Rational isogenies of prime degree $p\in\{11,17,19,37,43,67,163\}$

Question

Let $p\in\{11,17,19,37,43,67,163\}$ be a prime number. In [1], B. Mazur proves that there are only finite number of elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$.

Here is my question:

Is there a list of the $j$-invariant $j(E)$ for all the elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$?

I am interested in the case where $j(E)$ is not an integer, or $j(E)$ is of the form $2^a3^b$, so we can only consider my question in this special case.

Many thanks.

[1]: Mazur, B. "Rational isogenies of prime degree", Inventiones Mathematicae, 1978

Answer 1

Five of these $j(E)$ values, one each for $p=11,19,43,67,163$, are for curves $E$ with complex multiplication by the ring of integers in the quadratic field of discriminant $-p$. All of them are integral, and none is of the form $2^a 3^b$ though two are $-2^a 3^b$, for $p=11$ (with $j = -2^{15}$) and $p=19$ (with $j = -2^{15} 3^3$).

There is another $11$-isogenous pair of rational $j$-invariants, but those are $-11^2$ and $-11 \cdot 131^3$, so not within your scope; see https://www.lmfdb.org/EllipticCurve/Q/121/a (121.b is the CM case).

For $p=17$ these are the curves in LMFDB isogeny class 14450.b https://www.lmfdb.org/EllipticCurve/Q/14450/b, with $j=-297756989/2 = -17^2 101^3/2$ and
$-882216989/131072 = -17 \cdot 373^3 / 2^{17}$ (each with with several quadratic twists of the same conductor).

Finally, for $p=37$ the $j$-invariants are integral and not $\{2,3\}$-smooth: $9317 = -7 \cdot 11^3$ and $-162677523113838677 = -7 \cdot 137^3 2083^3$, as in isogeny class 1225.b https://www.lmfdb.org/EllipticCurve/Q/1225/b.

All this has been known for some 50 years(!), see the "Remarks on isogenies" in Modular Forms in One Variable IV, Springer LNM 476 (1975).