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SageMath
sage: E = EllipticCurve("d1")
sage: E.isogeny_class()
Elliptic curves in class 405.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 405.d1 | 405a2 | \([0, 0, 1, -162, -790]\) | \(884736/5\) | \(2657205\) | \([]\) | \(72\) | \(0.074670\) | |
| 405.d2 | 405a1 | \([0, 0, 1, -12, 15]\) | \(2359296/125\) | \(10125\) | \([3]\) | \(24\) | \(-0.47464\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 405.d have rank \(0\).
Complex multiplication
The elliptic curves in class 405.d do not have complex multiplication.Modular form 405.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
