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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 392784bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 392784.bd4 | 392784bd1 | \([0, -1, 0, 1168928, -1048686848]\) | \(366554400441263/1197281046528\) | \(-576958127484816064512\) | \([2]\) | \(18579456\) | \(2.6668\) | \(\Gamma_0(N)\)-optimal |
| 392784.bd3 | 392784bd2 | \([0, -1, 0, -11124192, -12328853760]\) | \(315922815546536017/46479778841664\) | \(22398154755862232825856\) | \([2, 2]\) | \(37158912\) | \(3.0134\) | |
| 392784.bd2 | 392784bd3 | \([0, -1, 0, -47784032, 114983438592]\) | \(25039399590518087377/2641281025170312\) | \(1272807716168753301454848\) | \([2]\) | \(74317824\) | \(3.3600\) | |
| 392784.bd1 | 392784bd4 | \([0, -1, 0, -171154272, -861768518400]\) | \(1150638118585800835537/31752757008504\) | \(15301345727666123145216\) | \([2]\) | \(74317824\) | \(3.3600\) |
Rank
sage: E.rank()
The elliptic curves in class 392784bd have rank \(1\).
Complex multiplication
The elliptic curves in class 392784bd do not have complex multiplication.Modular form 392784.2.a.bd
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.
