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Properties

Label 16.0.30400667142...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 5^{14}\cdot 41^{6}$
Root discriminant $39.15$
Ramified primes $2, 5, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![672400, 0, -672400, 0, 344400, 0, -112750, 0, 25535, 0, -4055, 0, 440, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 30*x^14 + 440*x^12 - 4055*x^10 + 25535*x^8 - 112750*x^6 + 344400*x^4 - 672400*x^2 + 672400)
 
gp: K = bnfinit(x^16 - 30*x^14 + 440*x^12 - 4055*x^10 + 25535*x^8 - 112750*x^6 + 344400*x^4 - 672400*x^2 + 672400, 1)
 

Normalized defining polynomial

\( x^{16} - 30 x^{14} + 440 x^{12} - 4055 x^{10} + 25535 x^{8} - 112750 x^{6} + 344400 x^{4} - 672400 x^{2} + 672400 \)

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:  \(30400667142400000000000000=2^{20}\cdot 5^{14}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.15$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2460} a^{12} + \frac{13}{615} a^{10} + \frac{28}{615} a^{8} + \frac{3}{164} a^{6} + \frac{35}{164} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{2460} a^{13} + \frac{13}{615} a^{11} + \frac{28}{615} a^{9} + \frac{3}{164} a^{7} + \frac{35}{164} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{13685496600} a^{14} + \frac{8173}{684274830} a^{12} - \frac{31789547}{684274830} a^{10} - \frac{159686899}{2737099320} a^{8} - \frac{351995433}{912366440} a^{6} - \frac{14007482}{68427483} a^{4} + \frac{417071}{3337926} a^{2} + \frac{448970}{1668963}$, $\frac{1}{13685496600} a^{15} + \frac{8173}{684274830} a^{13} - \frac{31789547}{684274830} a^{11} - \frac{159686899}{2737099320} a^{9} - \frac{351995433}{912366440} a^{7} - \frac{14007482}{68427483} a^{5} + \frac{417071}{3337926} a^{3} + \frac{448970}{1668963} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{150227}{1368549660} a^{14} - \frac{3797767}{1368549660} a^{12} + \frac{11693633}{342137415} a^{10} - \frac{118563291}{456183220} a^{8} + \frac{30165399}{22809161} a^{6} - \frac{1243923115}{273709932} a^{4} + \frac{16721053}{1668963} a^{2} - \frac{5921105}{556321} \) (order $10$)
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:  \( 2094284.04167 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

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 $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.5125.1, 4.0.1025.1, 8.0.26265625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.12.8$x^{8} + 2 x^{6} + 80$$2$$4$$12$$(C_8:C_2):C_2$$[2, 2, 3]^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{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}$