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Properties

Label 16.0.40198805069...5344.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 23^{4}\cdot 37^{4}\cdot 601^{2}\cdot 27457^{2}$
Root discriminant $344.96$
Ramified primes $2, 23, 37, 601, 27457$
Class number $1086472000$ (GRH)
Class group $[2, 2, 2, 135809000]$ (GRH)
Galois group 16T1765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1089218734982596, 0, 1742607190434464, 0, 100031244818584, 0, 2334616894288, 0, 28176729004, 0, 188299712, 0, 695390, 0, 1320, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1320*x^14 + 695390*x^12 + 188299712*x^10 + 28176729004*x^8 + 2334616894288*x^6 + 100031244818584*x^4 + 1742607190434464*x^2 + 1089218734982596)
 
gp: K = bnfinit(x^16 + 1320*x^14 + 695390*x^12 + 188299712*x^10 + 28176729004*x^8 + 2334616894288*x^6 + 100031244818584*x^4 + 1742607190434464*x^2 + 1089218734982596, 1)
 

Normalized defining polynomial

\( x^{16} + 1320 x^{14} + 695390 x^{12} + 188299712 x^{10} + 28176729004 x^{8} + 2334616894288 x^{6} + 100031244818584 x^{4} + 1742607190434464 x^{2} + 1089218734982596 \)

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:  \(40198805069552937123844028608838939705344=2^{48}\cdot 23^{4}\cdot 37^{4}\cdot 601^{2}\cdot 27457^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $344.96$
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:  $2, 23, 37, 601, 27457$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2506092563650457186672590531342775597579905025212028} a^{14} + \frac{119972637991144138360403040765649809645159391096664}{626523140912614296668147632835693899394976256303007} a^{12} - \frac{15120326160135588297053835872382319639470475758387}{89503305844659185238306804690813414199282322329001} a^{10} - \frac{110127966765354794620268465265461425258766652475890}{626523140912614296668147632835693899394976256303007} a^{8} - \frac{83987878783972214827791467595057910577228467712939}{179006611689318370476613609381626828398564644658002} a^{6} + \frac{38639746861325372016105973808575609103024905829608}{89503305844659185238306804690813414199282322329001} a^{4} - \frac{224080700984306882502287741392498295423182064962413}{626523140912614296668147632835693899394976256303007} a^{2} - \frac{4000064349490290271809624669446582175295228}{37967286613254311168154060700431108184770551}$, $\frac{1}{2506092563650457186672590531342775597579905025212028} a^{15} - \frac{146632588948037743226535469773094660814338691916351}{2506092563650457186672590531342775597579905025212028} a^{13} - \frac{15120326160135588297053835872382319639470475758387}{89503305844659185238306804690813414199282322329001} a^{11} - \frac{110127966765354794620268465265461425258766652475890}{626523140912614296668147632835693899394976256303007} a^{9} + \frac{2757713530343485205257668547877751811026927308031}{89503305844659185238306804690813414199282322329001} a^{7} - \frac{12223812122008441206094857073662195993232510669785}{179006611689318370476613609381626828398564644658002} a^{5} - \frac{224080700984306882502287741392498295423182064962413}{626523140912614296668147632835693899394976256303007} a^{3} - \frac{4000064349490290271809624669446582175295228}{37967286613254311168154060700431108184770551} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{135809000}$, which has order $1086472000$ (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:  \( 109226.925145 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12288
The 74 conjugacy class representatives for t16n1765 are not computed
Character table for t16n1765 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.8.47461236736.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
601Data not computed
27457Data not computed