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Clarify genus 2 minimal/simplified model snippets #21

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16 changes: 10 additions & 6 deletions lmfdb/genus2_curves/code.yaml
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Original file line number Diff line number Diff line change
Expand Up @@ -21,33 +21,37 @@ frontmatter:
{lang} code for working with genus 2 curve {label}.

curve:
comment: Define the curve
comment: Define the curve (minimal equation)
magma: |
R<x> := PolynomialRing(Rationals());
fh := %s;
f := R![a : a in fh[1]];
h := R![a : a in fh[2]];
C := HyperellipticCurve(f, h);
Cmin := HyperellipticCurve(f, h);

simple_curve:
comment: Simplified equation
magma: Csim := SimplifiedModel(Cmin);

aut:
comment: Automorphism group
magma: AutomorphismGroup(C);
magma: AutomorphismGroup(Cmin);

jacobian:
comment: Jacobian
magma: J := Jacobian(SimplifiedModel(C));
magma: J := Jacobian(Csim);

tors:
comment: Torsion subgroup
magma: TorsionSubgroup(J);

cond:
comment: Conductor
magma: Conductor(LSeries(C));
magma: Conductor(LSeries(Cmin));

disc:
comment: Discriminant
magma: Discriminant(C);
magma: Discriminant(Cmin);

ntors:
comment: Torsion order of Jacobian
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2 changes: 1 addition & 1 deletion lmfdb/genus2_curves/main.py
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Expand Up @@ -926,7 +926,7 @@ def labels_page():
learnmore=learnmore_list_remove("labels"),
)

sorted_code_names = ['curve', 'aut', 'jacobian', 'tors', 'cond', 'disc', 'ntors', 'mwgroup']
sorted_code_names = ['curve', 'simple_curve', 'aut', 'jacobian', 'tors', 'cond', 'disc', 'ntors', 'mwgroup']

Comment = {'magma': '//', 'sage': '#', 'gp': '\\\\', 'pari': '\\\\'}

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65 changes: 36 additions & 29 deletions lmfdb/genus2_curves/web_g2c.py
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Expand Up @@ -67,27 +67,33 @@ def simplify_hyperelliptic(fh):
return f.coefficients(sparse=False)


def simplify_hyperelliptic_point(fh, pt):
def simplify_hyperelliptic_scale(fh):
# The simplified model is y^2 = g with g = squarefree_part(n)*(4f+h^2)/n,
# where n is the content of 4f+h^2, so the y-coordinate of a point on the
# minimal model transforms as y -> (2y+h)/s, where s = sqrt(n/squarefree_part(n))
xR = PolynomialRing(QQ, 'x')
f = 4*xR(fh[0]) + xR(fh[1])**2
f1 = xR(fh[1])
n = gcd(f.coefficients())
n = ZZ(gcd(f.coefficients()))
return (n // integer_squarefree_part(n)).isqrt()


def simplify_hyperelliptic_point(fh, pt):
s = simplify_hyperelliptic_scale(fh)
f1 = PolynomialRing(QQ, 'x')(fh[1])
xzR = PolynomialRing(QQ,['x', 'z'])
z = xzR('z')
f1 = (xzR(f1)*z**(4-len(fh[1]))).homogenize(z)
return [pt[0], (2*pt[1] + f1([pt[0],pt[2]])) / n, pt[2]]
return [pt[0], (2*pt[1] + f1([pt[0],pt[2]])) / s, pt[2]]


def comp_poly(fh):
xR = PolynomialRing(QQ, 'x')
f = 4*xR(fh[0]) + xR(fh[1])**2
f1 = xR(fh[1])
n = gcd(f.coefficients())
s = simplify_hyperelliptic_scale(fh)
f1 = PolynomialRing(QQ, 'x')(fh[1])
xyzR = PolynomialRing(QQ,['x', 'y', 'z'])
y = xyzR('y')
z = xyzR('z')
f1 = (xyzR(f1)*z**(4-len(fh[1]))).homogenize(z)
return (2*y + f1) / n
return (2*y + f1) / s


def min_eqns_pretty(fh):
Expand Down Expand Up @@ -1120,34 +1126,35 @@ def make_object(self, curve, endo, tama, ratpts, clus, galrep, nonsurj, is_curve
code['show'] = {'sage':'', 'magma':''} # use default show names
f,h = fh = data['min_eqn']
g = simplify_hyperelliptic(fh)
code['curve'] = {'sage':'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s));' % (f, h),
'magma':'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);' % (f, h) }
code['simple_curve'] = {'sage':'X = HyperellipticCurve(R(%s))' % (g), 'magma':'X,pi:= SimplifiedModel(C);' }
code['curve'] = {'sage':'R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R(%s), R(%s)) # minimal equation' % (f, h),
'magma':'R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R!%s, R!%s); // minimal equation' % (f, h) }
code['simple_curve'] = {'sage':'Csim = HyperellipticCurve(R(%s)); Csim # simplified equation' % (g),
'magma':'Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation' }
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation('+str(data['cond'])+',2),R!'+str(bad2)+'>*]'
else:
magma_cond_option = ''
code['cond'] = {'magma': 'Conductor(LSeries(C%s)); Factorization($1);' % magma_cond_option}
code['disc'] = {'magma':'Discriminant(C); Factorization(Integers()!$1);'}
code['geom_inv'] = {'sage':'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':'IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);'}
code['aut'] = {'magma':'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {'magma':'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'}
code['num_rat_wpts'] = {'magma':'#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'}
code['cond'] = {'magma': 'Conductor(LSeries(Cmin%s)); Factorization($1);' % magma_cond_option}
code['disc'] = {'magma':'Discriminant(Cmin); Factorization(Integers()!$1);'}
code['geom_inv'] = {'sage':'Cmin.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':'IgusaClebschInvariants(Cmin); IgusaInvariants(Cmin); G2Invariants(Cmin);'}
code['aut'] = {'magma':'AutomorphismGroup(Cmin); IdentifyGroup($1);'}
code['autQbar'] = {'magma':'AutomorphismGroup(ChangeRing(Cmin,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'}
code['num_rat_wpts'] = {'magma':'#Roots(HyperellipticPolynomials(SimplifiedModel(Cmin)));'}
if ratpts:
code['rat_pts'] = {'magma': '[' + ','.join("C![%s,%s,%s]" % (p[0], p[1], p[2]) for p in ratpts['rat_pts']) + ']; // minimal model'}
code['rat_pts_simp'] = {'magma': '[' + ','.join(["C![%s,%s,%s]" % (p[0], p[1], p[2]) for p in [simplify_hyperelliptic_point(data['min_eqn'], pt) for pt in ratpts['rat_pts']]]) + ']; // simplified model'}
code['mw_group'] = {'magma':'MordellWeilGroupGenus2(Jacobian(C));'}
code['two_selmer'] = {'magma':'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'}
code['has_square_sha'] = {'magma':'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {'magma':'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'}
code['torsion_subgroup'] = {'magma':'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'}
code['decomp'] = {'magma':'HeuristicDecompositionFactors(C);'}
code['rat_pts'] = {'magma': '[' + ','.join("Cmin![%s,%s,%s]" % (p[0], p[1], p[2]) for p in ratpts['rat_pts']) + ']; // minimal model'}
code['rat_pts_simp'] = {'magma': '[' + ','.join(["Csim![%s,%s,%s]" % (p[0], p[1], p[2]) for p in [simplify_hyperelliptic_point(data['min_eqn'], pt) for pt in ratpts['rat_pts']]]) + ']; // simplified model'}
code['mw_group'] = {'magma':'MordellWeilGroupGenus2(Jacobian(Cmin));'}
code['two_selmer'] = {'magma':'TwoSelmerGroup(Jacobian(Cmin)); NumberOfGenerators($1);'}
code['has_square_sha'] = {'magma':'HasSquareSha(Jacobian(Cmin));'}
code['locally_solvable'] = {'magma':'f,h:=HyperellipticPolynomials(Cmin); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'}
code['torsion_subgroup'] = {'magma':'TorsionSubgroup(Jacobian(SimplifiedModel(Cmin))); AbelianInvariants($1);'}
code['decomp'] = {'magma':'HeuristicDecompositionFactors(Cmin);'}
code['endos0'] = {'magma':'//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code'}
code['endos1'] = {'magma':'HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);'}
code['endos2'] = {'magma':'HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);'}
code['endos1'] = {'magma':'HeuristicIsGL2(Cmin); HeuristicEndomorphismDescription(Cmin); HeuristicEndomorphismFieldOfDefinition(Cmin);'}
code['endos2'] = {'magma':'HeuristicIsGL2(Cmin : Geometric := true); HeuristicEndomorphismDescription(Cmin : Geometric := true); HeuristicEndomorphismLatticeDescription(Cmin);'}

self._code = None

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