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Properties

Label 16.0.12452113261...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{10}\cdot 41^{6}$
Root discriminant $37.02$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42025, 0, 0, 0, 0, 0, 1640, 0, 2294, 0, 416, 0, 32, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 32*x^12 + 416*x^10 + 2294*x^8 + 1640*x^6 + 42025)
 
gp: K = bnfinit(x^16 - 8*x^14 + 32*x^12 + 416*x^10 + 2294*x^8 + 1640*x^6 + 42025, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 32 x^{12} + 416 x^{10} + 2294 x^{8} + 1640 x^{6} + 42025 \)

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:  \(12452113261527040000000000=2^{28}\cdot 5^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.02$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{40} a^{8} - \frac{1}{10} a^{6} + \frac{1}{5} a^{4} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{40} a^{9} - \frac{1}{10} a^{7} + \frac{1}{5} a^{5} - \frac{1}{2} a^{3} + \frac{3}{8} a$, $\frac{1}{80} a^{10} - \frac{1}{80} a^{8} - \frac{1}{20} a^{6} + \frac{1}{20} a^{4} + \frac{7}{16} a^{2} - \frac{7}{16}$, $\frac{1}{80} a^{11} - \frac{1}{80} a^{9} - \frac{1}{20} a^{7} + \frac{1}{20} a^{5} + \frac{7}{16} a^{3} - \frac{7}{16} a$, $\frac{1}{82000} a^{12} - \frac{49}{82000} a^{10} + \frac{43}{5125} a^{8} + \frac{117}{2050} a^{6} + \frac{163}{656} a^{4} + \frac{1}{80} a^{2} + \frac{2}{5}$, $\frac{1}{164000} a^{13} - \frac{1}{164000} a^{12} + \frac{61}{10250} a^{11} - \frac{61}{10250} a^{10} + \frac{1713}{164000} a^{9} - \frac{1713}{164000} a^{8} + \frac{161}{2050} a^{7} - \frac{161}{2050} a^{6} + \frac{163}{1312} a^{5} - \frac{163}{1312} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{7}{160} a - \frac{7}{160}$, $\frac{1}{761452000} a^{14} - \frac{21}{761452000} a^{12} + \frac{473891}{761452000} a^{10} - \frac{19791}{18572000} a^{8} + \frac{648563}{152290400} a^{6} + \frac{1686477}{30458080} a^{4} - \frac{51639}{148576} a^{2} + \frac{162151}{742880}$, $\frac{1}{1522904000} a^{15} - \frac{1}{1522904000} a^{14} - \frac{21}{1522904000} a^{13} + \frac{21}{1522904000} a^{12} - \frac{9044259}{1522904000} a^{11} + \frac{9044259}{1522904000} a^{10} + \frac{212359}{37144000} a^{9} - \frac{212359}{37144000} a^{8} + \frac{8263083}{304580800} a^{7} - \frac{8263083}{304580800} a^{6} - \frac{15065467}{60916160} a^{5} + \frac{15065467}{60916160} a^{4} - \frac{116641}{297152} a^{3} + \frac{116641}{297152} a^{2} - \frac{627159}{1485760} a + \frac{627159}{1485760}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{221}{30458080} a^{14} - \frac{4641}{30458080} a^{12} + \frac{30261}{30458080} a^{10} - \frac{21}{148576} a^{8} - \frac{625669}{30458080} a^{6} - \frac{6955767}{30458080} a^{4} - \frac{11697}{148576} a^{2} + \frac{3485}{148576} \) (order $4$)
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:  \( 2233253.35718 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.1025.1, 4.4.16400.1, \(\Q(i, \sqrt{5})\), 8.0.268960000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.16.30$x^{8} + 8 x^{7} + 20$$4$$2$$16$$C_2^3: C_4$$[2, 2, 3]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$