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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 21294a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21294.r3 | 21294a1 | \([1, -1, 0, -36282, 1260468]\) | \(40530337875/18264064\) | \(2380243009277952\) | \([2]\) | \(96768\) | \(1.6458\) | \(\Gamma_0(N)\)-optimal |
| 21294.r4 | 21294a2 | \([1, -1, 0, 125958, 9340020]\) | \(1695802078125/1272491584\) | \(-165835993412037312\) | \([2]\) | \(193536\) | \(1.9923\) | |
| 21294.r1 | 21294a3 | \([1, -1, 0, -1455882, -675737308]\) | \(3592121380875/246064\) | \(23377576449781008\) | \([2]\) | \(290304\) | \(2.1951\) | |
| 21294.r2 | 21294a4 | \([1, -1, 0, -1364622, -764204752]\) | \(-2958077788875/946054564\) | \(-89880937055295530508\) | \([2]\) | \(580608\) | \(2.5416\) |
Rank
sage: E.rank()
The elliptic curves in class 21294a have rank \(1\).
Complex multiplication
The elliptic curves in class 21294a do not have complex multiplication.Modular form 21294.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
