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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 266805dh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 266805.dh4 | 266805dh1 | \([1, -1, 0, -187875, 227557336]\) | \(-4826809/144375\) | \(-21936325240379694375\) | \([2]\) | \(4423680\) | \(2.3917\) | \(\Gamma_0(N)\)-optimal |
| 266805.dh3 | 266805dh2 | \([1, -1, 0, -6858000, 6880340011]\) | \(234770924809/1334025\) | \(202691645221108376025\) | \([2, 2]\) | \(8847360\) | \(2.7382\) | |
| 266805.dh1 | 266805dh3 | \([1, -1, 0, -109577925, 441529430656]\) | \(957681397954009/31185\) | \(4738246251922013985\) | \([2]\) | \(17694720\) | \(3.0848\) | |
| 266805.dh2 | 266805dh4 | \([1, -1, 0, -10860075, -2070700934]\) | \(932288503609/527295615\) | \(80117250967730104096815\) | \([2]\) | \(17694720\) | \(3.0848\) |
Rank
sage: E.rank()
The elliptic curves in class 266805dh have rank \(0\).
Complex multiplication
The elliptic curves in class 266805dh do not have complex multiplication.Modular form 266805.2.a.dh
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.
