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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 493680et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 493680.et3 | 493680et1 | \([0, 1, 0, -115111, 12285260]\) | \(5951163357184/1129312125\) | \(32010325079634000\) | \([2]\) | \(4423680\) | \(1.8846\) | \(\Gamma_0(N)\)-optimal |
| 493680.et2 | 493680et2 | \([0, 1, 0, -556156, -148607956]\) | \(41948679809104/3291890625\) | \(1492936972164000000\) | \([2, 2]\) | \(8847360\) | \(2.2312\) | |
| 493680.et4 | 493680et3 | \([0, 1, 0, 554624, -667120060]\) | \(10400706415004/112060546875\) | \(-203286624750000000000\) | \([2]\) | \(17694720\) | \(2.5778\) | |
| 493680.et1 | 493680et4 | \([0, 1, 0, -8723656, -9920204956]\) | \(40472803590982276/281883375\) | \(511358559947136000\) | \([2]\) | \(17694720\) | \(2.5778\) |
Rank
sage: E.rank()
The elliptic curves in class 493680et have rank \(0\).
Complex multiplication
The elliptic curves in class 493680et do not have complex multiplication.Modular form 493680.2.a.et
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.
