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Properties

Label 16.0.36778727586...000.82
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $167.05$
Ramified primes $2, 3, 5, 13$
Class number $13312000$ (GRH)
Class group $[2, 8, 40, 40, 520]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1411816863601, 0, 369389779152, 0, 49972763916, 0, 3842403336, 0, 168912659, 0, 4194048, 0, 56904, 0, 384, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 384*x^14 + 56904*x^12 + 4194048*x^10 + 168912659*x^8 + 3842403336*x^6 + 49972763916*x^4 + 369389779152*x^2 + 1411816863601)
 
gp: K = bnfinit(x^16 + 384*x^14 + 56904*x^12 + 4194048*x^10 + 168912659*x^8 + 3842403336*x^6 + 49972763916*x^4 + 369389779152*x^2 + 1411816863601, 1)
 

Normalized defining polynomial

\( x^{16} + 384 x^{14} + 56904 x^{12} + 4194048 x^{10} + 168912659 x^{8} + 3842403336 x^{6} + 49972763916 x^{4} + 369389779152 x^{2} + 1411816863601 \)

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:  \(367787275866480643670016000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.05$
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:  \(3120=2^{4}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3120}(1,·)$, $\chi_{3120}(2183,·)$, $\chi_{3120}(781,·)$, $\chi_{3120}(467,·)$, $\chi_{3120}(469,·)$, $\chi_{3120}(1561,·)$, $\chi_{3120}(1247,·)$, $\chi_{3120}(1249,·)$, $\chi_{3120}(2341,·)$, $\chi_{3120}(2027,·)$, $\chi_{3120}(2029,·)$, $\chi_{3120}(623,·)$, $\chi_{3120}(2963,·)$, $\chi_{3120}(2807,·)$, $\chi_{3120}(2809,·)$, $\chi_{3120}(1403,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1188199} a^{11} - \frac{120918}{1188199} a^{9} - \frac{544212}{1188199} a^{7} - \frac{192563}{1188199} a^{5} + \frac{278149}{1188199} a^{3} + \frac{245890}{1188199} a$, $\frac{1}{1071219620251} a^{12} - \frac{36647742675}{1071219620251} a^{10} - \frac{346086078743}{1071219620251} a^{8} + \frac{560241988}{2375209801} a^{6} - \frac{26250602358}{1071219620251} a^{4} + \frac{118106038291}{1071219620251} a^{2} + \frac{245450}{901549}$, $\frac{1}{1071219620251} a^{13} + \frac{224175}{1071219620251} a^{11} - \frac{109372668656}{1071219620251} a^{9} - \frac{97065194}{2375209801} a^{7} + \frac{126167074680}{1071219620251} a^{5} - \frac{12468909575}{1071219620251} a^{3} - \frac{510452980613}{1071219620251} a$, $\frac{1}{48730809271579109815570309} a^{14} - \frac{15611671864730}{48730809271579109815570309} a^{12} - \frac{214009057108032557959840}{48730809271579109815570309} a^{10} + \frac{104159700094335264282240}{48730809271579109815570309} a^{8} - \frac{21322797848593916721873631}{48730809271579109815570309} a^{6} + \frac{20429080755017655453197336}{48730809271579109815570309} a^{4} + \frac{3145313985001283836873625}{48730809271579109815570309} a^{2} - \frac{3361027684193}{34516381357909}$, $\frac{1}{48730809271579109815570309} a^{15} - \frac{15611671864730}{48730809271579109815570309} a^{13} - \frac{6720143349205240602}{48730809271579109815570309} a^{11} - \frac{570698186979996772704165}{48730809271579109815570309} a^{9} - \frac{17328442998778207550785577}{48730809271579109815570309} a^{7} - \frac{10969097690955000064518553}{48730809271579109815570309} a^{5} - \frac{21367591494961035286713511}{48730809271579109815570309} a^{3} - \frac{10927910748134356507}{41012329813086115891} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{40}\times C_{40}\times C_{520}$, which has order $13312000$ (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:  \( -1 \) (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, 4.0.12168000.2, 4.0.3042000.1, 4.0.389376000.15, 4.0.389376000.1, 8.8.2621440000.1, 8.0.2368963584000000.146, 8.0.151613669376000000.27

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$