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Properties

Label 16.0.15765467071...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 71^{8}$
Root discriminant $28.17$
Ramified primes $5, 71$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![160000, 0, -24000, 0, -4400, 0, 1860, 0, -59, 0, 93, 0, -11, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000)
 
gp: K = bnfinit(x^16 - 3*x^14 - 11*x^12 + 93*x^10 - 59*x^8 + 1860*x^6 - 4400*x^4 - 24000*x^2 + 160000, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{14} - 11 x^{12} + 93 x^{10} - 59 x^{8} + 1860 x^{6} - 4400 x^{4} - 24000 x^{2} + 160000 \)

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:  \(157654670714297119140625=5^{12}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.17$
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:  $5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{1180} a^{10} + \frac{3}{20} a^{8} + \frac{1}{20} a^{6} + \frac{7}{20} a^{4} - \frac{1}{20} a^{2} - \frac{1}{2} a - \frac{25}{59}$, $\frac{1}{2360} a^{11} + \frac{3}{40} a^{9} - \frac{9}{40} a^{7} + \frac{7}{40} a^{5} - \frac{1}{2} a^{4} - \frac{1}{40} a^{3} - \frac{25}{118} a$, $\frac{1}{47200} a^{12} - \frac{1}{4720} a^{11} - \frac{3}{47200} a^{10} + \frac{17}{80} a^{9} - \frac{129}{800} a^{8} + \frac{9}{80} a^{7} - \frac{73}{800} a^{6} - \frac{7}{80} a^{5} + \frac{399}{800} a^{4} - \frac{39}{80} a^{3} - \frac{1087}{2360} a^{2} + \frac{25}{236} a - \frac{11}{118}$, $\frac{1}{472000} a^{13} + \frac{17}{472000} a^{11} - \frac{869}{8000} a^{9} - \frac{653}{8000} a^{7} - \frac{661}{8000} a^{5} + \frac{2377}{11800} a^{3} + \frac{277}{590} a - \frac{1}{2}$, $\frac{1}{944000} a^{14} - \frac{3}{944000} a^{12} - \frac{1}{4720} a^{11} - \frac{11}{944000} a^{10} - \frac{3}{80} a^{9} + \frac{3527}{16000} a^{8} - \frac{11}{80} a^{7} - \frac{1}{16000} a^{6} + \frac{13}{80} a^{5} + \frac{93}{47200} a^{4} + \frac{21}{80} a^{3} - \frac{11}{2360} a^{2} - \frac{17}{118} a - \frac{3}{118}$, $\frac{1}{9440000} a^{15} - \frac{3}{9440000} a^{13} + \frac{389}{9440000} a^{11} - \frac{1}{2360} a^{10} - \frac{3273}{160000} a^{9} - \frac{3}{40} a^{8} + \frac{12399}{160000} a^{7} + \frac{9}{40} a^{6} + \frac{55553}{472000} a^{5} + \frac{13}{40} a^{4} - \frac{951}{2360} a^{3} - \frac{19}{40} a^{2} - \frac{66}{295} a + \frac{25}{118}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( -\frac{93}{472000} a^{14} + \frac{279}{472000} a^{12} + \frac{1023}{472000} a^{10} - \frac{11}{8000} a^{8} + \frac{93}{8000} a^{6} - \frac{8649}{23600} a^{4} + \frac{1023}{1180} a^{2} + \frac{279}{59} \) (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:  \( 46337.4951728 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-71}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-355}) \), \(\Q(\sqrt{5}, \sqrt{-71})\), 4.2.8875.1 x2, 4.0.630125.1 x2, 4.0.25205.1 x2, 4.2.1775.1 x2, 4.4.630125.1, \(\Q(\zeta_{5})\), 8.0.397057515625.1, 8.0.15882300625.1, 8.0.397057515625.2, 8.4.397057515625.1 x2, 8.0.78765625.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$71$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$