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Properties

Label 16.0.93492089436...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 5^{4}$
Root discriminant $27.27$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73, -160, -44, 296, -112, -184, 208, -224, 280, -240, 196, -72, 76, -16, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 16*x^13 + 76*x^12 - 72*x^11 + 196*x^10 - 240*x^9 + 280*x^8 - 224*x^7 + 208*x^6 - 184*x^5 - 112*x^4 + 296*x^3 - 44*x^2 - 160*x + 73)
 
gp: K = bnfinit(x^16 + 16*x^14 - 16*x^13 + 76*x^12 - 72*x^11 + 196*x^10 - 240*x^9 + 280*x^8 - 224*x^7 + 208*x^6 - 184*x^5 - 112*x^4 + 296*x^3 - 44*x^2 - 160*x + 73, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 16 x^{13} + 76 x^{12} - 72 x^{11} + 196 x^{10} - 240 x^{9} + 280 x^{8} - 224 x^{7} + 208 x^{6} - 184 x^{5} - 112 x^{4} + 296 x^{3} - 44 x^{2} - 160 x + 73 \)

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:  \(93492089436304834560000=2^{48}\cdot 3^{12}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.27$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{109} a^{14} + \frac{47}{109} a^{13} + \frac{22}{109} a^{12} + \frac{46}{109} a^{11} - \frac{12}{109} a^{10} + \frac{50}{109} a^{9} - \frac{12}{109} a^{8} + \frac{8}{109} a^{7} - \frac{49}{109} a^{6} + \frac{14}{109} a^{5} + \frac{31}{109} a^{4} - \frac{30}{109} a^{3} + \frac{54}{109} a^{2} + \frac{36}{109} a - \frac{30}{109}$, $\frac{1}{75827915149393717} a^{15} - \frac{190054818910370}{75827915149393717} a^{14} - \frac{21076937215609444}{75827915149393717} a^{13} - \frac{14613316728266024}{75827915149393717} a^{12} - \frac{20366652895563719}{75827915149393717} a^{11} - \frac{30996198008481355}{75827915149393717} a^{10} - \frac{26263102837442164}{75827915149393717} a^{9} - \frac{27670767601694568}{75827915149393717} a^{8} - \frac{203641374439056}{418938757731457} a^{7} - \frac{4529568761172235}{75827915149393717} a^{6} - \frac{8143585322663507}{75827915149393717} a^{5} + \frac{35537634951366073}{75827915149393717} a^{4} - \frac{27586235968364247}{75827915149393717} a^{3} - \frac{22843714216343711}{75827915149393717} a^{2} + \frac{22689522093062017}{75827915149393717} a - \frac{15440004069830268}{75827915149393717}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28425.2702638 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.27648.1 x2, 4.0.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.3057647616.7, 8.0.8493465600.1, 8.8.19110297600.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$