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Properties

Label 16.0.22821050447...896.66
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 7^{8}\cdot 11^{8}$
Root discriminant $121.59$
Ramified primes $2, 3, 7, 11$
Class number $2211840$ (GRH)
Class group $[2, 4, 24, 48, 240]$ (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![51546056158, -472415544, 15254320436, -25740600, 2065793754, 1983264, 168416588, 183072, 9103527, 3528, 337124, -24, 8438, 0, 132, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 132*x^14 + 8438*x^12 - 24*x^11 + 337124*x^10 + 3528*x^9 + 9103527*x^8 + 183072*x^7 + 168416588*x^6 + 1983264*x^5 + 2065793754*x^4 - 25740600*x^3 + 15254320436*x^2 - 472415544*x + 51546056158)
 
gp: K = bnfinit(x^16 + 132*x^14 + 8438*x^12 - 24*x^11 + 337124*x^10 + 3528*x^9 + 9103527*x^8 + 183072*x^7 + 168416588*x^6 + 1983264*x^5 + 2065793754*x^4 - 25740600*x^3 + 15254320436*x^2 - 472415544*x + 51546056158, 1)
 

Normalized defining polynomial

\( x^{16} + 132 x^{14} + 8438 x^{12} - 24 x^{11} + 337124 x^{10} + 3528 x^{9} + 9103527 x^{8} + 183072 x^{7} + 168416588 x^{6} + 1983264 x^{5} + 2065793754 x^{4} - 25740600 x^{3} + 15254320436 x^{2} - 472415544 x + 51546056158 \)

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:  \(2282105044722241146451513778896896=2^{48}\cdot 3^{8}\cdot 7^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.59$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3696=2^{4}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3696}(1,·)$, $\chi_{3696}(2309,·)$, $\chi_{3696}(3079,·)$, $\chi_{3696}(461,·)$, $\chi_{3696}(1231,·)$, $\chi_{3696}(2003,·)$, $\chi_{3696}(2773,·)$, $\chi_{3696}(155,·)$, $\chi_{3696}(925,·)$, $\chi_{3696}(3233,·)$, $\chi_{3696}(1385,·)$, $\chi_{3696}(2155,·)$, $\chi_{3696}(2927,·)$, $\chi_{3696}(307,·)$, $\chi_{3696}(1079,·)$, $\chi_{3696}(1849,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47} a^{13} + \frac{18}{47} a^{12} + \frac{13}{47} a^{11} + \frac{23}{47} a^{10} - \frac{6}{47} a^{9} - \frac{18}{47} a^{8} + \frac{20}{47} a^{7} + \frac{17}{47} a^{6} + \frac{15}{47} a^{4} + \frac{14}{47} a^{3} + \frac{4}{47} a^{2} - \frac{13}{47} a + \frac{17}{47}$, $\frac{1}{2018262767} a^{14} + \frac{10733802}{2018262767} a^{13} - \frac{450888335}{2018262767} a^{12} - \frac{773240030}{2018262767} a^{11} + \frac{139572877}{2018262767} a^{10} + \frac{658562605}{2018262767} a^{9} + \frac{940023901}{2018262767} a^{8} - \frac{482993666}{2018262767} a^{7} + \frac{619412601}{2018262767} a^{6} + \frac{977874495}{2018262767} a^{5} + \frac{707036418}{2018262767} a^{4} + \frac{3257450}{2018262767} a^{3} + \frac{37194120}{2018262767} a^{2} + \frac{299779460}{2018262767} a - \frac{461616116}{2018262767}$, $\frac{1}{49667119565711671314639280707636332465775169} a^{15} - \frac{490486033048667380962032550892127}{49667119565711671314639280707636332465775169} a^{14} + \frac{45962075281646482547450651075162123979759}{49667119565711671314639280707636332465775169} a^{13} - \frac{23576557048297069560558709984452013682022221}{49667119565711671314639280707636332465775169} a^{12} - \frac{10452765269915339414272120923485566746986794}{49667119565711671314639280707636332465775169} a^{11} + \frac{126447245745154580279306017963448774418377}{699536895291713680487877193065300457264439} a^{10} + \frac{16060924256733591382934889361575869911789275}{49667119565711671314639280707636332465775169} a^{9} - \frac{9882112410213650712395437454930834386274130}{49667119565711671314639280707636332465775169} a^{8} - \frac{14123179047463552777718579664812694830473306}{49667119565711671314639280707636332465775169} a^{7} + \frac{6493298020829683768921833158651566626139817}{49667119565711671314639280707636332465775169} a^{6} - \frac{14570957763223407075431024790753086514721357}{49667119565711671314639280707636332465775169} a^{5} - \frac{1800561082978452127266867245729749761992816}{49667119565711671314639280707636332465775169} a^{4} - \frac{2923069816452510783556800678132084684225726}{49667119565711671314639280707636332465775169} a^{3} + \frac{14254036724946496099626974659241484009409159}{49667119565711671314639280707636332465775169} a^{2} + \frac{20217512224244086167347264102819999110450325}{49667119565711671314639280707636332465775169} a + \frac{10735091717911996941178440002015948677464657}{49667119565711671314639280707636332465775169}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{24}\times C_{48}\times C_{240}$, which has order $2211840$ (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

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-154}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-462}) \), \(\Q(\sqrt{3}, \sqrt{-77})\), \(\Q(\sqrt{2}, \sqrt{-77})\), \(\Q(\sqrt{6}, \sqrt{-77})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-154})\), \(\Q(\sqrt{2}, \sqrt{-231})\), \(\Q(\sqrt{6}, \sqrt{-154})\), 4.4.18432.1, 4.0.109283328.8, \(\Q(\zeta_{16})^+\), 4.0.12142592.8, 8.0.186606965293056.201, 8.0.47771383115022336.189, 8.0.589770161913856.75, \(\Q(\zeta_{48})^+\), 8.0.47771383115022336.288, 8.0.11942845778755584.325, 8.0.11942845778755584.528

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$