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Properties

Label 16.0.11476349427...9184.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 41^{14}$
Root discriminant $654.99$
Ramified primes $2, 3, 41$
Class number $1064534016$ (GRH)
Class group $[2, 4, 4, 4, 4, 28, 74256]$ (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![44116164, 0, 196071840, 0, 831126744, 0, 174770208, 0, 27131422, 0, 1931264, 0, 101516, 0, 328, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 328*x^14 + 101516*x^12 + 1931264*x^10 + 27131422*x^8 + 174770208*x^6 + 831126744*x^4 + 196071840*x^2 + 44116164)
 
gp: K = bnfinit(x^16 + 328*x^14 + 101516*x^12 + 1931264*x^10 + 27131422*x^8 + 174770208*x^6 + 831126744*x^4 + 196071840*x^2 + 44116164, 1)
 

Normalized defining polynomial

\( x^{16} + 328 x^{14} + 101516 x^{12} + 1931264 x^{10} + 27131422 x^{8} + 174770208 x^{6} + 831126744 x^{4} + 196071840 x^{2} + 44116164 \)

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:  \(1147634942730979008947338773912344095761629184=2^{62}\cdot 3^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $654.99$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3936=2^{5}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(2627,·)$, $\chi_{3936}(2633,·)$, $\chi_{3936}(2705,·)$, $\chi_{3936}(1555,·)$, $\chi_{3936}(875,·)$, $\chi_{3936}(3353,·)$, $\chi_{3936}(2651,·)$, $\chi_{3936}(1313,·)$, $\chi_{3936}(1315,·)$, $\chi_{3936}(1321,·)$, $\chi_{3936}(3499,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(2867,·)$, $\chi_{3936}(2041,·)$, $\chi_{3936}(1339,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{82} a^{8}$, $\frac{1}{246} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{14022} a^{10} + \frac{13}{14022} a^{8} - \frac{13}{171} a^{6} + \frac{62}{171} a^{4} - \frac{68}{171} a^{2} + \frac{1}{19}$, $\frac{1}{42066} a^{11} + \frac{13}{42066} a^{9} - \frac{184}{513} a^{7} - \frac{109}{513} a^{5} - \frac{239}{513} a^{3} - \frac{6}{19} a$, $\frac{1}{156864114} a^{12} - \frac{14}{1912977} a^{10} - \frac{4666}{1912977} a^{8} - \frac{848908}{1912977} a^{6} - \frac{5018}{1912977} a^{4} - \frac{1472}{23617} a^{2} - \frac{72}{23617}$, $\frac{1}{470592342} a^{13} - \frac{14}{5738931} a^{11} - \frac{4666}{5738931} a^{9} - \frac{2761885}{5738931} a^{7} - \frac{1917995}{5738931} a^{5} - \frac{8363}{23617} a^{3} + \frac{23545}{70851} a$, $\frac{1}{3665433843814124142486} a^{14} - \frac{81828773495}{166610629264278370113} a^{12} + \frac{84782509540953935}{3665433843814124142486} a^{10} - \frac{4851537525362432525}{1832716921907062071243} a^{8} + \frac{846440726742989858}{4063673884494594393} a^{6} - \frac{86307997972178066}{451519320499399377} a^{4} + \frac{25130368687290055}{551856947277043683} a^{2} - \frac{1464147919148507}{61317438586338187}$, $\frac{1}{10996301531442372427458} a^{15} - \frac{81828773495}{499831887792835110339} a^{13} + \frac{84782509540953935}{10996301531442372427458} a^{11} - \frac{4851537525362432525}{5498150765721186213729} a^{9} + \frac{846440726742989858}{12191021653483783179} a^{7} - \frac{537827318471577443}{1354557961498198131} a^{5} + \frac{576987315964333738}{1655570841831131049} a^{3} + \frac{59853290667189680}{183952315759014561} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{28}\times C_{74256}$, which has order $1064534016$ (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:  \( \frac{8023862542}{1474430347471490001} a^{14} + \frac{2629364650171}{1474430347471490001} a^{12} + \frac{19847682152248}{35961715791987561} a^{10} + \frac{30500327990298151}{2948860694942980002} a^{8} + \frac{5223381783311984}{35961715791987561} a^{6} + \frac{3666698840459018}{3995746199109729} a^{4} + \frac{1974798922703368}{443971799901081} a^{2} + \frac{51764031839705}{49330199989009} \) (order $6$)
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:  \( 38574398.31621447 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{-246}) \), \(\Q(\sqrt{-3}, \sqrt{82})\), 4.4.141150208.2, 4.0.1270351872.2, 8.0.1613793878693904384.10, 8.0.33876761101542440828928.1, 8.8.418231618537560997888.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
41Data not computed