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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 43120cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43120.ce1 | 43120cm1 | \([0, 1, 0, -69400, 7081108]\) | \(-76711450249/851840\) | \(-410493436559360\) | \([]\) | \(157248\) | \(1.6189\) | \(\Gamma_0(N)\)-optimal |
43120.ce2 | 43120cm2 | \([0, 1, 0, 232440, 36902900]\) | \(2882081488391/2883584000\) | \(-1389571170369536000\) | \([]\) | \(471744\) | \(2.1682\) |
Rank
sage: E.rank()
The elliptic curves in class 43120cm have rank \(0\).
Complex multiplication
The elliptic curves in class 43120cm do not have complex multiplication.Modular form 43120.2.a.cm
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.