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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 259920.eq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259920.eq1 | 259920eq3 | \([0, 0, 0, -134292, 18937699]\) | \(488095744/125\) | \(68592894498000\) | \([2]\) | \(995328\) | \(1.6409\) | |
| 259920.eq2 | 259920eq4 | \([0, 0, 0, -118047, 23690986]\) | \(-20720464/15625\) | \(-137185788996000000\) | \([2]\) | \(1990656\) | \(1.9875\) | |
| 259920.eq3 | 259920eq1 | \([0, 0, 0, -4332, -75449]\) | \(16384/5\) | \(2743715779920\) | \([2]\) | \(331776\) | \(1.0916\) | \(\Gamma_0(N)\)-optimal |
| 259920.eq4 | 259920eq2 | \([0, 0, 0, 11913, -507566]\) | \(21296/25\) | \(-219497262393600\) | \([2]\) | \(663552\) | \(1.4381\) |
Rank
sage: E.rank()
The elliptic curves in class 259920.eq have rank \(1\).
Complex multiplication
The elliptic curves in class 259920.eq do not have complex multiplication.Modular form 259920.2.a.eq
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
