↑

Properties

Label 16.0.11352029626...1824.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}\cdot 3^{8}\cdot 17^{15}$
Root discriminant $755.86$
Ramified primes $2, 3, 17$
Class number $3479804032$ (GRH)
Class group $[2, 2, 2, 434975504]$ (GRH)
Galois group $C_{16}$ (as 16T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127157756850, 0, 173471860224, 0, 57355398864, 0, 6765454368, 0, 366177348, 0, 10046592, 0, 138312, 0, 816, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 816*x^14 + 138312*x^12 + 10046592*x^10 + 366177348*x^8 + 6765454368*x^6 + 57355398864*x^4 + 173471860224*x^2 + 127157756850)
 
gp: K = bnfinit(x^16 + 816*x^14 + 138312*x^12 + 10046592*x^10 + 366177348*x^8 + 6765454368*x^6 + 57355398864*x^4 + 173471860224*x^2 + 127157756850, 1)
 

Normalized defining polynomial

\( x^{16} + 816 x^{14} + 138312 x^{12} + 10046592 x^{10} + 366177348 x^{8} + 6765454368 x^{6} + 57355398864 x^{4} + 173471860224 x^{2} + 127157756850 \)

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:  \(11352029626874238391016464929669692733284941824=2^{79}\cdot 3^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $755.86$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3264=2^{6}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3264}(1,·)$, $\chi_{3264}(1091,·)$, $\chi_{3264}(2185,·)$, $\chi_{3264}(11,·)$, $\chi_{3264}(2257,·)$, $\chi_{3264}(2579,·)$, $\chi_{3264}(2905,·)$, $\chi_{3264}(1115,·)$, $\chi_{3264}(2209,·)$, $\chi_{3264}(1187,·)$, $\chi_{3264}(2473,·)$, $\chi_{3264}(1451,·)$, $\chi_{3264}(1585,·)$, $\chi_{3264}(1331,·)$, $\chi_{3264}(121,·)$, $\chi_{3264}(1979,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{45} a^{5} + \frac{1}{5} a$, $\frac{1}{135} a^{6} + \frac{1}{15} a^{2}$, $\frac{1}{135} a^{7} + \frac{1}{15} a^{3}$, $\frac{1}{405} a^{8} + \frac{1}{45} a^{4}$, $\frac{1}{405} a^{9} - \frac{1}{5} a$, $\frac{1}{30375} a^{10} - \frac{1}{2025} a^{8} + \frac{2}{3375} a^{6} + \frac{1}{25} a^{4} + \frac{1}{375} a^{2} + \frac{2}{5}$, $\frac{1}{30375} a^{11} - \frac{1}{2025} a^{9} + \frac{2}{3375} a^{7} - \frac{1}{225} a^{5} + \frac{1}{375} a^{3}$, $\frac{1}{85384125} a^{12} - \frac{278}{28461375} a^{10} + \frac{8842}{9487125} a^{8} + \frac{2308}{1054125} a^{6} - \frac{3951}{117125} a^{4} + \frac{234}{117125} a^{2} - \frac{156}{4685}$, $\frac{1}{85384125} a^{13} - \frac{278}{28461375} a^{11} + \frac{8842}{9487125} a^{9} + \frac{2308}{1054125} a^{7} + \frac{11291}{1054125} a^{5} + \frac{234}{117125} a^{3} + \frac{1718}{4685} a$, $\frac{1}{141607070507535890625} a^{14} - \frac{4996983073}{3146823789056353125} a^{12} - \frac{69759604131464}{5244706315093921875} a^{10} + \frac{72149550887338}{116549029224309375} a^{8} - \frac{4320850000325237}{1748235438364640625} a^{6} - \frac{625050966061868}{38849676408103125} a^{4} + \frac{2226314537627906}{194248382040515625} a^{2} + \frac{1087800270283627}{2589978427206875}$, $\frac{1}{21382667646637919484375} a^{15} - \frac{4769273654444}{1425511176442527965625} a^{13} - \frac{4478735246400517}{2375851960737546609375} a^{11} - \frac{3965563242417587}{17598903412870715625} a^{9} + \frac{392075210077794388}{263983551193060734375} a^{7} - \frac{176766749316162829}{17598903412870715625} a^{5} - \frac{418526293358415448}{9777168562705953125} a^{3} - \frac{148012586424846498}{391086742508238125} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{434975504}$, which has order $3479804032$ (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:  \( 241880713.68251026 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{34}) \), 4.4.10061824.1, 8.8.881195590409519104.4

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ $16$ $16$

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$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17Data not computed