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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 47040ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.ba3 | 47040ef1 | \([0, -1, 0, -5486301, -4943406915]\) | \(151591373397612544/32558203125\) | \(3922370600400000000\) | \([2]\) | \(1474560\) | \(2.5626\) | \(\Gamma_0(N)\)-optimal |
| 47040.ba2 | 47040ef2 | \([0, -1, 0, -6098801, -3770714415]\) | \(13015144447800784/4341909875625\) | \(8369297079622133760000\) | \([2, 2]\) | \(2949120\) | \(2.9092\) | |
| 47040.ba4 | 47040ef3 | \([0, -1, 0, 17715199, -26027278815]\) | \(79743193254623804/84085819746075\) | \(-648322259832404552908800\) | \([2]\) | \(5898240\) | \(3.2558\) | |
| 47040.ba1 | 47040ef4 | \([0, -1, 0, -39712801, 93528369985]\) | \(898353183174324196/29899176238575\) | \(230529970031303732428800\) | \([2]\) | \(5898240\) | \(3.2558\) |
Rank
sage: E.rank()
The elliptic curves in class 47040ef have rank \(1\).
Complex multiplication
The elliptic curves in class 47040ef do not have complex multiplication.Modular form 47040.2.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.
