First known Mordell curves of given rank
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Numbers
$r$
$k$
0
$k > 0$:
1
0
$k < 0$:
-1
1
$k > 0$:
2
1
$k < 0$:
-2
2
$k > 0$:
15
2
$k < 0$:
-11
3
$k > 0$:
113
3
$k < 0$:
-174
4
$k > 0$:
2089
4
$k < 0$:
-2351
5
$k > 0$:
66265
5
$k < 0$:
-28279
6
$k > 0$:
1358556
6
$k < 0$:
-975379
7
$k > 0$:
47550317
7
$k < 0$:
-56877643
8
$k > 0$:
1632201497
8
$k < 0$:
-2520963512
9
$k > 0$:
185418133372
9
$k < 0$:
-463066403167
10
$k > 0$:
68513487607153
10
$k < 0$:
-56736325657288
11
$k > 0$:
35470887868736225
11
$k < 0$:
-46111487743732324
12
$k > 0$:
176415071705787247056
12
$k < 0$:
-6533891544658786928
Definition
A Mordell curve is an elliptic curve with model $E_k: y^2 = x^3 + k$, for a non-zero integer $k$. Given $r \geq 0$, this table list the smallest positive known $k$ and the largest negative known $k$ for which $E_k$ has Mordell-Weil rank $r$.
Parameters
$r$
—   rank of $E_k$ ($r \geq 0$)
(1)
Up to $\mathbb{Q}$-isomorphism, Mordell curves are exactly the ellpitic curves/$\mathbb{Q}$ with $j=0$.
References

J. Gebel, A. Petho, H.G. Zimmer, "On Mordell’s Equation", Compositio Mathematica 110 (1998), 335-367.

P. Llorente, J. Quer, "On the 3-Sylow subgroup of the class group of quadratic fields", Math. Comp. 50 (1988), 321-333.

J. Quer, "Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12", C.R. Acad Sc. Paris I 305 (1987), 215-218.

Thomas Womack, "Explicit Descent on Elliptic Curves", PhD thesis, University of Nottingham, 2003.
Reliability: For $r \leq 6$, the given $k$ are proved to be the smallest positive and largest negative $k$ with $\rm{rank}(E_k) = r$ . For $r \geq 7$, this is unknown.