Here's a challenge for elliptic curve descenders/programmers. It seems no public software or public tables can determine if the rank is zero for the following curves (over rational x,y):



```
 y^2 = x^3 - 9122*x + 106889
 y^2 = x^3 - x^2 - 42144*x + 66420
 y^2 = x^3 - x^2 - 168615*x + 21827700
 y^2 = x^3 - 210386*x + 32627329

```

Can anybody definitely say if any of their ranks are zero?


By the way, these arose from Heronian triangles for a given base and height, so there are equivalent quartic forms which might be easier to analyze...for example, the rank of the first curve above is zero iff there are no rational s,t solutions to this equation:



```
 ( s^2 - t^2 )^2   =   25*( 2*( s^2 + t^2 ) - 509 )

```

Also, I have tried both Magma and Sage. Sage seems to be better at determining the rank in about 20% of similar cases. For example, "y^2 = x^3 + x^2 - 58055\*x + 4135350" has rank 0 according to Sage, but Magma only bounds the rank from 0 to 2 (limited to one minute). Anyway, these 4 cases are unsolved.



Specific Elliptic Curves: Rank
Your elliptic curves $E$ all (provably) satisfy $L(E,1) \neq 0$, so by Kolyvagin's theorem , they have rank $0$. You can prove that $L(E,1) \neq 0$ by using the command *ellanalyticrank* in Pari/GP (there are similar commands in Magma and Sage).


By the way, your first elliptic curve has conductor 223960, so is likely to be soon in Cremona's tables : <http://www.warwick.ac.uk/staff/J.E.Cremona/ftp/data/INDEX.html>


