Following the comments, we start with the $\mathbb{Z}/2\mathbb{Z}$ curve
$$
E=[ 0, 0, 0, -15650411093524454493683423178813553233328484724599030637228,
492395613211229713687666165349208268497505680211462943972719303000171030884176146690352 ]
$$
of rank $0$ and extend it over $\mathbb{Q}(\sqrt{2})$ to obtain the $\mathbb{Z}/2\mathbb{Z}$ curve found by Elkies - Klagsbrun (2020) of rank $20$.

The use of Magma's `TwoPowerIsogenyDescentRankBound`

significantly speeds up the process.

```
SetClassGroupBounds("GRH");
E := EllipticCurve([ 0, 0, 0, -15650411093524454493683423178813553233328484724599030637228, 492395613211229713687666165349208268497505680211462943972719303000171030884176146690352 ]);
Coefficients(E);
TwoPowerIsogenyDescentRankBound(E);
QT := MinimalModel(QuadraticTwist(E, 2));
Coefficients(QT);
QT eq EllipticCurve([1,-1,1,-244537673336319601463803487168961769270757573821859853707,961710182053183034546222979258806817743270682028964434238957830989898438151121499931]);
[ 0, 0, 0, -15650411093524454493683423178813553233328484724599030637228,
4923956132112297136876661653492082684975056802114629439727193030001710308841761\
46690352 ]
0 [ 11, 5, 2, 2, 1 ]
[ 7, 5, 2, 2, 1 ]
[ 1, -1, 1, -244537673336319601463803487168961769270757573821859853707,
9617101820531830345462229792588068177432706820289644342389578309898984381511214\
99931 ]
true
```

A great collection of references on the topic is provided on Andrej Dujella's page High rank elliptic curves with prescribed torsion over quadratic fields. Some of the papers also discuss the extensions over cubic and/or quartic number fields, e.g.

J. Bosman, P. Bruin, A. Dujella and F. Najman, *Ranks of elliptic curves with prescribed torsion over number fields*, Int. Math. Res. Not. 2014 (11) (2014), 2885-2923, doi:10.1093/imrn/rnt013, arXiv:1201.0252, PDF.

D. Jeon, C. H. Kim, and Y. Lee, *Families of elliptic curves over cubic number fields with prescribed torsion subgroups*, Mathematics of Computation, Volume 80, Number 273, January 2011, pp. 579–591, doi:10.1090/S0025-5718-10-02369-0.

quicklyshow that the jump in rank could be at least $20$ starting with Elkies - Klagsbrun (2020) Z/2Z curve of rank $20$ posted on Andrej Dujella's page. I'll post the Magma code when I find the suitable quadratic extension. $\endgroup$