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Properties

Label 16.16.7700400667...1424.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{56}\cdot 7^{4}\cdot 17^{4}\cdot 73^{2}$
Root discriminant $63.89$
Ramified primes $2, 7, 17, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-44927, 1107856, -441080, -3218552, 749036, 2427744, -264112, -747688, 56250, 117840, -8888, -10104, 900, 448, -48, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 48*x^14 + 448*x^13 + 900*x^12 - 10104*x^11 - 8888*x^10 + 117840*x^9 + 56250*x^8 - 747688*x^7 - 264112*x^6 + 2427744*x^5 + 749036*x^4 - 3218552*x^3 - 441080*x^2 + 1107856*x - 44927)
 
gp: K = bnfinit(x^16 - 8*x^15 - 48*x^14 + 448*x^13 + 900*x^12 - 10104*x^11 - 8888*x^10 + 117840*x^9 + 56250*x^8 - 747688*x^7 - 264112*x^6 + 2427744*x^5 + 749036*x^4 - 3218552*x^3 - 441080*x^2 + 1107856*x - 44927, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 48 x^{14} + 448 x^{13} + 900 x^{12} - 10104 x^{11} - 8888 x^{10} + 117840 x^{9} + 56250 x^{8} - 747688 x^{7} - 264112 x^{6} + 2427744 x^{5} + 749036 x^{4} - 3218552 x^{3} - 441080 x^{2} + 1107856 x - 44927 \)

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:  \(77004006676572437563964391424=2^{56}\cdot 7^{4}\cdot 17^{4}\cdot 73^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.89$
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:  $2, 7, 17, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{10} + \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{12}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{12} a^{5} - \frac{1}{6} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{12} a$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{11} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{12} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{24317261250385525621428} a^{15} - \frac{86477997364213419583}{24317261250385525621428} a^{14} - \frac{165426420383499971753}{12158630625192762810714} a^{13} - \frac{291736439606713712927}{12158630625192762810714} a^{12} - \frac{157470550461068948345}{24317261250385525621428} a^{11} - \frac{180287443980105221133}{4052876875064254270238} a^{10} - \frac{1245833703841916442091}{24317261250385525621428} a^{9} + \frac{555598118227792989317}{4052876875064254270238} a^{8} + \frac{7253167735527056367511}{24317261250385525621428} a^{7} + \frac{6385886464891458410327}{24317261250385525621428} a^{6} + \frac{3191756572028790247}{6079315312596381405357} a^{5} + \frac{1192729083274408341845}{6079315312596381405357} a^{4} + \frac{2053881512375677783897}{24317261250385525621428} a^{3} - \frac{2020188640149766045526}{6079315312596381405357} a^{2} + \frac{10070949085277264519819}{24317261250385525621428} a - \frac{1588125330858913755173}{6079315312596381405357}$

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:  \( 2837611194.44 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.7168.1, 4.4.14336.1, 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$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$$[\ ]_{2}^{4}$
$17$$\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$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$