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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 21294bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21294.u2 | 21294bc1 | \([1, -1, 0, -792, 1080]\) | \(2640625/1512\) | \(31481305128\) | \([]\) | \(16128\) | \(0.70410\) | \(\Gamma_0(N)\)-optimal |
| 21294.u1 | 21294bc2 | \([1, -1, 0, -46422, 3861378]\) | \(531373116625/2058\) | \(42849554202\) | \([3]\) | \(48384\) | \(1.2534\) |
Rank
sage: E.rank()
The elliptic curves in class 21294bc have rank \(1\).
Complex multiplication
The elliptic curves in class 21294bc do not have complex multiplication.Modular form 21294.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
