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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 121680z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.c3 | 121680z1 | \([0, 0, 0, -169338, 26757263]\) | \(9538484224/26325\) | \(1482094872133200\) | \([2]\) | \(1032192\) | \(1.7840\) | \(\Gamma_0(N)\)-optimal |
| 121680.c2 | 121680z2 | \([0, 0, 0, -237783, 3088982]\) | \(1650587344/950625\) | \(856321481676960000\) | \([2, 2]\) | \(2064384\) | \(2.1305\) | |
| 121680.c4 | 121680z3 | \([0, 0, 0, 948597, 24681098]\) | \(26198797244/15234375\) | \(-54892402671600000000\) | \([2]\) | \(4128768\) | \(2.4771\) | |
| 121680.c1 | 121680z4 | \([0, 0, 0, -2519283, -1533273118]\) | \(490757540836/2142075\) | \(7718310954848332800\) | \([2]\) | \(4128768\) | \(2.4771\) |
Rank
sage: E.rank()
The elliptic curves in class 121680z have rank \(2\).
Complex multiplication
The elliptic curves in class 121680z do not have complex multiplication.Modular form 121680.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
