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Properties

Label 16.0.22538752331...2544.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 73^{14}$
Root discriminant $591.64$
Ramified primes $2, 3, 73$
Class number $1469026304$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 34, 68, 9928]$ (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![5650252604676, 0, 2354400118368, 0, 325634279004, 0, 20830421520, 0, 701820613, 0, 12925088, 0, 126144, 0, 584, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 584*x^14 + 126144*x^12 + 12925088*x^10 + 701820613*x^8 + 20830421520*x^6 + 325634279004*x^4 + 2354400118368*x^2 + 5650252604676)
 
gp: K = bnfinit(x^16 + 584*x^14 + 126144*x^12 + 12925088*x^10 + 701820613*x^8 + 20830421520*x^6 + 325634279004*x^4 + 2354400118368*x^2 + 5650252604676, 1)
 

Normalized defining polynomial

\( x^{16} + 584 x^{14} + 126144 x^{12} + 12925088 x^{10} + 701820613 x^{8} + 20830421520 x^{6} + 325634279004 x^{4} + 2354400118368 x^{2} + 5650252604676 \)

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:  \(225387523315734662028879718967317456742252544=2^{48}\cdot 3^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $591.64$
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:  $2, 3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3504=2^{4}\cdot 3\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{3504}(1,·)$, $\chi_{3504}(1607,·)$, $\chi_{3504}(265,·)$, $\chi_{3504}(3275,·)$, $\chi_{3504}(1871,·)$, $\chi_{3504}(145,·)$, $\chi_{3504}(2387,·)$, $\chi_{3504}(1751,·)$, $\chi_{3504}(1487,·)$, $\chi_{3504}(3421,·)$, $\chi_{3504}(2723,·)$, $\chi_{3504}(2533,·)$, $\chi_{3504}(1835,·)$, $\chi_{3504}(2869,·)$, $\chi_{3504}(3385,·)$, $\chi_{3504}(1981,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{657} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{657} a^{9} - \frac{1}{9} a^{3}$, $\frac{1}{12483} a^{10} - \frac{1}{12483} a^{8} - \frac{7}{57} a^{6} - \frac{58}{171} a^{4} + \frac{61}{171} a^{2} + \frac{2}{19}$, $\frac{1}{37449} a^{11} - \frac{1}{37449} a^{9} + \frac{4}{57} a^{7} - \frac{1}{513} a^{5} + \frac{118}{513} a^{3} - \frac{17}{57} a$, $\frac{1}{898776} a^{12} - \frac{7}{224694} a^{10} + \frac{5}{49932} a^{8} - \frac{143}{1539} a^{6} + \frac{1513}{12312} a^{4} + \frac{5}{38} a^{2} + \frac{35}{76}$, $\frac{1}{180653976} a^{13} - \frac{115}{45163494} a^{11} - \frac{7115}{10036332} a^{9} + \frac{991}{309339} a^{7} + \frac{186625}{2474712} a^{5} - \frac{1903}{68742} a^{3} + \frac{1997}{5092} a$, $\frac{1}{50393131392263653965828888} a^{14} - \frac{1955438513606030030}{6299141424032956745728611} a^{12} - \frac{3544734554610316399}{311068712297923789912524} a^{10} - \frac{1748097534558913832762}{6299141424032956745728611} a^{8} + \frac{85772878287356674560313}{690316868387173341997656} a^{6} + \frac{2503951094138346173069}{9587734283155185305523} a^{4} + \frac{72593381305084504121}{224274486155676849252} a^{2} + \frac{281845907690518851}{588890994604458283}$, $\frac{1}{907076365060745771384919984} a^{15} - \frac{7414429431958204}{2983803832436663721660921} a^{13} + \frac{174339557167716789379}{16797710464087884655276296} a^{11} + \frac{21577285870729074298585}{113384545632593221423114998} a^{9} + \frac{1463561341729877607483025}{12425703630969120155957808} a^{7} - \frac{20540903237877802383571}{172579217096793335499414} a^{5} + \frac{4164666660749945419175}{76701874265241482444184} a^{3} + \frac{304774756564292299985}{710202539492976689298} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{34}\times C_{68}\times C_{9928}$, which has order $1469026304$ (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:  \( 259264854.2777166 \) (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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{438}) \), \(\Q(\sqrt{6}, \sqrt{73})\), 4.4.24897088.1, 4.4.56018448.2, 8.8.803345028180148224.1, 8.0.3753227971657652502528.2, 8.0.46336147798242623488.23

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$