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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 122304ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122304.fr3 | 122304ef1 | \([0, 1, 0, -177249, 27870975]\) | \(19968681097/628992\) | \(19398729349988352\) | \([2]\) | \(884736\) | \(1.9002\) | \(\Gamma_0(N)\)-optimal |
| 122304.fr2 | 122304ef2 | \([0, 1, 0, -428129, -68918529]\) | \(281397674377/96589584\) | \(2978917375807586304\) | \([2, 2]\) | \(1769472\) | \(2.2467\) | |
| 122304.fr4 | 122304ef3 | \([0, 1, 0, 1265311, -477714945]\) | \(7264187703863/7406095788\) | \(-228411248046876131328\) | \([2]\) | \(3538944\) | \(2.5933\) | |
| 122304.fr1 | 122304ef4 | \([0, 1, 0, -6135649, -5850636289]\) | \(828279937799497/193444524\) | \(5966018590559698944\) | \([2]\) | \(3538944\) | \(2.5933\) |
Rank
sage: E.rank()
The elliptic curves in class 122304ef have rank \(0\).
Complex multiplication
The elliptic curves in class 122304ef do not have complex multiplication.Modular form 122304.2.a.ef
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
