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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 8112.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.w1 | 8112bd2 | \([0, 1, 0, -202856, 195456756]\) | \(-276301129/4782969\) | \(-15981014019648098304\) | \([]\) | \(139776\) | \(2.3660\) | |
| 8112.w2 | 8112bd1 | \([0, 1, 0, -27096, -1745964]\) | \(-658489/9\) | \(-30071097298944\) | \([]\) | \(19968\) | \(1.3930\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8112.w have rank \(1\).
Complex multiplication
The elliptic curves in class 8112.w do not have complex multiplication.Modular form 8112.2.a.w
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
