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Properties

Label 16.4.11456651943...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{10}\cdot 71^{8}\cdot 101^{8}$
Root discriminant $654.92$
Ramified primes $2, 5, 71, 101$
Class number $256$ (GRH)
Class group $[2, 2, 2, 2, 4, 4]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-444209861085895, 2187279234337270, -669485660464680, 25818046094220, 12138022544004, -778584634754, 120057079608, -18844746598, -1092134344, -34026026, 13090654, -1032102, 73695, -5214, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 - 5214*x^13 + 73695*x^12 - 1032102*x^11 + 13090654*x^10 - 34026026*x^9 - 1092134344*x^8 - 18844746598*x^7 + 120057079608*x^6 - 778584634754*x^5 + 12138022544004*x^4 + 25818046094220*x^3 - 669485660464680*x^2 + 2187279234337270*x - 444209861085895)
 
gp: K = bnfinit(x^16 - 18*x^14 - 5214*x^13 + 73695*x^12 - 1032102*x^11 + 13090654*x^10 - 34026026*x^9 - 1092134344*x^8 - 18844746598*x^7 + 120057079608*x^6 - 778584634754*x^5 + 12138022544004*x^4 + 25818046094220*x^3 - 669485660464680*x^2 + 2187279234337270*x - 444209861085895, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} - 5214 x^{13} + 73695 x^{12} - 1032102 x^{11} + 13090654 x^{10} - 34026026 x^{9} - 1092134344 x^{8} - 18844746598 x^{7} + 120057079608 x^{6} - 778584634754 x^{5} + 12138022544004 x^{4} + 25818046094220 x^{3} - 669485660464680 x^{2} + 2187279234337270 x - 444209861085895 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1145665194381286185627968624158474240000000000=2^{24}\cdot 5^{10}\cdot 71^{8}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $654.92$
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:  $2, 5, 71, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{15} - \frac{7616331809158737092468607347095951858774502125021613943463739936482584850088326577922591361769701030292108638402566}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{14} - \frac{12573878907605926645243596110796320221723187007751672865883486346358917810683336321217835752770164695809694441890007}{29350684993927091294151048173432530062360916681338241427168285527718609357225763237566254722891506324397818668234577} a^{13} + \frac{66737870066049313407180445799090172509768317570509376514618756653830442702541169947892774604096659848242311996401816}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{12} - \frac{54737799918245124176758893545975663903047209502662156939213803227497143913987313257624139209325077142681818329443126}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{11} + \frac{26953848140375200798212676711244204411136854340998979520601899718511725025045325746768634020488357363691768476402956}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{10} + \frac{37489827298257981113434574666514795122168317686114481396885569931766713627304254571336165093937190619247060632856043}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{9} + \frac{72868915485831666534051521071747340200085813336851599134153735852355638355290748202390240681536977786609450962982408}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{8} - \frac{51234341553271558030236970416403449134522866422741012978407473680513401740609781846540491141047384793874714480174944}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{7} - \frac{13835024992003260310441422003134820155791779548269578765583623876786435543479568971061597886437518050879619653872853}{29350684993927091294151048173432530062360916681338241427168285527718609357225763237566254722891506324397818668234577} a^{6} + \frac{12404989231199757889673099366120319422709132784882158601986697322997861576914621356198840093174764767553216984334769}{146753424969635456470755240867162650311804583406691207135841427638593046786128816187831273614457531621989093341172885} a^{5} + \frac{5018331988728356670332303273679480812330529091270043839142863041736243040119245965302041990206530884830550492550001}{11288724997664265882365787759012511562446506415899323625833955972199465137394524322140867201112117817076084103167145} a^{4} + \frac{8293800704873154447008304936221910488634418757801071358225970387482096559442224712304861434481413002687380205080186}{29350684993927091294151048173432530062360916681338241427168285527718609357225763237566254722891506324397818668234577} a^{3} + \frac{6106639062120578856026984877619775345122159389294993325064984338497741821762861178309647950171789781592031389659100}{29350684993927091294151048173432530062360916681338241427168285527718609357225763237566254722891506324397818668234577} a^{2} - \frac{1213223195906515662948119087331713718391243424952207765203315231734565055223267818314848867895605126545746014835446}{2668244090357008299468277106675684551123719698303476493378935047974419032475069385233295883899227847672528969839507} a - \frac{279058478576621707899310973253307238612006370129320381385202250759437797025936006335048056275957310544428200718696}{29350684993927091294151048173432530062360916681338241427168285527718609357225763237566254722891506324397818668234577}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}$, which has order $256$ (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:  $9$
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:  \( 157908636599000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{355}) \), \(\Q(\sqrt{71}) \), 4.4.2525.1, 4.4.203656400.1, \(\Q(\sqrt{5}, \sqrt{71})\), 8.8.41475929260960000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71Data not computed
101Data not computed