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Properties

Label 16.0.83854078791...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 11^{6}\cdot 59^{4}$
Root discriminant $15.23$
Ramified primes $5, 11, 59$
Class number $1$
Class group Trivial
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -8, -17, 75, -20, -114, 297, -361, 305, -143, 15, 42, -26, 3, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 7*x^14 + 3*x^13 - 26*x^12 + 42*x^11 + 15*x^10 - 143*x^9 + 305*x^8 - 361*x^7 + 297*x^6 - 114*x^5 - 20*x^4 + 75*x^3 - 17*x^2 - 8*x + 19)
 
gp: K = bnfinit(x^16 - 4*x^15 + 7*x^14 + 3*x^13 - 26*x^12 + 42*x^11 + 15*x^10 - 143*x^9 + 305*x^8 - 361*x^7 + 297*x^6 - 114*x^5 - 20*x^4 + 75*x^3 - 17*x^2 - 8*x + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 7 x^{14} + 3 x^{13} - 26 x^{12} + 42 x^{11} + 15 x^{10} - 143 x^{9} + 305 x^{8} - 361 x^{7} + 297 x^{6} - 114 x^{5} - 20 x^{4} + 75 x^{3} - 17 x^{2} - 8 x + 19 \)

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:  \(8385407879109765625=5^{8}\cdot 11^{6}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.23$
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:  $5, 11, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{15} a^{14} + \frac{7}{15} a^{13} - \frac{1}{3} a^{12} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{3} a^{8} - \frac{2}{15} a^{7} - \frac{7}{15} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{2}{5} a^{3} - \frac{4}{15} a^{2} + \frac{7}{15} a - \frac{4}{15}$, $\frac{1}{2125612520565} a^{15} + \frac{27092503619}{2125612520565} a^{14} + \frac{447040797614}{2125612520565} a^{13} + \frac{61029798629}{425122504113} a^{12} - \frac{736072325416}{2125612520565} a^{11} + \frac{280439187598}{708537506855} a^{10} - \frac{14332314043}{2125612520565} a^{9} + \frac{802329483008}{2125612520565} a^{8} - \frac{90640207287}{708537506855} a^{7} - \frac{323274420368}{708537506855} a^{6} + \frac{85090844719}{425122504113} a^{5} + \frac{841660663006}{2125612520565} a^{4} + \frac{588010158563}{2125612520565} a^{3} + \frac{62927419068}{708537506855} a^{2} - \frac{13509176215}{141707501371} a + \frac{361826814347}{2125612520565}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 516.752679802 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.4461875.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59Data not computed