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Properties

Label 16.0.15064566819...5536.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $49.96$
Ramified primes $2, 3, 13$
Class number $2160$ (GRH)
Class group $[3, 3, 240]$ (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![747358, -261816, 639668, -65208, 222298, 3360, 56524, 2592, 9255, 840, 1124, -24, 118, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 118*x^12 - 24*x^11 + 1124*x^10 + 840*x^9 + 9255*x^8 + 2592*x^7 + 56524*x^6 + 3360*x^5 + 222298*x^4 - 65208*x^3 + 639668*x^2 - 261816*x + 747358)
 
gp: K = bnfinit(x^16 + 4*x^14 + 118*x^12 - 24*x^11 + 1124*x^10 + 840*x^9 + 9255*x^8 + 2592*x^7 + 56524*x^6 + 3360*x^5 + 222298*x^4 - 65208*x^3 + 639668*x^2 - 261816*x + 747358, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} + 118 x^{12} - 24 x^{11} + 1124 x^{10} + 840 x^{9} + 9255 x^{8} + 2592 x^{7} + 56524 x^{6} + 3360 x^{5} + 222298 x^{4} - 65208 x^{3} + 639668 x^{2} - 261816 x + 747358 \)

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:  \(1506456681949104716472385536=2^{48}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.96$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(624=2^{4}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{624}(1,·)$, $\chi_{624}(259,·)$, $\chi_{624}(389,·)$, $\chi_{624}(77,·)$, $\chi_{624}(131,·)$, $\chi_{624}(469,·)$, $\chi_{624}(599,·)$, $\chi_{624}(157,·)$, $\chi_{624}(415,·)$, $\chi_{624}(545,·)$, $\chi_{624}(443,·)$, $\chi_{624}(103,·)$, $\chi_{624}(233,·)$, $\chi_{624}(287,·)$, $\chi_{624}(313,·)$, $\chi_{624}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{47} a^{12} - \frac{16}{47} a^{11} - \frac{10}{47} a^{10} - \frac{6}{47} a^{9} - \frac{7}{47} a^{8} - \frac{20}{47} a^{7} - \frac{19}{47} a^{6} + \frac{7}{47} a^{5} - \frac{14}{47} a^{4} + \frac{7}{47} a^{3} + \frac{10}{47} a^{2} + \frac{2}{47} a - \frac{16}{47}$, $\frac{1}{3431} a^{13} - \frac{5}{3431} a^{12} - \frac{844}{3431} a^{11} + \frac{307}{3431} a^{10} + \frac{1525}{3431} a^{9} + \frac{1031}{3431} a^{8} - \frac{1132}{3431} a^{7} - \frac{249}{3431} a^{6} + \frac{1003}{3431} a^{5} - \frac{53}{3431} a^{4} + \frac{1027}{3431} a^{3} + \frac{1099}{3431} a^{2} + \frac{523}{3431} a + \frac{153}{3431}$, $\frac{1}{78913} a^{14} - \frac{7}{78913} a^{13} + \frac{5}{3431} a^{12} + \frac{17690}{78913} a^{11} + \frac{15438}{78913} a^{10} + \frac{9442}{78913} a^{9} + \frac{17611}{78913} a^{8} + \frac{1500}{3431} a^{7} + \frac{7487}{78913} a^{6} + \frac{38894}{78913} a^{5} - \frac{12153}{78913} a^{4} + \frac{29705}{78913} a^{3} - \frac{23064}{78913} a^{2} + \frac{1005}{78913} a + \frac{18820}{78913}$, $\frac{1}{154067688291483677945949241} a^{15} - \frac{687232459076617612209}{154067688291483677945949241} a^{14} - \frac{17572803499688304367800}{154067688291483677945949241} a^{13} - \frac{933245836625728743770274}{154067688291483677945949241} a^{12} - \frac{37296829461460622490224714}{154067688291483677945949241} a^{11} + \frac{62022201783630869188835326}{154067688291483677945949241} a^{10} - \frac{30923541828879340621105349}{154067688291483677945949241} a^{9} - \frac{16591178036575488614094600}{154067688291483677945949241} a^{8} + \frac{43311159688850004091859590}{154067688291483677945949241} a^{7} + \frac{17650716855537133141639370}{154067688291483677945949241} a^{6} - \frac{96624032391983020721079}{3278035921095397403105303} a^{5} + \frac{53492803163625144745166424}{154067688291483677945949241} a^{4} - \frac{33469457474538307845161180}{154067688291483677945949241} a^{3} + \frac{49370295364144611758041882}{154067688291483677945949241} a^{2} + \frac{28664777482737997682856467}{154067688291483677945949241} a + \frac{70234998850730065856401313}{154067688291483677945949241}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{240}$, which has order $2160$ (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:  \( 11964.310642723332 \) (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

\(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-78}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}, \sqrt{-26})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{6}, \sqrt{-26})\), \(\Q(\sqrt{2}, \sqrt{-39})\), \(\Q(\sqrt{6}, \sqrt{-13})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-13})\), 4.0.3115008.1, 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.346112.2, 8.0.151613669376.5, 8.0.38813099360256.28, 8.0.479174066176.2, 8.0.9703274840064.5, 8.0.9703274840064.2, 8.0.38813099360256.32, \(\Q(\zeta_{48})^+\)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$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}$
$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}$