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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 80275a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80275.m3 | 80275a1 | \([0, -1, 1, 2817, -3482]\) | \(32768/19\) | \(-1432958921875\) | \([]\) | \(77760\) | \(1.0220\) | \(\Gamma_0(N)\)-optimal |
| 80275.m2 | 80275a2 | \([0, -1, 1, -39433, -3193357]\) | \(-89915392/6859\) | \(-517298170796875\) | \([]\) | \(233280\) | \(1.5713\) | |
| 80275.m1 | 80275a3 | \([0, -1, 1, -3250433, -2254505732]\) | \(-50357871050752/19\) | \(-1432958921875\) | \([]\) | \(699840\) | \(2.1206\) |
Rank
sage: E.rank()
The elliptic curves in class 80275a have rank \(1\).
Complex multiplication
The elliptic curves in class 80275a do not have complex multiplication.Modular form 80275.2.a.a
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
