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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 82810bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82810.z2 | 82810bc1 | \([1, -1, 0, -7259849, -12267898707]\) | \(-2609064081/2500000\) | \(-40547284268857502500000\) | \([]\) | \(7207200\) | \(3.0343\) | \(\Gamma_0(N)\)-optimal |
| 82810.z1 | 82810bc2 | \([1, -1, 0, -637029899, 6689549044933]\) | \(-1762712152495281/171798691840\) | \(-2786388158021731912347811840\) | \([]\) | \(50450400\) | \(4.0072\) |
Rank
sage: E.rank()
The elliptic curves in class 82810bc have rank \(1\).
Complex multiplication
The elliptic curves in class 82810bc do not have complex multiplication.Modular form 82810.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
