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Properties

Label 16.0.11019960576...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 3^{16}\cdot 5^{12}$
Root discriminant $23.86$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $A_4:C_4$ (as 16T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, 875, 0, 725, 0, 490, 0, 266, 0, 98, 0, 29, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 29*x^12 + 98*x^10 + 266*x^8 + 490*x^6 + 725*x^4 + 875*x^2 + 625)
 
gp: K = bnfinit(x^16 + 7*x^14 + 29*x^12 + 98*x^10 + 266*x^8 + 490*x^6 + 725*x^4 + 875*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{16} + 7 x^{14} + 29 x^{12} + 98 x^{10} + 266 x^{8} + 490 x^{6} + 725 x^{4} + 875 x^{2} + 625 \)

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:  \(11019960576000000000000=2^{20}\cdot 3^{16}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.86$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{4} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{150} a^{12} + \frac{1}{75} a^{10} - \frac{31}{150} a^{8} - \frac{1}{2} a^{7} + \frac{1}{50} a^{6} - \frac{37}{75} a^{4} + \frac{7}{30} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{750} a^{13} - \frac{13}{750} a^{11} - \frac{61}{750} a^{9} + \frac{81}{250} a^{7} - \frac{119}{750} a^{5} - \frac{11}{150} a^{3} + \frac{7}{15} a$, $\frac{1}{43500} a^{14} + \frac{28}{10875} a^{12} + \frac{63}{14500} a^{10} - \frac{3257}{43500} a^{8} - \frac{1}{2} a^{7} - \frac{3869}{43500} a^{6} - \frac{1111}{8700} a^{4} + \frac{99}{290} a^{2} - \frac{1}{2} a + \frac{11}{348}$, $\frac{1}{217500} a^{15} + \frac{28}{54375} a^{13} + \frac{63}{72500} a^{11} - \frac{46757}{217500} a^{9} - \frac{69119}{217500} a^{7} - \frac{1}{2} a^{6} + \frac{7589}{43500} a^{5} - \frac{481}{1450} a^{3} + \frac{37}{348} a - \frac{1}{2}$

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

Class group and class number

$C_{4}$, 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:  \( 7336.89054062 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 16T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.40500.1, \(\Q(\zeta_{20})^+\), 8.0.1640250000.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$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}$