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Properties

Label 16.4.77740662152...1872.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 17^{15}\cdot 103^{2}$
Root discriminant $35.95$
Ramified primes $2, 17, 103$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8093, 13797, -12242, -8605, 3268, -3564, 3171, -2392, 2158, -1268, 614, -269, 132, -57, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 25*x^14 - 57*x^13 + 132*x^12 - 269*x^11 + 614*x^10 - 1268*x^9 + 2158*x^8 - 2392*x^7 + 3171*x^6 - 3564*x^5 + 3268*x^4 - 8605*x^3 - 12242*x^2 + 13797*x + 8093)
 
gp: K = bnfinit(x^16 - 5*x^15 + 25*x^14 - 57*x^13 + 132*x^12 - 269*x^11 + 614*x^10 - 1268*x^9 + 2158*x^8 - 2392*x^7 + 3171*x^6 - 3564*x^5 + 3268*x^4 - 8605*x^3 - 12242*x^2 + 13797*x + 8093, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 25 x^{14} - 57 x^{13} + 132 x^{12} - 269 x^{11} + 614 x^{10} - 1268 x^{9} + 2158 x^{8} - 2392 x^{7} + 3171 x^{6} - 3564 x^{5} + 3268 x^{4} - 8605 x^{3} - 12242 x^{2} + 13797 x + 8093 \)

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:  \(7774066215287714751471872=2^{8}\cdot 17^{15}\cdot 103^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.95$
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:  $2, 17, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5767021202212432089229837874173843} a^{15} + \frac{1829611326532379133824863128859194}{5767021202212432089229837874173843} a^{14} + \frac{215821188304397958323435330051949}{5767021202212432089229837874173843} a^{13} + \frac{936193944950549762928642409698726}{5767021202212432089229837874173843} a^{12} + \frac{2444964016869638713843932447204136}{5767021202212432089229837874173843} a^{11} - \frac{1905129910716344848381427491348461}{5767021202212432089229837874173843} a^{10} + \frac{2661517904864468903054820990552954}{5767021202212432089229837874173843} a^{9} + \frac{411887177357486039223025504558326}{5767021202212432089229837874173843} a^{8} - \frac{702870423102922667548590268926541}{5767021202212432089229837874173843} a^{7} + \frac{2251906666125323267746145712976709}{5767021202212432089229837874173843} a^{6} - \frac{2625452562625474981766102986035494}{5767021202212432089229837874173843} a^{5} + \frac{1806673015862624223312255567294170}{5767021202212432089229837874173843} a^{4} - \frac{2423958009601857114043796352696577}{5767021202212432089229837874173843} a^{3} + \frac{11074291178567332271036684957134}{5767021202212432089229837874173843} a^{2} + \frac{17975515979257966315403670928929}{5767021202212432089229837874173843} a - \frac{1188909344238333415119939481489852}{5767021202212432089229837874173843}$

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:  $9$
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:  \( 442141.544744 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$