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Properties

Label 16.8.42832921613...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{8}\cdot 17^{2}\cdot 19^{10}\cdot 331^{8}$
Root discriminant $1460.45$
Ramified primes $2, 5, 17, 19, 331$
Class number $48$ (GRH)
Class group $[2, 2, 2, 6]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![286123551244969, 0, 72152735659128, 0, 4340249387320, 0, -5670878472, 0, -3915519483, 0, -30852792, 0, 435112, 0, -1224, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1224*x^14 + 435112*x^12 - 30852792*x^10 - 3915519483*x^8 - 5670878472*x^6 + 4340249387320*x^4 + 72152735659128*x^2 + 286123551244969)
 
gp: K = bnfinit(x^16 - 1224*x^14 + 435112*x^12 - 30852792*x^10 - 3915519483*x^8 - 5670878472*x^6 + 4340249387320*x^4 + 72152735659128*x^2 + 286123551244969, 1)
 

Normalized defining polynomial

\( x^{16} - 1224 x^{14} + 435112 x^{12} - 30852792 x^{10} - 3915519483 x^{8} - 5670878472 x^{6} + 4340249387320 x^{4} + 72152735659128 x^{2} + 286123551244969 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(428329216131304888705048916200034767627878400000000=2^{32}\cdot 5^{8}\cdot 17^{2}\cdot 19^{10}\cdot 331^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1460.45$
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:  $2, 5, 17, 19, 331$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{18} a^{8} - \frac{1}{18} a^{4} + \frac{7}{18}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{5} + \frac{7}{18} a$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{6} + \frac{7}{18} a^{2}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{7} + \frac{7}{18} a^{3}$, $\frac{1}{3078} a^{12} + \frac{5}{513} a^{10} + \frac{2}{513} a^{8} + \frac{85}{513} a^{6} + \frac{26}{513} a^{4} - \frac{10}{27} a^{2} + \frac{73}{162}$, $\frac{1}{3078} a^{13} + \frac{5}{513} a^{11} + \frac{2}{513} a^{9} + \frac{85}{513} a^{7} + \frac{26}{513} a^{5} - \frac{10}{27} a^{3} + \frac{73}{162} a$, $\frac{1}{89048243514685173711238303887625378681373154} a^{14} - \frac{2151562273357126101244764467438025293}{238096907793275865538070331250335237115971} a^{12} - \frac{751849658011438708191064718232916900308}{549680515522747985871841382022378880749217} a^{10} + \frac{85563389468818606916987619507589193013581}{14841373919114195618539717314604229780228859} a^{8} + \frac{1495197991020713484153099072299604962910265}{14841373919114195618539717314604229780228859} a^{6} - \frac{413097167548410069550747219567381670499764}{4947124639704731872846572438201409926742953} a^{4} + \frac{791213881485918740359555976112986862628501}{4686749658667640721644121257243440983230166} a^{2} + \frac{66819226737442840331101969876629349048475}{137845578196107080048356507565983558330299}$, $\frac{1}{4663367464620547862083838736291053456164830701826} a^{15} - \frac{89863076731658912532613689417949084252199}{923621997350078800175052235351763409816761874} a^{13} + \frac{649660394501640280554000791468786051334771650}{40906732145794279491963497686763626808463427209} a^{11} + \frac{38120138781086679103540963611229606825425818753}{1554455821540182620694612912097017818721610233942} a^{9} + \frac{47338997555992826023529256992155949821298874443}{777227910770091310347306456048508909360805116971} a^{7} - \frac{197557304908968864034874295895135910701523333}{1554455821540182620694612912097017818721610233942} a^{5} + \frac{109502357405633626527435211878453145410976910641}{245440392874765676951780986120581760850780563254} a^{3} - \frac{1045841626517487727841677922816012006713543324}{2406278361517310558350793981574330988733142777} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (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:  $11$
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:  \( 2421860231180000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{31445}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6289}) \), 4.4.7600.1, 4.4.832663600.1, \(\Q(\sqrt{5}, \sqrt{6289})\), 8.8.250291650146150560000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.51$x^{8} + 3$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
2.8.16.49$x^{8} + 14 x^{4} + 12 x^{2} + 8 x + 4$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
331Data not computed