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Properties

Label 16.0.22795311112...2496.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 7^{8}\cdot 11^{8}$
Root discriminant $140.40$
Ramified primes $2, 7, 11$
Class number $3101184$ (GRH)
Class group $[2, 48, 32304]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![142701862081, 0, 57207791296, 0, 15807416016, 0, 1663938528, 0, 86011860, 0, 2414368, 0, 37544, 0, 304, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 304*x^14 + 37544*x^12 + 2414368*x^10 + 86011860*x^8 + 1663938528*x^6 + 15807416016*x^4 + 57207791296*x^2 + 142701862081)
 
gp: K = bnfinit(x^16 + 304*x^14 + 37544*x^12 + 2414368*x^10 + 86011860*x^8 + 1663938528*x^6 + 15807416016*x^4 + 57207791296*x^2 + 142701862081, 1)
 

Normalized defining polynomial

\( x^{16} + 304 x^{14} + 37544 x^{12} + 2414368 x^{10} + 86011860 x^{8} + 1663938528 x^{6} + 15807416016 x^{4} + 57207791296 x^{2} + 142701862081 \)

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:  \(22795311112775003166262217194602496=2^{64}\cdot 7^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.40$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2464=2^{5}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2464}(1,·)$, $\chi_{2464}(1539,·)$, $\chi_{2464}(1541,·)$, $\chi_{2464}(1231,·)$, $\chi_{2464}(1233,·)$, $\chi_{2464}(923,·)$, $\chi_{2464}(925,·)$, $\chi_{2464}(2463,·)$, $\chi_{2464}(615,·)$, $\chi_{2464}(617,·)$, $\chi_{2464}(2155,·)$, $\chi_{2464}(2157,·)$, $\chi_{2464}(307,·)$, $\chi_{2464}(309,·)$, $\chi_{2464}(1847,·)$, $\chi_{2464}(1849,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{31161} a^{8} + \frac{152}{31161} a^{6} + \frac{7220}{31161} a^{4} - \frac{14900}{31161} a^{2} + \frac{11354}{31161}$, $\frac{1}{11771348199} a^{9} - \frac{2030076676}{11771348199} a^{7} + \frac{740797673}{11771348199} a^{5} + \frac{4607450560}{11771348199} a^{3} - \frac{3315394402}{11771348199} a$, $\frac{1}{11771348199} a^{10} + \frac{190}{11771348199} a^{8} + \frac{1085809377}{3923782733} a^{6} - \frac{5337432874}{11771348199} a^{4} + \frac{53186684}{692432247} a^{2} + \frac{3158}{31161}$, $\frac{1}{11771348199} a^{11} + \frac{517506004}{11771348199} a^{7} - \frac{1610937452}{3923782733} a^{5} - \frac{1143888682}{3923782733} a^{3} - \frac{4534903444}{11771348199} a$, $\frac{1}{11771348199} a^{12} - \frac{7942}{3923782733} a^{8} - \frac{1097909123}{11771348199} a^{6} + \frac{3291688636}{11771348199} a^{4} - \frac{3573506789}{11771348199} a^{2} - \frac{5641}{31161}$, $\frac{1}{11771348199} a^{13} - \frac{95003216}{905488323} a^{7} - \frac{268096382}{905488323} a^{5} + \frac{5721580096}{11771348199} a^{3} + \frac{2799802918}{11771348199} a$, $\frac{1}{11771348199} a^{14} - \frac{147637}{11771348199} a^{8} - \frac{1374303386}{3923782733} a^{6} - \frac{341480042}{3923782733} a^{4} + \frac{1493890055}{11771348199} a^{2} + \frac{3475}{31161}$, $\frac{1}{11771348199} a^{15} + \frac{270362936}{905488323} a^{7} + \frac{525491666}{11771348199} a^{5} + \frac{257947054}{3923782733} a^{3} + \frac{202322713}{905488323} a$

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

Class group and class number

$C_{2}\times C_{48}\times C_{32304}$, which has order $3101184$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-154}) \), \(\Q(\sqrt{2}, \sqrt{-77})\), \(\Q(\zeta_{16})^+\), 4.0.12142592.8, 8.0.589770161913856.75, \(\Q(\zeta_{32})^+\), 8.0.75490580724973568.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
11Data not computed