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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 68544ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68544.bh3 | 68544ex1 | \([0, 0, 0, -1191711, 500725384]\) | \(4011705594213827392/52680152007\) | \(2457845172038592\) | \([2]\) | \(589824\) | \(2.0973\) | \(\Gamma_0(N)\)-optimal |
| 68544.bh2 | 68544ex2 | \([0, 0, 0, -1224516, 471699520]\) | \(68003243639904448/7163272192041\) | \(21389416153079353344\) | \([2, 2]\) | \(1179648\) | \(2.4439\) | |
| 68544.bh4 | 68544ex3 | \([0, 0, 0, 1584564, 2320074160]\) | \(18419405270942584/108003564029403\) | \(-2579975313078183100416\) | \([4]\) | \(2359296\) | \(2.7904\) | |
| 68544.bh1 | 68544ex4 | \([0, 0, 0, -4558476, -3234330416]\) | \(438536015880092936/64602489661101\) | \(1543216003905704067072\) | \([2]\) | \(2359296\) | \(2.7904\) |
Rank
sage: E.rank()
The elliptic curves in class 68544ex have rank \(1\).
Complex multiplication
The elliptic curves in class 68544ex do not have complex multiplication.Modular form 68544.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
