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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 81225.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81225.d1 | 81225bl2 | \([0, 0, 1, -676875, -219702344]\) | \(-102400/3\) | \(-1004778727998046875\) | \([]\) | \(1620000\) | \(2.2331\) | |
| 81225.d2 | 81225bl1 | \([0, 0, 1, 5415, 677326]\) | \(20480/243\) | \(-208350917037675\) | \([]\) | \(324000\) | \(1.4284\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 81225.d have rank \(1\).
Complex multiplication
The elliptic curves in class 81225.d do not have complex multiplication.Modular form 81225.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
