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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 80688bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80688.bb3 | 80688bd1 | \([0, 1, 0, -242624, -43092684]\) | \(81182737/5904\) | \(114870744837586944\) | \([2]\) | \(967680\) | \(2.0199\) | \(\Gamma_0(N)\)-optimal |
| 80688.bb2 | 80688bd2 | \([0, 1, 0, -780544, 214033076]\) | \(2703045457/544644\) | \(10596826211267395584\) | \([2, 2]\) | \(1935360\) | \(2.3665\) | |
| 80688.bb4 | 80688bd3 | \([0, 1, 0, 1640096, 1280082932]\) | \(25076571983/50863698\) | \(-989625825618916220928\) | \([2]\) | \(3870720\) | \(2.7131\) | |
| 80688.bb1 | 80688bd4 | \([0, 1, 0, -11807904, 15612638580]\) | \(9357915116017/538002\) | \(10467596623325110272\) | \([4]\) | \(3870720\) | \(2.7131\) |
Rank
sage: E.rank()
The elliptic curves in class 80688bd have rank \(1\).
Complex multiplication
The elliptic curves in class 80688bd do not have complex multiplication.Modular form 80688.2.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.
