac{2007540397060132975148261423431991512282306804069501}{10631542643587071991544314804255333176403146756630519}$

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:  \( 2471383441.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.14336.1, 4.4.7168.1, 8.8.3288334336.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1567Data not computed