> E:=EllipticCurve(CremonaDatabase(),"4862D");
> E;
Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 10730944*x - 
13560247581 over Rational Field
> factor(Discriminant(E));
[ <2, 17>, <11, 5>, <13, 7>, <17, 2> ]
> modular_degree:=247520;  // database lookup
> factor($1);
[ <2, 1>, <11, 1>, <13, 1> ]  // possible level-lowering congruences. 
> // Thus the mod 2, mod 11, and mod 13 representations have a reasonable
> // a chance to be unramified at a prime in {2,11,13,17}.
> // Here's an example:
> WeierstrassModel(E);
Elliptic Curve defined by y^2 = x^3 - 13907303019*x - 632792076853914 
> K:=NumberField(x^3 - 13907303019*x - 632792076853914 );
> factor(Discriminant(K));
[ <2, 3>, <11, 1>, <13, 1> ]
> // So, the mod 2 representations in unramified at 17 -- this means we
> // should be able to knock 17 out of the level.

Yep, 286B satisfies a mod-2 congruence.

>>>> The real monster is to lower the level to level 2431.  

At level 2431, I computed mod 17 and, at least for the first few Fourier coefficient,
*did* find a form that is congruent to the newform attached to E.   I didn't bother
to decompose this space.   The mod 17 congruence is not with a Q-rational newform. 

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There are no other congruences because the other geometric component
groups are divisible by 5 and 7, which have nothing to do with the
level.