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Properties

Label 16.16.4179504042...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 31^{2}$
Root discriminant $46.11$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, 700, -160, -8540, 2729, 22714, -8627, -22916, 9606, 9380, -4106, -1428, 652, 90, -43, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 43*x^14 + 90*x^13 + 652*x^12 - 1428*x^11 - 4106*x^10 + 9380*x^9 + 9606*x^8 - 22916*x^7 - 8627*x^6 + 22714*x^5 + 2729*x^4 - 8540*x^3 - 160*x^2 + 700*x - 25)
 
gp: K = bnfinit(x^16 - 2*x^15 - 43*x^14 + 90*x^13 + 652*x^12 - 1428*x^11 - 4106*x^10 + 9380*x^9 + 9606*x^8 - 22916*x^7 - 8627*x^6 + 22714*x^5 + 2729*x^4 - 8540*x^3 - 160*x^2 + 700*x - 25, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 43 x^{14} + 90 x^{13} + 652 x^{12} - 1428 x^{11} - 4106 x^{10} + 9380 x^{9} + 9606 x^{8} - 22916 x^{7} - 8627 x^{6} + 22714 x^{5} + 2729 x^{4} - 8540 x^{3} - 160 x^{2} + 700 x - 25 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(417950404226786329600000000=2^{16}\cdot 5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.11$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{575} a^{14} + \frac{28}{575} a^{13} - \frac{38}{575} a^{12} + \frac{13}{115} a^{11} + \frac{62}{575} a^{10} - \frac{1}{25} a^{9} + \frac{129}{575} a^{8} + \frac{28}{115} a^{7} - \frac{8}{25} a^{6} - \frac{206}{575} a^{5} - \frac{87}{575} a^{4} + \frac{64}{575} a^{3} - \frac{101}{575} a^{2} + \frac{16}{115} a + \frac{19}{115}$, $\frac{1}{25662409787563430075} a^{15} - \frac{11802979228100073}{25662409787563430075} a^{14} - \frac{344403470802987221}{25662409787563430075} a^{13} + \frac{2211409118145703708}{25662409787563430075} a^{12} + \frac{6004101347551396467}{25662409787563430075} a^{11} - \frac{57833697310541343}{5132481957512686015} a^{10} + \frac{2188733761789198262}{25662409787563430075} a^{9} + \frac{5510713216516739061}{25662409787563430075} a^{8} - \frac{10173399006088731519}{25662409787563430075} a^{7} - \frac{9728059561108036842}{25662409787563430075} a^{6} - \frac{1532802830920416056}{25662409787563430075} a^{5} + \frac{10169653475909640306}{25662409787563430075} a^{4} - \frac{7501986029068616}{1026496391502537203} a^{3} + \frac{1049625248484321851}{25662409787563430075} a^{2} - \frac{150372858165734972}{5132481957512686015} a + \frac{1540670296906061446}{5132481957512686015}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 105723118.854 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.4171360000.1, 8.8.2576088125.1, 8.8.20443835360000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$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}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$