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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 122304ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122304.ij3 | 122304ek1 | \([0, 1, 0, 42271, -31794561]\) | \(270840023/14329224\) | \(-441927303004422144\) | \([]\) | \(1990656\) | \(2.0655\) | \(\Gamma_0(N)\)-optimal |
| 122304.ij2 | 122304ek2 | \([0, 1, 0, -381089, 866998719]\) | \(-198461344537/10417365504\) | \(-321281755494507085824\) | \([]\) | \(5971968\) | \(2.6148\) | |
| 122304.ij1 | 122304ek3 | \([0, 1, 0, -81713249, 284283892479]\) | \(-1956469094246217097/36641439744\) | \(-1130057890382965899264\) | \([]\) | \(17915904\) | \(3.1641\) |
Rank
sage: E.rank()
The elliptic curves in class 122304ek have rank \(0\).
Complex multiplication
The elliptic curves in class 122304ek do not have complex multiplication.Modular form 122304.2.a.ek
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
