= L.absolute_field(); L1
///
Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
}}}
{{{id=43|
E2 = E.change_ring(L1); E2
///
Elliptic Curve defined by y^2 = x^3 + 18*x + (-9) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
}}}
{{{id=44|
E2.local_data()
///
[Local data at Fractional ideal (1/8*b1^3 + 1/4*b1^2 - 1/2*b1):
Reduction type: bad additive
Local minimal model: Elliptic Curve defined by y^2 + 1/4*b1^3*y = x^3 + 3/8*b1^3*x^2 + 15*x + (17/8*b1^3-8) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
Minimal discriminant valuation: 8
Conductor exponent: 3
Kodaira Symbol: I1*
Tamagawa Number: 4, Local data at Fractional ideal (-1/4*b1^2 - 1):
Reduction type: bad additive
Local minimal model: Elliptic Curve defined by y^2 = x^3 + 18*x + (-9) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
Minimal discriminant valuation: 12
Conductor exponent: 6
Kodaira Symbol: IV*
Tamagawa Number: 3, Local data at Fractional ideal (-1/2*b1^2 + 1/2*b1 + 2):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 = x^3 + 18*x + (-9) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1, Local data at Fractional ideal (1/2*b1^2 + 1/2*b1 - 2):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 = x^3 + 18*x + (-9) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1, Local data at Fractional ideal (-3/4*b1^2 + 1):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 = x^3 + 18*x + (-9) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1, Local data at Fractional ideal (3/4*b1^2 - 2):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 = x^3 + 18*x + (-9) over Number Field in b1 with defining polynomial x^4 - 4*x^2 + 16
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1]
}}}
3. local_data
In $p = 2, d = 6$ computations, local_data sometimes crashes during Tate's algorithm. (ticket #9389)
To-Do List
1. Speed
Currently, the speed of the code is mostly restricted by calculations done in local_data and hilbert_symbol.
{{{id=26|
for E in cremona_curves([42, 84]):
time root_number(E)
///
1
Time: CPU 0.02 s, Wall: 0.02 s
1
Time: CPU 0.01 s, Wall: 0.01 s
1
Time: CPU 0.02 s, Wall: 0.01 s
1
Time: CPU 0.01 s, Wall: 0.01 s
1
Time: CPU 0.01 s, Wall: 0.02 s
1
Time: CPU 0.00 s, Wall: 0.02 s
1
Time: CPU 0.37 s, Wall: 0.38 s
1
Time: CPU 0.86 s, Wall: 0.87 s
1
Time: CPU 0.42 s, Wall: 0.42 s
1
Time: CPU 0.90 s, Wall: 0.90 s
1
Time: CPU 0.39 s, Wall: 0.39 s
1
Time: CPU 0.85 s, Wall: 0.85 s
}}}
But once these have been optimized, the speed of our code should be of no concern.
2. $p = 2$
REALLY messed up. All parts ($d = 1$, $2$, $3$, and $6$) of $p = 2$ still need work!
On to Adam and $L$-functions ...
{{{id=37|
H. = NumberField(x^2 - 2); D = EllipticCurve(H, [0,2,0, 2*u +4, 2*u + 3]); D
print root_number(D)
L. = NumberField(x^2 - 2); F = EllipticCurve(L, [0,2,0, 2*s +4, 2*s + 3]); F
print root_number(F)
J. = NumberField(x^2 + 1); G = EllipticCurve(J, [0, 1, 0, -2*r - 2, 2*r]); G
print root_number(G)
K. = NumberField(x^2 -2); E = EllipticCurve(K, [t,0,1,0,0]); E
print root_number(E)
M. = NumberField(x^2 +1); A = EllipticCurve(M, [s,0,s,0,0]); A
print root_number(A)
///
1
1
1
1
1
}}}
{{{id=47|
///
}}}