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Properties

Label 16.0.18240547258...000.28
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 61^{12}$
Root discriminant $583.87$
Ramified primes $2, 5, 61$
Class number $1115920000$ (GRH)
Class group $[2, 20, 20, 1394900]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![156220976347921, 0, 76635733034448, 0, 9436607057716, 0, 400872052024, 0, 7707154739, 0, 74539072, 0, 379664, 0, 976, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 976*x^14 + 379664*x^12 + 74539072*x^10 + 7707154739*x^8 + 400872052024*x^6 + 9436607057716*x^4 + 76635733034448*x^2 + 156220976347921)
 
gp: K = bnfinit(x^16 + 976*x^14 + 379664*x^12 + 74539072*x^10 + 7707154739*x^8 + 400872052024*x^6 + 9436607057716*x^4 + 76635733034448*x^2 + 156220976347921, 1)
 

Normalized defining polynomial

\( x^{16} + 976 x^{14} + 379664 x^{12} + 74539072 x^{10} + 7707154739 x^{8} + 400872052024 x^{6} + 9436607057716 x^{4} + 76635733034448 x^{2} + 156220976347921 \)

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:  \(182405472588468433947695572320256000000000000=2^{48}\cdot 5^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $583.87$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4880=2^{4}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(2307,·)$, $\chi_{4880}(3283,·)$, $\chi_{4880}(1109,·)$, $\chi_{4880}(987,·)$, $\chi_{4880}(4381,·)$, $\chi_{4880}(487,·)$, $\chi_{4880}(3049,·)$, $\chi_{4880}(1963,·)$, $\chi_{4880}(367,·)$, $\chi_{4880}(2929,·)$, $\chi_{4880}(3061,·)$, $\chi_{4880}(1463,·)$, $\chi_{4880}(121,·)$, $\chi_{4880}(2429,·)$, $\chi_{4880}(1343,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{61} a^{4}$, $\frac{1}{61} a^{5}$, $\frac{1}{61} a^{6}$, $\frac{1}{61} a^{7}$, $\frac{1}{40931} a^{8} - \frac{3}{671} a^{6} + \frac{5}{671} a^{4} - \frac{1}{11} a^{2} + \frac{4}{11}$, $\frac{1}{40931} a^{9} - \frac{3}{671} a^{7} + \frac{5}{671} a^{5} - \frac{1}{11} a^{3} + \frac{4}{11} a$, $\frac{1}{736717069} a^{10} + \frac{10}{12077329} a^{8} + \frac{74131}{12077329} a^{6} + \frac{6060}{12077329} a^{4} + \frac{57027}{197989} a^{2} - \frac{68009}{197989}$, $\frac{1}{2474632634771} a^{11} + \frac{504582}{2474632634771} a^{9} + \frac{306831088}{40567748111} a^{7} - \frac{258027604}{40567748111} a^{5} - \frac{226352394}{665045051} a^{3} - \frac{110905851}{665045051} a$, $\frac{1}{150952590721031} a^{12} + \frac{12}{2474632634771} a^{10} - \frac{500617}{2474632634771} a^{8} - \frac{4024456}{40567748111} a^{6} - \frac{130711587}{40567748111} a^{4} + \frac{10263930}{60458641} a^{2} + \frac{89633}{197989}$, $\frac{1}{150952590721031} a^{13} - \frac{6555601}{2474632634771} a^{9} + \frac{304272794}{40567748111} a^{7} + \frac{305439457}{40567748111} a^{5} + \frac{168951754}{665045051} a^{3} + \frac{301857357}{665045051} a$, $\frac{1}{150952590721031} a^{14} + \frac{1167}{2474632634771} a^{10} + \frac{9195821}{2474632634771} a^{8} + \frac{35899861}{40567748111} a^{6} - \frac{201059597}{40567748111} a^{4} - \frac{266156261}{665045051} a^{2} - \frac{28942}{197989}$, $\frac{1}{150952590721031} a^{15} + \frac{24935037}{2474632634771} a^{9} - \frac{60366474}{40567748111} a^{7} + \frac{14498014}{40567748111} a^{5} - \frac{75339069}{665045051} a^{3} + \frac{69337481}{665045051} a$

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

Class group and class number

$C_{2}\times C_{20}\times C_{20}\times C_{1394900}$, which has order $1115920000$ (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:  \( 4361877.087043648 \) (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

\(\Q(\sqrt{122}) \), \(\Q(\sqrt{610}) \), \(\Q(\sqrt{5}) \), 4.0.11621427200.4, \(\Q(\sqrt{5}, \sqrt{122})\), 4.0.464857088.4, 4.0.58107136000.4, 4.0.58107136000.6, \(\Q(\zeta_{20})^+\), 4.4.29768000.2, 8.0.135057570164899840000.3, 8.0.3376439254122496000000.6, 8.8.14178141184000000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
5Data not computed
$61$61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$