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Properties

Label 16.0.31633787341...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $339.83$
Ramified primes $2, 5, 11$
Class number $613599496$ (GRH)
Class group $[2, 306799748]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![77437145761250, 0, 97435855000000, 0, 9300695250000, 0, 338207100000, 0, 6039412500, 0, 58564000, 0, 314600, 0, 880, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 880*x^14 + 314600*x^12 + 58564000*x^10 + 6039412500*x^8 + 338207100000*x^6 + 9300695250000*x^4 + 97435855000000*x^2 + 77437145761250)
 
gp: K = bnfinit(x^16 + 880*x^14 + 314600*x^12 + 58564000*x^10 + 6039412500*x^8 + 338207100000*x^6 + 9300695250000*x^4 + 97435855000000*x^2 + 77437145761250, 1)
 

Normalized defining polynomial

\( x^{16} + 880 x^{14} + 314600 x^{12} + 58564000 x^{10} + 6039412500 x^{8} + 338207100000 x^{6} + 9300695250000 x^{4} + 97435855000000 x^{2} + 77437145761250 \)

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:  \(31633787341870088027340013568000000000000=2^{79}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $339.83$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3520=2^{6}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3520}(1,·)$, $\chi_{3520}(2947,·)$, $\chi_{3520}(969,·)$, $\chi_{3520}(1803,·)$, $\chi_{3520}(2641,·)$, $\chi_{3520}(2067,·)$, $\chi_{3520}(89,·)$, $\chi_{3520}(923,·)$, $\chi_{3520}(1761,·)$, $\chi_{3520}(1187,·)$, $\chi_{3520}(2729,·)$, $\chi_{3520}(43,·)$, $\chi_{3520}(881,·)$, $\chi_{3520}(307,·)$, $\chi_{3520}(1849,·)$, $\chi_{3520}(2683,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{605} a^{4}$, $\frac{1}{605} a^{5}$, $\frac{1}{6655} a^{6}$, $\frac{1}{6655} a^{7}$, $\frac{1}{11346775} a^{8} + \frac{8}{206305} a^{6} + \frac{7}{18755} a^{4} - \frac{13}{341} a^{2} - \frac{12}{31}$, $\frac{1}{192895175} a^{9} + \frac{163}{3507185} a^{7} + \frac{3}{3751} a^{5} - \frac{137}{5797} a^{3} - \frac{229}{527} a$, $\frac{1}{2121846925} a^{10} - \frac{1}{192895175} a^{8} + \frac{3}{206305} a^{6} - \frac{73}{318835} a^{4} - \frac{161}{5797} a^{2} - \frac{13}{31}$, $\frac{1}{2121846925} a^{11} + \frac{214}{3507185} a^{7} + \frac{182}{318835} a^{5} + \frac{229}{5797} a^{3} + \frac{77}{527} a$, $\frac{1}{116701580875} a^{12} - \frac{7}{192895175} a^{8} - \frac{256}{3507185} a^{6} + \frac{263}{318835} a^{4} + \frac{148}{5797} a^{2} + \frac{1}{31}$, $\frac{1}{116701580875} a^{13} - \frac{169}{3507185} a^{7} - \frac{12}{63767} a^{5} + \frac{243}{5797} a^{3} - \frac{5}{527} a$, $\frac{1}{1283717389625} a^{14} + \frac{1}{38579035} a^{8} - \frac{111}{3507185} a^{6} - \frac{213}{318835} a^{4} + \frac{12}{5797} a^{2} - \frac{11}{31}$, $\frac{1}{1283717389625} a^{15} + \frac{128}{3507185} a^{7} + \frac{3}{10285} a^{5} + \frac{10}{341} a^{3} - \frac{96}{527} a$

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

Class group and class number

$C_{2}\times C_{306799748}$, which has order $613599496$ (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:  \( 320942.0117381313 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.1342177280000.1

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 R $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ $16$ $16$

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
5Data not computed
11Data not computed