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Properties

Label 16.0.65437948737...3125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{7}$
Root discriminant $30.80$
Ramified primes $5, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![701, -1521, 7578, -20657, 34621, -38916, 32530, -21391, 11819, -5577, 2315, -823, 266, -69, 17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 17*x^14 - 69*x^13 + 266*x^12 - 823*x^11 + 2315*x^10 - 5577*x^9 + 11819*x^8 - 21391*x^7 + 32530*x^6 - 38916*x^5 + 34621*x^4 - 20657*x^3 + 7578*x^2 - 1521*x + 701)
 
gp: K = bnfinit(x^16 - 3*x^15 + 17*x^14 - 69*x^13 + 266*x^12 - 823*x^11 + 2315*x^10 - 5577*x^9 + 11819*x^8 - 21391*x^7 + 32530*x^6 - 38916*x^5 + 34621*x^4 - 20657*x^3 + 7578*x^2 - 1521*x + 701, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 17 x^{14} - 69 x^{13} + 266 x^{12} - 823 x^{11} + 2315 x^{10} - 5577 x^{9} + 11819 x^{8} - 21391 x^{7} + 32530 x^{6} - 38916 x^{5} + 34621 x^{4} - 20657 x^{3} + 7578 x^{2} - 1521 x + 701 \)

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:  \(654379487370001220703125=5^{14}\cdot 101^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.80$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{2}{9} a^{10} - \frac{4}{9} a^{9} + \frac{2}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{5465847820006030755393} a^{15} - \frac{225379892572252904411}{5465847820006030755393} a^{14} + \frac{248102231058316350646}{1821949273335343585131} a^{13} + \frac{245536044060727173488}{1821949273335343585131} a^{12} + \frac{357433959613887255458}{5465847820006030755393} a^{11} - \frac{2096772991224856884410}{5465847820006030755393} a^{10} + \frac{79399258389195935915}{607316424445114528377} a^{9} - \frac{38367200768491450763}{1821949273335343585131} a^{8} + \frac{57803668716450580118}{176317671613097766303} a^{7} - \frac{1954113075569195925170}{5465847820006030755393} a^{6} + \frac{1271942025159235259558}{5465847820006030755393} a^{5} - \frac{2715379412544356974930}{5465847820006030755393} a^{4} - \frac{446704091748504580811}{1821949273335343585131} a^{3} + \frac{995044291094786803759}{5465847820006030755393} a^{2} - \frac{2517669700122533714132}{5465847820006030755393} a + \frac{2581976248509179554477}{5465847820006030755393}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{1468871579280168038}{176317671613097766303} a^{15} - \frac{2266248528344232631}{176317671613097766303} a^{14} + \frac{2248447499912440304}{19590852401455307367} a^{13} - \frac{23642341309029810043}{58772557204365922101} a^{12} + \frac{268767563203614909076}{176317671613097766303} a^{11} - \frac{764794449959545829569}{176317671613097766303} a^{10} + \frac{691517884794978633985}{58772557204365922101} a^{9} - \frac{513524269739173977904}{19590852401455307367} a^{8} + \frac{9134923603267573574503}{176317671613097766303} a^{7} - \frac{15044371359084591359665}{176317671613097766303} a^{6} + \frac{19996946817857582532289}{176317671613097766303} a^{5} - \frac{19220043826242715227557}{176317671613097766303} a^{4} + \frac{4141749262592793061259}{58772557204365922101} a^{3} - \frac{4651879645182349864456}{176317671613097766303} a^{2} + \frac{1214391494859244690604}{176317671613097766303} a - \frac{291361345867215621487}{176317671613097766303} \) (order $10$)
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:  \( 349244.95735 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.159390625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$