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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 8281.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 8281.d1 | 8281g4 | \([1, -1, 1, -307950, 65849606]\) | \(16581375\) | \(194779153450063\) | \([2]\) | \(32928\) | \(1.8029\) | \(-28\) | |
| 8281.d2 | 8281g3 | \([1, -1, 1, -18115, 1158434]\) | \(-3375\) | \(-194779153450063\) | \([2]\) | \(16464\) | \(1.4563\) | \(-7\) | |
| 8281.d3 | 8281g2 | \([1, -1, 1, -6285, -190186]\) | \(16581375\) | \(1655595487\) | \([2]\) | \(4704\) | \(0.82991\) | \(-28\) | |
| 8281.d4 | 8281g1 | \([1, -1, 1, -370, -3272]\) | \(-3375\) | \(-1655595487\) | \([2]\) | \(2352\) | \(0.48334\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
Rank
sage: E.rank()
The elliptic curves in class 8281.d have rank \(0\).
Complex multiplication
Each elliptic curve in class 8281.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 8281.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
