↑

Properties

Label 16.0.55086646241...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{12}$
Root discriminant $54.18$
Ramified primes $5, 41$
Class number $80$ (GRH)
Class group $[4, 20]$ (GRH)
Galois group $C_4^2$ (as 16T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 1152, 4224, 14752, 50896, 52344, 49432, 42566, 31987, 9763, 4288, 1323, 311, -49, 16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 16*x^14 - 49*x^13 + 311*x^12 + 1323*x^11 + 4288*x^10 + 9763*x^9 + 31987*x^8 + 42566*x^7 + 49432*x^6 + 52344*x^5 + 50896*x^4 + 14752*x^3 + 4224*x^2 + 1152*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + 16*x^14 - 49*x^13 + 311*x^12 + 1323*x^11 + 4288*x^10 + 9763*x^9 + 31987*x^8 + 42566*x^7 + 49432*x^6 + 52344*x^5 + 50896*x^4 + 14752*x^3 + 4224*x^2 + 1152*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 16 x^{14} - 49 x^{13} + 311 x^{12} + 1323 x^{11} + 4288 x^{10} + 9763 x^{9} + 31987 x^{8} + 42566 x^{7} + 49432 x^{6} + 52344 x^{5} + 50896 x^{4} + 14752 x^{3} + 4224 x^{2} + 1152 x + 256 \)

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:  \(5508664624112838398681640625=5^{12}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.18$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(205=5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(132,·)$, $\chi_{205}(9,·)$, $\chi_{205}(204,·)$, $\chi_{205}(81,·)$, $\chi_{205}(83,·)$, $\chi_{205}(196,·)$, $\chi_{205}(91,·)$, $\chi_{205}(32,·)$, $\chi_{205}(163,·)$, $\chi_{205}(42,·)$, $\chi_{205}(173,·)$, $\chi_{205}(114,·)$, $\chi_{205}(73,·)$, $\chi_{205}(122,·)$, $\chi_{205}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{7}{16} a^{8} - \frac{5}{16} a^{7} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{1147079529632} a^{13} + \frac{19624763883}{1147079529632} a^{12} + \frac{2189779431}{286769882408} a^{11} - \frac{1157541905}{1147079529632} a^{10} - \frac{220701442125}{1147079529632} a^{9} - \frac{153046022777}{1147079529632} a^{8} + \frac{70215279045}{286769882408} a^{7} + \frac{327365762451}{1147079529632} a^{6} - \frac{264593908833}{1147079529632} a^{5} - \frac{246301918375}{573539764816} a^{4} + \frac{28857959545}{71692470602} a^{3} + \frac{28219660059}{71692470602} a^{2} + \frac{1492527788}{35846235301} a + \frac{5358071884}{35846235301}$, $\frac{1}{2294159059264} a^{14} - \frac{1}{2294159059264} a^{13} - \frac{4443514209}{573539764816} a^{12} - \frac{58704482725}{2294159059264} a^{11} - \frac{207906369793}{2294159059264} a^{10} + \frac{517132408767}{2294159059264} a^{9} - \frac{88996108673}{573539764816} a^{8} - \frac{70436960049}{2294159059264} a^{7} - \frac{200584001381}{2294159059264} a^{6} + \frac{368900941189}{1147079529632} a^{5} - \frac{232715150319}{573539764816} a^{4} - \frac{91557549155}{286769882408} a^{3} + \frac{10630080236}{35846235301} a^{2} + \frac{11696116470}{35846235301} a + \frac{8810902681}{35846235301}$, $\frac{1}{4588318118528} a^{15} - \frac{1}{4588318118528} a^{14} - \frac{73543177649}{4588318118528} a^{12} + \frac{265022776807}{4588318118528} a^{11} - \frac{498310184741}{4588318118528} a^{10} - \frac{33907754477}{143384941204} a^{9} + \frac{2157957281315}{4588318118528} a^{8} - \frac{2002735921181}{4588318118528} a^{7} - \frac{852008057157}{2294159059264} a^{6} + \frac{34512399687}{573539764816} a^{5} - \frac{225337092383}{573539764816} a^{4} - \frac{36848537919}{286769882408} a^{3} + \frac{54245597751}{143384941204} a^{2} + \frac{23403854937}{71692470602} a + \frac{2615041441}{35846235301}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  \( -\frac{420676101}{286769882408} a^{15} + \frac{514159679}{286769882408} a^{14} - \frac{6823618665}{286769882408} a^{13} + \frac{22108866197}{286769882408} a^{12} - \frac{135410962733}{286769882408} a^{11} - \frac{527481088865}{286769882408} a^{10} - \frac{840090173697}{143384941204} a^{9} - \frac{1852196748749}{143384941204} a^{8} - \frac{12543486270673}{286769882408} a^{7} - \frac{1864529963210}{35846235301} a^{6} - \frac{4203909760871}{71692470602} a^{5} - \frac{2174848700381}{35846235301} a^{4} - \frac{16689412472737}{286769882408} a^{3} - \frac{180984207008}{35846235301} a^{2} - \frac{49733263496}{35846235301} a - \frac{11218029360}{35846235301} \) (order $10$)
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:  \( 1050930.24893 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{41}) \), 4.0.8615125.2, \(\Q(\sqrt{5}, \sqrt{41})\), 4.0.8615125.1, 4.0.210125.1, \(\Q(\zeta_{5})\), 4.4.68921.1, 4.4.1723025.1, 8.0.74220378765625.1, 8.0.44152515625.1, 8.8.2968815150625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$