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Properties

Label 16.0.77587910560...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 19^{4}\cdot 61^{2}$
Root discriminant $23.34$
Ramified primes $2, 5, 19, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2861, -5188, 9306, -13606, 19534, -21072, 17604, -10482, 4080, -438, -632, 494, -175, 18, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 + 18*x^13 - 175*x^12 + 494*x^11 - 632*x^10 - 438*x^9 + 4080*x^8 - 10482*x^7 + 17604*x^6 - 21072*x^5 + 19534*x^4 - 13606*x^3 + 9306*x^2 - 5188*x + 2861)
 
gp: K = bnfinit(x^16 - 6*x^15 + 12*x^14 + 18*x^13 - 175*x^12 + 494*x^11 - 632*x^10 - 438*x^9 + 4080*x^8 - 10482*x^7 + 17604*x^6 - 21072*x^5 + 19534*x^4 - 13606*x^3 + 9306*x^2 - 5188*x + 2861, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 12 x^{14} + 18 x^{13} - 175 x^{12} + 494 x^{11} - 632 x^{10} - 438 x^{9} + 4080 x^{8} - 10482 x^{7} + 17604 x^{6} - 21072 x^{5} + 19534 x^{4} - 13606 x^{3} + 9306 x^{2} - 5188 x + 2861 \)

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:  \(7758791056000000000000=2^{16}\cdot 5^{12}\cdot 19^{4}\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.34$
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:  $2, 5, 19, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{232130323031359398445554} a^{15} - \frac{1921296076742660998595}{232130323031359398445554} a^{14} + \frac{7416148502002598805340}{38688387171893233074259} a^{13} - \frac{12197887113475522029475}{116065161515679699222777} a^{12} + \frac{27886386760274174396726}{116065161515679699222777} a^{11} - \frac{13746420327228807236175}{77376774343786466148518} a^{10} + \frac{8790663016810423281547}{77376774343786466148518} a^{9} - \frac{32153311813775662467427}{232130323031359398445554} a^{8} + \frac{5656027503181700406241}{232130323031359398445554} a^{7} + \frac{8524823011952464358189}{116065161515679699222777} a^{6} - \frac{12044701287995467619619}{77376774343786466148518} a^{5} - \frac{71805381144714460555523}{232130323031359398445554} a^{4} - \frac{33872829205795623564653}{77376774343786466148518} a^{3} + \frac{17991015538339408911697}{116065161515679699222777} a^{2} - \frac{23085981224292358613906}{116065161515679699222777} a + \frac{116038960168975062011015}{232130323031359398445554}$

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:  \( \frac{38445933891767961922}{116065161515679699222777} a^{15} + \frac{91304958748619955908}{116065161515679699222777} a^{14} - \frac{1050740734311239030662}{116065161515679699222777} a^{13} + \frac{640120180897502021687}{38688387171893233074259} a^{12} + \frac{3577774304321975160814}{116065161515679699222777} a^{11} - \frac{25249950075965771295992}{116065161515679699222777} a^{10} + \frac{54168592004952937709866}{116065161515679699222777} a^{9} - \frac{11345311182332749533306}{38688387171893233074259} a^{8} - \frac{130515637105813418729066}{116065161515679699222777} a^{7} + \frac{497949576781061748766868}{116065161515679699222777} a^{6} - \frac{906392044307212280079136}{116065161515679699222777} a^{5} + \frac{395932798067744996322766}{38688387171893233074259} a^{4} - \frac{321847388492081003984514}{38688387171893233074259} a^{3} + \frac{789768887476272176422898}{116065161515679699222777} a^{2} - \frac{384174085701378475601108}{116065161515679699222777} a + \frac{128511364233635722045116}{38688387171893233074259} \) (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:  \( 55644.4925419 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.7600.1, 4.4.38000.1, 8.0.1444000000.5

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$