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Properties

Label 16.0.58710405163...8777.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 73^{7}$
Root discriminant $14.90$
Ramified primes $3, 73$
Class number $1$
Class group Trivial
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 43, -109, 185, -241, 283, -233, 205, -87, 97, -33, 25, -7, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 - 7*x^13 + 25*x^12 - 33*x^11 + 97*x^10 - 87*x^9 + 205*x^8 - 233*x^7 + 283*x^6 - 241*x^5 + 185*x^4 - 109*x^3 + 43*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^16 + 7*x^14 - 7*x^13 + 25*x^12 - 33*x^11 + 97*x^10 - 87*x^9 + 205*x^8 - 233*x^7 + 283*x^6 - 241*x^5 + 185*x^4 - 109*x^3 + 43*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 7 x^{14} - 7 x^{13} + 25 x^{12} - 33 x^{11} + 97 x^{10} - 87 x^{9} + 205 x^{8} - 233 x^{7} + 283 x^{6} - 241 x^{5} + 185 x^{4} - 109 x^{3} + 43 x^{2} - 10 x + 1 \)

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:  \(5871040516387428777=3^{12}\cdot 73^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.90$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{86} a^{14} - \frac{39}{86} a^{13} + \frac{17}{86} a^{12} + \frac{37}{86} a^{11} - \frac{15}{86} a^{10} + \frac{29}{86} a^{9} - \frac{41}{86} a^{8} + \frac{5}{86} a^{7} + \frac{41}{86} a^{6} - \frac{13}{86} a^{5} - \frac{15}{86} a^{4} + \frac{35}{86} a^{3} - \frac{15}{86} a^{2} - \frac{35}{86} a - \frac{7}{86}$, $\frac{1}{11037727706} a^{15} - \frac{18728671}{5518863853} a^{14} - \frac{100932973}{5518863853} a^{13} + \frac{2137226585}{5518863853} a^{12} - \frac{1277686110}{5518863853} a^{11} - \frac{221979809}{5518863853} a^{10} - \frac{1782587768}{5518863853} a^{9} + \frac{184425000}{5518863853} a^{8} + \frac{1771208756}{5518863853} a^{7} - \frac{527932783}{5518863853} a^{6} - \frac{2167426340}{5518863853} a^{5} + \frac{1920050789}{5518863853} a^{4} + \frac{2472726894}{5518863853} a^{3} - \frac{375211717}{5518863853} a^{2} + \frac{1161513770}{5518863853} a + \frac{3214527801}{11037727706}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{624081332}{128345671} a^{15} + \frac{208910181}{128345671} a^{14} + \frac{4430959902}{128345671} a^{13} - \frac{2893222129}{128345671} a^{12} + \frac{14576152993}{128345671} a^{11} - \frac{15717677993}{128345671} a^{10} + \frac{55110705098}{128345671} a^{9} - \frac{35760391060}{128345671} a^{8} + \frac{115389046944}{128345671} a^{7} - \frac{106741857420}{128345671} a^{6} + \frac{139599640800}{128345671} a^{5} - \frac{103144298623}{128345671} a^{4} + \frac{79814866837}{128345671} a^{3} - \frac{40661349514}{128345671} a^{2} + \frac{12646711503}{128345671} a - \frac{1711425666}{128345671} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1279.51275205 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.657.1, 8.0.31510377.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $16$ $16$ $16$

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
3Data not computed
$73$73.8.7.6$x^{8} + 9125$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.0.1$x^{8} - x + 40$$1$$8$$0$$C_8$$[\ ]^{8}$