2011-02-04-sqrt5_talk

import psage.modform.hilbert.sqrt5 as H F = H.tables.F

N = F.factor(31)[0][0] time S = H.HilbertModularForms(N); S

Time: CPU 0.29 s, Wall: 0.29 s
Hilbert modular forms of dimension 2, level 5*a-2 (of norm 31=31) over
QQ(sqrt(5))

Time: CPU 0.29 s, Wall: 0.29 s
Hilbert modular forms of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))

T2 = S.hecke_matrix(F.factor(2)[0][0]); T2

[0 5]
[3 2]

[0 5]
[3 2]

T2.charpoly().factor()

(x - 5) * (x + 3)

(x - 5) * (x + 3)

N = F.factor(100019)[0][0]; N

Fractional ideal (65*a + 292)

Fractional ideal (65*a + 292)

time S = H.HilbertModularForms(N); S

Time: CPU 0.27 s, Wall: 0.27 s
Hilbert modular forms of dimension 1667, level 65*a+292 (of norm
100019=100019) over QQ(sqrt(5))

Time: CPU 0.27 s, Wall: 0.27 s
Hilbert modular forms of dimension 1667, level 65*a+292 (of norm 100019=100019) over QQ(sqrt(5))

time T5 = S.hecke_matrix(F.factor(5)[0][0])

Time: CPU 0.13 s, Wall: 0.13 s

Time: CPU 0.13 s, Wall: 0.13 s

len(T5.nonzero_positions())/1667.0^2

0.00359460201540976

0.00359460201540976

T5.visualize_structure(maxsize=4096)

# 7 is an inert prime -- norm 49 time T7 = S.hecke_matrix(F.factor(7)[0][0])

Time: CPU 0.00 s, Wall: 0.00 s

Time: CPU 0.00 s, Wall: 0.00 s

# 11 is split time T11 = S.hecke_matrix(F.factor(11)[0][0])

Time: CPU 0.22 s, Wall: 0.22 s

Time: CPU 0.22 s, Wall: 0.22 s

# 13 is inert -- norm 169 time T13 = S.hecke_matrix(F.factor(13)[0][0])

Time: CPU 13.25 s, Wall: 13.51 s

Time: CPU 13.25 s, Wall: 13.51 s

a = F.0

E = EllipticCurve([1,a+1,a,a,0]); show(E)

\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + xy + ay = x^3 + \left(a + 1\right)x^2 + ax

\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + xy + ay = x^3 + \left(a + 1\right)x^2 + ax

E.j_invariant()

-106208/31*a + 51455/31

-106208/31*a + 51455/31

E.torsion_subgroup()

Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve
defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in
a with defining polynomial x^2 - x - 1

Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1

E.conductor()

Fractional ideal (5*a - 2)

Fractional ideal (5*a - 2)

E.local_data(E.conductor())

Local data at Fractional ideal (5*a - 2):
Reduction type: bad non-split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + x*y + a*y = x^3 +
(a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x
- 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1

Local data at Fractional ideal (5*a - 2):
Reduction type: bad non-split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1

time L = E.lseries().dokchitser(53)

Time: CPU 43.54 s, Wall: 46.52 s

Time: CPU 43.54 s, Wall: 46.52 s

L(1)

0.359928959498039

0.359928959498039

phi = F.embeddings(RR) Omega0 = E.period_lattice(phi[0]); Omega0

Period lattice associated to Elliptic Curve defined by y^2 + x*y + a*y =
x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial
x^2 - x - 1 with respect to the embedding Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Algebraic Real Field
  Defn: a |--> -0.618033988749895?

Period lattice associated to Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1 with respect to the embedding Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Algebraic Real Field
  Defn: a |--> -0.618033988749895?

Omega0.real_period()

3.05217315335726

3.05217315335726

Omega1 = E.period_lattice(phi[1]); Omega1.real_period()

8.43805988789973

8.43805988789973

( (math.sqrt(F.disc()) * L(1) * E.torsion_order()^2 ) / (Omega0.real_period()*Omega1.real_period() * E.tamagawa_product_bsd()) )

Lplot = line([(s, L(s)) for s in [.1,.15, .., 2]]); Lplot

set_verbose(-2) c = (line([(s, L(1+I*s).real()) for s in [0,0.05,..,5]]) + line([(s, L(1+I*s).imag()) for s in [0,0.05,..,5]], color='red')) c.show() c.save('critical_zero-31.pdf')

critical_zero-31.pdf

find_root(lambda s:L(1+I*s).real(), 3.5,4)

3.678991475792357

3.678991475792357

L(1 + 3.67899147*I)

-1.39082483086184e-8 - 1.11224155167065e-8*I

-1.39082483086184e-8 - 1.11224155167065e-8*I

E = EllipticCurve([0,-a-1,1,a,0]); show(E)

\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + \left(-a - 1\right)x^2 + ax

\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + \left(-a - 1\right)x^2 + ax

E.j_invariant()

-524288/199*a + 622592/199

-524288/199*a + 622592/199

E.torsion_subgroup()

Torsion Subgroup isomorphic to Z/3 associated to the Elliptic Curve
defined by y^2 + y = x^3 + (-a-1)*x^2 + a*x over Number Field in a with
defining polynomial x^2 - x - 1

Torsion Subgroup isomorphic to Z/3 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-a-1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1

E.conductor()

Fractional ideal (3*a + 13)

Fractional ideal (3*a + 13)

E.tamagawa_numbers()

[1]

[1]

E.local_data()

[Local data at Fractional ideal (3*a + 13):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + y = x^3 +
(-a-1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x
- 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1]

[Local data at Fractional ideal (3*a + 13):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + y = x^3 + (-a-1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1]

E.gens()

[(0 : 0 : 1)]

[(0 : 0 : 1)]

E.regulator_of_points(E.gens())

0.154308568543030

0.154308568543030

time L = E.lseries().dokchitser()

Time: CPU 55.21 s, Wall: 55.56 s

Time: CPU 55.21 s, Wall: 55.56 s

L(1)

L.derivative(1,1)

0.657814883009960

0.657814883009960

phi = F.embeddings(RR) Omega0 = E.period_lattice(phi[0]); O0 = Omega0.real_period() Omega1 = E.period_lattice(phi[1]); O1 = Omega1.real_period() O0, O1

(3.53489274657737, 6.06743219455559)

(3.53489274657737, 6.06743219455559)

print ( math.sqrt(F.disc()) * L.derivative(1,1) * E.torsion_order()^2 / (Omega0.real_period()*Omega1.real_period()* E.tamagawa_product_bsd() *E.regulator_of_points(E.gens())) )

4.00000000000002

4.00000000000002

E = EllipticCurve([0,-a,1,-a-1,2*a+1]); show(E)

\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + \left(-a\right)x^2 + \left(-a - 1\right)x + \left(2 a + 1\right)

\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + \left(-a\right)x^2 + \left(-a - 1\right)x + \left(2 a + 1\right)

E.conductor()

Fractional ideal (7*a + 40)

Fractional ideal (7*a + 40)

E.torsion_subgroup()

Torsion Subgroup isomorphic to Trivial group associated to the Elliptic
Curve defined by y^2 + y = x^3 + (-a)*x^2 + (-a-1)*x + (2*a+1) over
Number Field in a with defining polynomial x^2 - x - 1

Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + y = x^3 + (-a)*x^2 + (-a-1)*x + (2*a+1) over Number Field in a with defining polynomial x^2 - x - 1

E.tamagawa_numbers()

[1]

[1]

E.local_data()

[Local data at Fractional ideal (7*a + 40):
Reduction type: bad non-split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + y = x^3 + (-a)*x^2
+ (-a-1)*x + (2*a+1) over Number Field in a with defining polynomial x^2
- x - 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1]

[Local data at Fractional ideal (7*a + 40):
Reduction type: bad non-split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + y = x^3 + (-a)*x^2 + (-a-1)*x + (2*a+1) over Number Field in a with defining polynomial x^2 - x - 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1]

show(E.gens())

\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(0 : -a - 1 : 1\right), \left(-\frac{3}{4} a + \frac{1}{4} : -\frac{5}{4} a - \frac{5}{8} : 1\right)\right]

\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(0 : -a - 1 : 1\right), \left(-\frac{3}{4} a + \frac{1}{4} : -\frac{5}{4} a - \frac{5}{8} : 1\right)\right]

reg = E.regulator_of_points(E.gens()); reg

0.191946627694056

0.191946627694056

time L = E.lseries().dokchitser()

Time: CPU 205.79 s, Wall: 207.57 s

Time: CPU 205.79 s, Wall: 207.57 s

L(1)

-2.31497738102376e-20

-2.31497738102376e-20

L.derivative(1,1)

7.76867974285369e-22

7.76867974285369e-22

L.derivative(1,2)

2.88288222151816

2.88288222151816

phi = F.embeddings(RR) Omega0 = E.period_lattice(phi[0]); O0 = Omega0.real_period() Omega1 = E.period_lattice(phi[1]); O1 = Omega1.real_period() O0, O1

(3.75830925418163, 5.02645072067941)

(3.75830925418163, 5.02645072067941)

print ( math.sqrt(F.disc()) * (L.derivative(1,2)/2) * E.torsion_order()^2 / (Omega0.real_period()*Omega1.real_period()* E.tamagawa_product_bsd() *E.regulator_of_points(E.gens())) )

0.888888888888870

0.888888888888870

8/9.0

0.888888888888889

0.888888888888889

2011-02-04-sqrt5_talk_demo

476 days ago by wstein

A Bigger Example

Example Curve: Norm Conductor 31

Example Curve: Norm Conductor 199 (rank 1)

Example Curve: Norm Conductor 1831