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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 52416ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52416.j2 | 52416ex1 | \([0, 0, 0, -264, 1654]\) | \(-43614208/91\) | \(-4245696\) | \([]\) | \(13824\) | \(0.15737\) | \(\Gamma_0(N)\)-optimal |
| 52416.j3 | 52416ex2 | \([0, 0, 0, 456, 8206]\) | \(224755712/753571\) | \(-35158608576\) | \([]\) | \(41472\) | \(0.70668\) | |
| 52416.j1 | 52416ex3 | \([0, 0, 0, -4224, -260426]\) | \(-178643795968/524596891\) | \(-24475592546496\) | \([]\) | \(124416\) | \(1.2560\) |
Rank
sage: E.rank()
The elliptic curves in class 52416ex have rank \(1\).
Complex multiplication
The elliptic curves in class 52416ex do not have complex multiplication.Modular form 52416.2.a.ex
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
