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Properties

Label 16.16.1009760059...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{14}\cdot 41^{6}$
Root discriminant $75.03$
Ramified primes $2, 3, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times Q_8).C_2^3$ (as 16T226)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3404025, 0, -45082575, 0, 31353750, 0, -8710200, 0, 1245555, 0, -99540, 0, 4470, 0, -105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 105*x^14 + 4470*x^12 - 99540*x^10 + 1245555*x^8 - 8710200*x^6 + 31353750*x^4 - 45082575*x^2 + 3404025)
 
gp: K = bnfinit(x^16 - 105*x^14 + 4470*x^12 - 99540*x^10 + 1245555*x^8 - 8710200*x^6 + 31353750*x^4 - 45082575*x^2 + 3404025, 1)
 

Normalized defining polynomial

\( x^{16} - 105 x^{14} + 4470 x^{12} - 99540 x^{10} + 1245555 x^{8} - 8710200 x^{6} + 31353750 x^{4} - 45082575 x^{2} + 3404025 \)

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:  \(1009760059176512400000000000000=2^{16}\cdot 3^{12}\cdot 5^{14}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.03$
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:  $2, 3, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{45} a^{8}$, $\frac{1}{45} a^{9}$, $\frac{1}{135} a^{10} - \frac{1}{9} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{135} a^{11} - \frac{1}{9} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{270} a^{12} - \frac{1}{90} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2}$, $\frac{1}{270} a^{13} - \frac{1}{90} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a$, $\frac{1}{6116377469558310} a^{14} - \frac{1917515330389}{2038792489852770} a^{12} - \frac{632293821323}{226532498872530} a^{10} - \frac{3055321258531}{339798748308795} a^{8} - \frac{2328847002767}{67959749661759} a^{6} + \frac{5853565760327}{45306499774506} a^{4} + \frac{10011167843681}{45306499774506} a^{2} + \frac{131236346735}{368345526622}$, $\frac{1}{6116377469558310} a^{15} - \frac{1917515330389}{2038792489852770} a^{13} - \frac{632293821323}{226532498872530} a^{11} - \frac{3055321258531}{339798748308795} a^{9} - \frac{2328847002767}{67959749661759} a^{7} + \frac{5853565760327}{45306499774506} a^{5} + \frac{10011167843681}{45306499774506} a^{3} + \frac{131236346735}{368345526622} a$

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:  $15$
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:  \( 3271776077.15 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times Q_8).C_2^3$ (as 16T226):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times Q_8).C_2^3$
Character table for $(C_2\times Q_8).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.9225.1, \(\Q(\zeta_{15})^+\), 4.4.5125.1, 8.8.2127515625.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$