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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 68770d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68770.a2 | 68770d1 | \([1, 0, 1, -17204, -864798]\) | \(3803721481/26000\) | \(3848933114000\) | \([2]\) | \(285120\) | \(1.2489\) | \(\Gamma_0(N)\)-optimal |
| 68770.a3 | 68770d2 | \([1, 0, 1, -6624, -1914334]\) | \(-217081801/10562500\) | \(-1563629077562500\) | \([2]\) | \(570240\) | \(1.5955\) | |
| 68770.a1 | 68770d3 | \([1, 0, 1, -109779, 13428782]\) | \(988345570681/44994560\) | \(6660809689763840\) | \([2]\) | \(855360\) | \(1.7982\) | |
| 68770.a4 | 68770d4 | \([1, 0, 1, 59501, 51144366]\) | \(157376536199/7722894400\) | \(-1143265538157121600\) | \([2]\) | \(1710720\) | \(2.1448\) |
Rank
sage: E.rank()
The elliptic curves in class 68770d have rank \(1\).
Complex multiplication
The elliptic curves in class 68770d do not have complex multiplication.Modular form 68770.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
