Modular symbols using eclib newforms

class sage.libs.eclib.newforms.ECModularSymbol

Bases: object

Modular symbol associated with an elliptic curve, using John Cremona’s newforms class.

EXAMPLES:

sage: from sage.libs.eclib.newforms import ECModularSymbol
sage: E = EllipticCurve('11a')
sage: M = ECModularSymbol(E,1); M
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field

By default, symbols are based at the cusp \(\infty\), i.e. we evaluate \(\{\infty,r\}\):

sage: [M(1/i) for i in range(1,11)]
[2/5, -8/5, -3/5, 7/5, 12/5, 12/5, 7/5, -3/5, -8/5, 2/5]

We can also switch the base point to the cusp \(0\):

sage: [M(1/i, base_at_infinity=False) for i in range(1,11)]
[0, -2, -1, 1, 2, 2, 1, -1, -2, 0]

For the minus symbols this makes no difference since \(\{0,\infty\}\) is in the plus space. Note that to evaluate minus symbols the space must be defined with sign 0, which makes both signs available:

sage: M = ECModularSymbol(E,0); M
Modular symbol with sign 0 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: [M(1/i, -1) for i in range(1,11)]
[0, 0, 1, 1, 0, 0, -1, -1, 0, 0]
sage: [M(1/i, -1, base_at_infinity=False) for i in range(1,11)]
[0, 0, 1, 1, 0, 0, -1, -1, 0, 0]

If the ECModularSymbol is created with sign 0 then as well as asking for both + and - symbols, we can also obtain both (as a tuple). However it is more work to create the full modular symbol space:

sage: E = EllipticCurve('11a1')
sage: M = ECModularSymbol(E,0); M
Modular symbol with sign 0 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: [M(1/i) for i in range(2,11)]
[[-8/5, 0],
 [-3/5, 1],
 [7/5, 1],
 [12/5, 0],
 [12/5, 0],
 [7/5, -1],
 [-3/5, -1],
 [-8/5, 0],
 [2/5, 0]]

The curve is automatically converted to its minimal model:

sage: E = EllipticCurve([0,0,0,0,1/4])
sage: ECModularSymbol(E)
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 over Rational Field

Non-optimal curves are handled correctly in eclib, by comparing the ratios of real and/or imaginary periods:

sage: from sage.libs.eclib.newforms import ECModularSymbol
sage: E1 = EllipticCurve('11a1') # optimal
sage: E1.period_lattice().basis()
(1.26920930427955, 0.634604652139777 + 1.45881661693850*I)
sage: M1 = ECModularSymbol(E1,0)
sage: M1(0)
[2/5, 0]
sage: M1(1/3)
[-3/5, 1]

One non-optimal curve has real period 1/5 that of the optimal one, so plus symbols scale up by a factor of 5 while minus symbols are unchanged:

sage: E2 = EllipticCurve('11a2') # not optimal
sage: E2.period_lattice().basis()
(0.253841860855911, 0.126920930427955 + 1.45881661693850*I)
sage: M2 = ECModularSymbol(E2,0)
sage: M2(0)
[2, 0]
sage: M2(1/3)
[-3, 1]
sage: all((M2(r,1)==5*M1(r,1)) for r in QQ.range_by_height(10))
True
sage: all((M2(r,-1)==M1(r,-1)) for r in QQ.range_by_height(10))
True

The other non-optimal curve has real period 5 times that of the optimal one, so plus symbols scale down by a factor of 5; again, minus symbols are unchanged:

sage: E3 = EllipticCurve('11a3') # not optimal
sage: E3.period_lattice().basis()
(6.34604652139777, 3.17302326069888 + 1.45881661693850*I)
sage: M3 = ECModularSymbol(E3,0)
sage: M3(0)
[2/25, 0]
sage: M3(1/3)
[-3/25, 1]
sage: all((5*M3(r,1)==M1(r,1)) for r in QQ.range_by_height(10))
True
sage: all((M3(r,-1)==M1(r,-1)) for r in QQ.range_by_height(10))
True