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Properties

Label 16.0.75853444258...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $47.86$
Ramified primes $2, 5, 41$
Class number $736$ (GRH)
Class group $[2, 2, 2, 92]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![212321, -294108, 344084, -318432, 270644, -170828, 109376, -50784, 28019, -9716, 5176, -840, 632, -24, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 - 24*x^13 + 632*x^12 - 840*x^11 + 5176*x^10 - 9716*x^9 + 28019*x^8 - 50784*x^7 + 109376*x^6 - 170828*x^5 + 270644*x^4 - 318432*x^3 + 344084*x^2 - 294108*x + 212321)
 
gp: K = bnfinit(x^16 + 40*x^14 - 24*x^13 + 632*x^12 - 840*x^11 + 5176*x^10 - 9716*x^9 + 28019*x^8 - 50784*x^7 + 109376*x^6 - 170828*x^5 + 270644*x^4 - 318432*x^3 + 344084*x^2 - 294108*x + 212321, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} - 24 x^{13} + 632 x^{12} - 840 x^{11} + 5176 x^{10} - 9716 x^{9} + 28019 x^{8} - 50784 x^{7} + 109376 x^{6} - 170828 x^{5} + 270644 x^{4} - 318432 x^{3} + 344084 x^{2} - 294108 x + 212321 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(758534442582016000000000000=2^{40}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.86$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{205} a^{14} - \frac{4}{41} a^{13} + \frac{12}{205} a^{12} + \frac{14}{205} a^{11} - \frac{28}{205} a^{10} + \frac{1}{205} a^{9} + \frac{43}{205} a^{8} - \frac{57}{205} a^{7} - \frac{28}{205} a^{6} - \frac{16}{41} a^{5} - \frac{8}{41} a^{4} - \frac{37}{205} a^{3} - \frac{54}{205} a^{2} - \frac{2}{205} a - \frac{42}{205}$, $\frac{1}{2752262742865537974498798490461721645} a^{15} - \frac{831540337129481978983413418575424}{550452548573107594899759698092344329} a^{14} + \frac{65002843185534472226056327622817812}{2752262742865537974498798490461721645} a^{13} - \frac{221448881864030671411372528137989169}{2752262742865537974498798490461721645} a^{12} + \frac{1103748067155543855738126221984624864}{2752262742865537974498798490461721645} a^{11} - \frac{1318287725645401267495653275893275304}{2752262742865537974498798490461721645} a^{10} - \frac{85527074319292702670263357667345159}{550452548573107594899759698092344329} a^{9} - \frac{1013647899721259768286903319815990536}{2752262742865537974498798490461721645} a^{8} - \frac{58537512352238385810988782508605766}{2752262742865537974498798490461721645} a^{7} + \frac{158837087843133051000047709794615930}{550452548573107594899759698092344329} a^{6} - \frac{88925935664480514934004423004907017}{550452548573107594899759698092344329} a^{5} - \frac{1109410297500854309312069111453782763}{2752262742865537974498798490461721645} a^{4} + \frac{15810816600551356899715036937896124}{550452548573107594899759698092344329} a^{3} - \frac{57003540032134913197539901178602863}{550452548573107594899759698092344329} a^{2} - \frac{1374705959298936407405531507849892961}{2752262742865537974498798490461721645} a - \frac{862585372007064018619597641303023}{11420177356288539313273022781998845}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{92}$, which has order $736$ (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:  $7$
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:  \( 7114.13535725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed