by catid posted (>30 days ago) 12:34am Fri. Oct 11th 2013 PDT
Here's my MAGMA script (first I've ever written, and all from scratch, and it worked! :) to test the claim of http://www.chesworkshop.org/ches2010/presentations/CHES2010_Session02_Talk03.pdf that the curve they specify for "ted1271gls" on slide 20 has secure group order (though not on its twist):

p := 2^127-1;
K<w> := GF(p^2);

mu := 2 + w;
aa := -mu;
dd := 109*mu;

A2 := 2 * (aa + dd) / (aa - dd);
A4 := 4 / (aa - dd);
A := 1 / (A4 * A4) - A2 * A2 / (3 * A4 * A4);
B := -A2 * A2 * A2 / (27 * A4 * A4 * A4) - A2 * A / (3 * A4);

E := EllipticCurve([K | A, B]);
ETwists := Twists(E);
print ETwists;
time [ SEA(F : MaxSmooth := 1) : F in ETwists ];

EP := \$1[1];
TP := \$1[2];

if EP mod 8 eq 0 then
TEST := IsPrime(EP div 8);
else
if EP mod 4 eq 0 then
TEST := IsPrime(EP div 4);
else
if EP mod 2 eq 0 then
TEST := IsPrime(EP div 2);
else
TEST := IsPrime(EP);
end if;
end if;
end if;

if TEST eq true then
print "TWISTED EDWARDS CURVE IS GOOD";
else
print "TWISTED EDWARDS CURVE SUCKS";
end if;

Factorization(EP):Hex;

if TP mod 8 eq 0 then
TTEST := IsPrime(TP div 8);
else
if TP mod 4 eq 0 then
TTEST := IsPrime(TP div 4);
else
if TP mod 2 eq 0 then
TTEST := IsPrime(TP div 2);
else
TTEST := IsPrime(TP);
end if;
end if;
end if;

if TTEST eq true then
else
end if;

Factorization(TP):Hex;

You can plug it into the Wolfram Alpha-like web calculator for free!

http://magma.maths.usyd.edu.au/calc/

The result?

[
Elliptic Curve defined by y^2 = x^3 + (170141183460469231731687303715884104\
864*w + 127605887595351923798765477786913078648)*x +
(85070591730234615865843651857942031430*w +
170141183460469231731687303715884101830) over
GF(170141183460469231731687303715884105727^2),
Elliptic Curve defined by y^2 = x^3 + (108862929589523551023699670699203009\
961*w + 21468635481913102246467482002211501024)*x +
(17218003551755020253627573229434217032*w +
98219829738602315410342339074910081680) over
GF(170141183460469231731687303715884105727^2)
]
[ 28948022309329048855892746252171976962839764946219840790663900086538002237076\
, 28948022309329048855892746252171976963114662652758564302138142702555026159984
]
Time: 0.700
TWISTED EDWARDS CURVE IS GOOD
[ <0x2, 0x2>, <0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFA6261414C0DC87D3CE9B68E3B09E01A\
5, 0x1> ]