Properties

Label 16.0.60516574977...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $30.65$
Ramified primes $2, 3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22801, 0, -15881, 0, 11675, 0, -3294, 0, 1259, 0, -114, 0, 50, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801)
 
gp: K = bnfinit(x^16 + 4*x^14 + 50*x^12 - 114*x^10 + 1259*x^8 - 3294*x^6 + 11675*x^4 - 15881*x^2 + 22801, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} + 50 x^{12} - 114 x^{10} + 1259 x^{8} - 3294 x^{6} + 11675 x^{4} - 15881 x^{2} + 22801 \)

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:  \(605165749776000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(323,·)$, $\chi_{420}(197,·)$, $\chi_{420}(391,·)$, $\chi_{420}(139,·)$, $\chi_{420}(83,·)$, $\chi_{420}(407,·)$, $\chi_{420}(349,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(169,·)$, $\chi_{420}(113,·)$, $\chi_{420}(211,·)$, $\chi_{420}(181,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{24573148562793} a^{14} - \frac{2552008684120}{24573148562793} a^{12} + \frac{114174152058}{264227403901} a^{10} - \frac{8115531350084}{24573148562793} a^{8} + \frac{1684474300064}{8191049520931} a^{6} - \frac{343598590231}{24573148562793} a^{4} - \frac{5135159655305}{24573148562793} a^{2} + \frac{1516452978428}{8191049520931}$, $\frac{1}{3710545432981743} a^{15} + \frac{407000467362430}{3710545432981743} a^{13} + \frac{40240860445225}{119695013967153} a^{11} + \frac{1466273382417496}{3710545432981743} a^{9} - \frac{838624677755701}{3710545432981743} a^{7} + \frac{1105448086735454}{3710545432981743} a^{5} + \frac{568238306809865}{3710545432981743} a^{3} - \frac{716262998906644}{3710545432981743} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( -\frac{437437}{3838986401} a^{15} + \frac{1584030}{3838986401} a^{13} + \frac{34600}{123838271} a^{11} + \frac{261130710}{3838986401} a^{9} - \frac{455030440}{3838986401} a^{7} + \frac{4579621900}{3838986401} a^{5} - \frac{7029315280}{3838986401} a^{3} + \frac{18471719510}{3838986401} a \) (order $4$)
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:  \( 67203.8653181 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{35})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, 8.0.384160000.1, 8.0.777924000000.8, 8.0.324000000.1, 8.8.777924000000.3, 8.0.777924000000.3, 8.0.777924000000.4, 8.0.3038765625.2

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

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$