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Properties

Label 16.0.13701143706...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $27.93$
Ramified primes $2, 3, 5, 13$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59536, 0, -109540, 0, 63833, 0, -12936, 0, 1544, 0, -294, 0, 56, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 56*x^12 - 294*x^10 + 1544*x^8 - 12936*x^6 + 63833*x^4 - 109540*x^2 + 59536)
 
gp: K = bnfinit(x^16 - 4*x^14 + 56*x^12 - 294*x^10 + 1544*x^8 - 12936*x^6 + 63833*x^4 - 109540*x^2 + 59536, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 56 x^{12} - 294 x^{10} + 1544 x^{8} - 12936 x^{6} + 63833 x^{4} - 109540 x^{2} + 59536 \)

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:  \(137011437068313600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.93$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(389,·)$, $\chi_{780}(391,·)$, $\chi_{780}(521,·)$, $\chi_{780}(779,·)$, $\chi_{780}(79,·)$, $\chi_{780}(209,·)$, $\chi_{780}(259,·)$, $\chi_{780}(469,·)$, $\chi_{780}(599,·)$, $\chi_{780}(649,·)$, $\chi_{780}(181,·)$, $\chi_{780}(311,·)$, $\chi_{780}(571,·)$, $\chi_{780}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{12} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{24} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} + \frac{1}{6} a^{6} - \frac{7}{24} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{11} + \frac{1}{6} a^{7} - \frac{7}{24} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{120} a^{12} + \frac{1}{60} a^{10} + \frac{1}{30} a^{8} - \frac{23}{120} a^{6} - \frac{5}{12} a^{4} + \frac{7}{15} a^{2} - \frac{7}{15}$, $\frac{1}{7320} a^{13} + \frac{11}{610} a^{11} + \frac{37}{3660} a^{9} - \frac{261}{2440} a^{7} - \frac{25}{183} a^{5} - \frac{199}{1220} a^{3} - \frac{157}{915} a$, $\frac{1}{4092313856400} a^{14} - \frac{38331659}{14667791600} a^{12} - \frac{70318949857}{4092313856400} a^{10} + \frac{3126654101}{163692554256} a^{8} - \frac{654380324141}{4092313856400} a^{6} - \frac{44649111929}{4092313856400} a^{4} - \frac{9003093947}{341026154700} a^{2} - \frac{1028010832}{4192944525}$, $\frac{1}{4092313856400} a^{15} - \frac{778567}{44003374800} a^{13} - \frac{32302919497}{4092313856400} a^{11} + \frac{2327659879}{818462771280} a^{9} + \frac{577227247669}{4092313856400} a^{7} - \frac{1800095219729}{4092313856400} a^{5} - \frac{227746590689}{682052309400} a^{3} - \frac{2703811739}{511539232050} a$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( -\frac{3662947}{15800439600} a^{15} + \frac{99469}{169897200} a^{13} - \frac{192237371}{15800439600} a^{11} + \frac{31812145}{632017584} a^{9} - \frac{4525875223}{15800439600} a^{7} + \frac{40754362613}{15800439600} a^{5} - \frac{4834022097}{438901100} a^{3} + \frac{18885779213}{1975054950} a \) (order $12$)
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:  \( 115751.484499 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

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 $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{195}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{195})\), \(\Q(i, \sqrt{65})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{-65})\), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{-3}, \sqrt{-65})\), \(\Q(\sqrt{3}, \sqrt{65})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{39})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-13}, \sqrt{15})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{-5}, \sqrt{39})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{-13}, \sqrt{-15})\), \(\Q(\sqrt{-5}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{15}, \sqrt{-39})\), \(\Q(\sqrt{-15}, \sqrt{39})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{15}, \sqrt{39})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.370150560000.10, 8.0.370150560000.2, 8.0.370150560000.8, 8.0.370150560000.7, 8.0.4569760000.1, 8.0.12960000.1, 8.0.592240896.1, 8.0.370150560000.5, 8.0.370150560000.3, 8.0.370150560000.1, 8.0.1445900625.1, 8.0.370150560000.4, 8.0.370150560000.9, 8.8.370150560000.1, 8.0.370150560000.6

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$