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Properties

Label 16.16.1739116855...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 5^{14}\cdot 19^{8}$
Root discriminant $50.41$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3025, 0, -22800, 0, 55300, 0, -57850, 0, 29210, 0, -7380, 0, 905, 0, -50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 50*x^14 + 905*x^12 - 7380*x^10 + 29210*x^8 - 57850*x^6 + 55300*x^4 - 22800*x^2 + 3025)
 
gp: K = bnfinit(x^16 - 50*x^14 + 905*x^12 - 7380*x^10 + 29210*x^8 - 57850*x^6 + 55300*x^4 - 22800*x^2 + 3025, 1)
 

Normalized defining polynomial

\( x^{16} - 50 x^{14} + 905 x^{12} - 7380 x^{10} + 29210 x^{8} - 57850 x^{6} + 55300 x^{4} - 22800 x^{2} + 3025 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1739116855398400000000000000=2^{24}\cdot 5^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.41$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{380} a^{12} - \frac{1}{20} a^{11} + \frac{9}{380} a^{10} - \frac{1}{20} a^{9} - \frac{9}{380} a^{8} + \frac{31}{76} a^{6} - \frac{7}{76} a^{4} - \frac{1}{4} a^{3} + \frac{8}{19} a^{2} + \frac{35}{76}$, $\frac{1}{380} a^{13} - \frac{1}{38} a^{11} + \frac{1}{38} a^{9} - \frac{1}{20} a^{8} - \frac{7}{76} a^{7} - \frac{1}{4} a^{6} - \frac{7}{76} a^{5} + \frac{13}{76} a^{3} - \frac{3}{76} a - \frac{1}{4}$, $\frac{1}{555286780} a^{14} - \frac{175887}{555286780} a^{12} - \frac{1}{20} a^{11} - \frac{1282833}{111057356} a^{10} + \frac{10309479}{277643390} a^{8} + \frac{1}{4} a^{7} + \frac{11828015}{27764339} a^{6} - \frac{75016}{677179} a^{4} - \frac{1}{4} a^{3} - \frac{54571953}{111057356} a^{2} + \frac{1}{4} a - \frac{4989665}{111057356}$, $\frac{1}{30540772900} a^{15} - \frac{2373227}{6108154580} a^{13} - \frac{141655033}{3054077290} a^{11} - \frac{1}{20} a^{10} - \frac{280241491}{6108154580} a^{9} + \frac{32285949}{1527038645} a^{7} + \frac{1}{4} a^{6} + \frac{491099}{7448969} a^{5} + \frac{182025157}{610815458} a^{3} - \frac{1}{4} a^{2} - \frac{82829669}{1221630916} a + \frac{1}{4}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 465694862.702 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), 4.4.45125.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{5}, \sqrt{19})\), 8.8.2606420000000.1 x2, 8.8.115520000000.1 x2, 8.8.521284000000.1

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 8 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$5.8.7.2$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}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$