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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 21294k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21294.x1 | 21294k1 | \([1, -1, 0, -135147, 20098053]\) | \(-354003515818875/20661046784\) | \(-15932704244341248\) | \([3]\) | \(233280\) | \(1.8651\) | \(\Gamma_0(N)\)-optimal |
| 21294.x2 | 21294k2 | \([1, -1, 0, 734358, 34995572]\) | \(77908020328125/46036680704\) | \(-25880264148623818752\) | \([]\) | \(699840\) | \(2.4144\) |
Rank
sage: E.rank()
The elliptic curves in class 21294k have rank \(0\).
Complex multiplication
The elliptic curves in class 21294k do not have complex multiplication.Modular form 21294.2.a.k
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.
