The result of LMFDB claims (<https://www.lmfdb.org/EllipticCurve/Q/1640/c/1> )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. In particular, the order of $2$-Selmer group is larger than $16$.


But Magma calculates as following,



```
A:=EllipticCurve([0,0,0,-8747,-314874]); 
Sel2:=TwoSelmerGroup(A); Sel2;

```

The output of this is,



```
Abelian Group isomorphic to Z/2 + Z/2 + Z/2 Defined on 3 generators Relations:
    2*Sel2.1 = 0
    2*Sel2.2 = 0
    2*Sel2.3 = 0

```

It seems contradiction because $2$-Selmer group is larger than $2$-part of Tate-Shafarevich group.


Where did I go wrong and how can I correctly calculate order of $2$-Selmer group using Magma ?



Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
I don't see any contradiction: the Selmer group also has a contribution of rational points. Indeed, the group of 2-torsion rational points on this elliptic curve is isomorphic to $\mathbb Z/2\mathbb Z$ and generated by $(-54,0)$, as Magma will readily confirm (probably, I used GP/Pari).


UPDATE: As David suspected, the Tate-Shafarevic group of this elliptic curve is isomorphic to $(\mathbb Z/4\mathbb Z)^2$ so 2-descent misses the elements of order 4 in Sha.


