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Properties

Label 16.8.24278623839...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 29^{8}\cdot 89^{6}$
Root discriminant $79.26$
Ramified primes $5, 29, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4828349, 9493947, 6569944, 3229665, 2548935, 1234391, -207505, -532478, -260184, -36704, 21331, 7402, -73, -206, -33, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 33*x^14 - 206*x^13 - 73*x^12 + 7402*x^11 + 21331*x^10 - 36704*x^9 - 260184*x^8 - 532478*x^7 - 207505*x^6 + 1234391*x^5 + 2548935*x^4 + 3229665*x^3 + 6569944*x^2 + 9493947*x + 4828349)
 
gp: K = bnfinit(x^16 - x^15 - 33*x^14 - 206*x^13 - 73*x^12 + 7402*x^11 + 21331*x^10 - 36704*x^9 - 260184*x^8 - 532478*x^7 - 207505*x^6 + 1234391*x^5 + 2548935*x^4 + 3229665*x^3 + 6569944*x^2 + 9493947*x + 4828349, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 33 x^{14} - 206 x^{13} - 73 x^{12} + 7402 x^{11} + 21331 x^{10} - 36704 x^{9} - 260184 x^{8} - 532478 x^{7} - 207505 x^{6} + 1234391 x^{5} + 2548935 x^{4} + 3229665 x^{3} + 6569944 x^{2} + 9493947 x + 4828349 \)

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:  \(2427862383905930688921103515625=5^{10}\cdot 29^{8}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.26$
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:  $5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{6} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{5}{12} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{495825730076604427806223240607145509308247401498596} a^{15} - \frac{64569437889950744862328579516781321898275657333}{2739368674456378054178028953630638172973742549716} a^{14} - \frac{3791162811407201540452863525470520010134421665103}{41318810839717035650518603383928792442353950124883} a^{13} + \frac{1248230856682952496659113530516299607864435631408}{11268766592650100631959619104707852484278350034059} a^{12} - \frac{5353047243172954951611629606721695287472353473049}{165275243358868142602074413535715169769415800499532} a^{11} - \frac{718416113458919323979739603139313483772275285155}{165275243358868142602074413535715169769415800499532} a^{10} - \frac{77994967383265612900468579376063636736778357179665}{495825730076604427806223240607145509308247401498596} a^{9} + \frac{19571010362636395407903594096531626788358402397713}{82637621679434071301037206767857584884707900249766} a^{8} - \frac{62075659032198824194484151674473607352239857579273}{247912865038302213903111620303572754654123700749298} a^{7} + \frac{80116305379830711032543542312021190698554095602875}{165275243358868142602074413535715169769415800499532} a^{6} + \frac{62414495152193613939131612251537948467227141575391}{165275243358868142602074413535715169769415800499532} a^{5} - \frac{172281220882258994603993101271022504930217434961333}{495825730076604427806223240607145509308247401498596} a^{4} - \frac{19953456218889561411563491924308624501690374667260}{123956432519151106951555810151786377327061850374649} a^{3} + \frac{58697402077585497010850975538692848550476791060085}{123956432519151106951555810151786377327061850374649} a^{2} + \frac{48038339934722092275652444277707158652791162357763}{495825730076604427806223240607145509308247401498596} a + \frac{6820288072085467483306853156631472228668468955235}{82637621679434071301037206767857584884707900249766}$

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:  \( 981160292.352 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.2, 4.4.725.1, 4.4.2225.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$