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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 8712u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8712.y2 | 8712u1 | \([0, 0, 0, -1002243, -384911890]\) | \(63253004/243\) | \(427728240248398848\) | \([2]\) | \(253440\) | \(2.2415\) | \(\Gamma_0(N)\)-optimal |
| 8712.y1 | 8712u2 | \([0, 0, 0, -1481403, 20936630]\) | \(102129622/59049\) | \(207875924760721840128\) | \([2]\) | \(506880\) | \(2.5881\) |
Rank
sage: E.rank()
The elliptic curves in class 8712u have rank \(1\).
Complex multiplication
The elliptic curves in class 8712u do not have complex multiplication.Modular form 8712.2.a.u
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
