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Properties

Label 16.0.31713911056...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 211^{4}$
Root discriminant $25.49$
Ramified primes $2, 5, 211$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1027

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7621, -386, 12298, -2118, 12178, -2518, 7356, -914, 2558, 18, 534, 44, 83, -4, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 4*x^13 + 83*x^12 + 44*x^11 + 534*x^10 + 18*x^9 + 2558*x^8 - 914*x^7 + 7356*x^6 - 2518*x^5 + 12178*x^4 - 2118*x^3 + 12298*x^2 - 386*x + 7621)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 4*x^13 + 83*x^12 + 44*x^11 + 534*x^10 + 18*x^9 + 2558*x^8 - 914*x^7 + 7356*x^6 - 2518*x^5 + 12178*x^4 - 2118*x^3 + 12298*x^2 - 386*x + 7621, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 4 x^{13} + 83 x^{12} + 44 x^{11} + 534 x^{10} + 18 x^{9} + 2558 x^{8} - 914 x^{7} + 7356 x^{6} - 2518 x^{5} + 12178 x^{4} - 2118 x^{3} + 12298 x^{2} - 386 x + 7621 \)

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:  \(31713911056000000000000=2^{16}\cdot 5^{12}\cdot 211^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.49$
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:  $2, 5, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} + \frac{5}{19} a^{12} - \frac{1}{19} a^{11} + \frac{8}{19} a^{10} + \frac{9}{19} a^{8} + \frac{9}{19} a^{7} - \frac{9}{19} a^{6} + \frac{5}{19} a^{5} + \frac{3}{19} a^{4} - \frac{1}{19} a^{3} - \frac{6}{19} a^{2} + \frac{3}{19} a + \frac{3}{19}$, $\frac{1}{361} a^{14} + \frac{9}{361} a^{13} + \frac{9}{19} a^{12} - \frac{110}{361} a^{11} + \frac{108}{361} a^{10} + \frac{47}{361} a^{9} - \frac{50}{361} a^{8} + \frac{122}{361} a^{7} + \frac{178}{361} a^{6} + \frac{61}{361} a^{5} - \frac{65}{361} a^{4} + \frac{66}{361} a^{3} - \frac{21}{361} a^{2} + \frac{167}{361} a - \frac{121}{361}$, $\frac{1}{425552519985524764253201} a^{15} + \frac{46798909883400204388}{425552519985524764253201} a^{14} - \frac{5238350356004751241230}{425552519985524764253201} a^{13} - \frac{110450495400558006529278}{425552519985524764253201} a^{12} - \frac{80700054811414181867342}{425552519985524764253201} a^{11} + \frac{193941344487354352211952}{425552519985524764253201} a^{10} - \frac{144204866502929738693901}{425552519985524764253201} a^{9} + \frac{112411890342513446601608}{425552519985524764253201} a^{8} - \frac{162401138698541600801237}{425552519985524764253201} a^{7} + \frac{116538877339245406105959}{425552519985524764253201} a^{6} - \frac{152339669590430804937951}{425552519985524764253201} a^{5} - \frac{35147529834983203045421}{425552519985524764253201} a^{4} - \frac{58748434190548579073019}{425552519985524764253201} a^{3} + \frac{160440480386301146397103}{425552519985524764253201} a^{2} + \frac{3249049227744535157528}{425552519985524764253201} a - \frac{170396728178521181925239}{425552519985524764253201}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{28704915579745508904}{425552519985524764253201} a^{15} - \frac{417067178815543714632}{425552519985524764253201} a^{14} + \frac{579664893122859297608}{425552519985524764253201} a^{13} - \frac{4548234288592611093736}{425552519985524764253201} a^{12} - \frac{2687303102444181936864}{425552519985524764253201} a^{11} - \frac{30322976787747768427860}{425552519985524764253201} a^{10} - \frac{39104140816801814036908}{425552519985524764253201} a^{9} - \frac{190372643490836273157825}{425552519985524764253201} a^{8} - \frac{43588599916851587665160}{425552519985524764253201} a^{7} - \frac{775727968186984339040168}{425552519985524764253201} a^{6} + \frac{101126513155806603756080}{425552519985524764253201} a^{5} - \frac{1589187228503776158697862}{425552519985524764253201} a^{4} - \frac{154067579857752533146844}{425552519985524764253201} a^{3} - \frac{1712521165597581064774464}{425552519985524764253201} a^{2} - \frac{456641606952406708733928}{425552519985524764253201} a - \frac{1357263052733646621969005}{425552519985524764253201} \) (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:  \( 93585.6905262 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1027:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 28 conjugacy class representatives for t16n1027
Character table for t16n1027 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.8.178084000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
211Data not computed