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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 169065.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169065.u1 | 169065bb3 | \([1, -1, 0, -40591260, 99549932575]\) | \(420339554066191969/244298925\) | \(4298754193774913925\) | \([2]\) | \(9437184\) | \(2.9001\) | |
| 169065.u2 | 169065bb2 | \([1, -1, 0, -2551635, 1537034800]\) | \(104413920565969/2472575625\) | \(43508152307237450625\) | \([2, 2]\) | \(4718592\) | \(2.5535\) | |
| 169065.u3 | 169065bb1 | \([1, -1, 0, -353790, -45853169]\) | \(278317173889/109245825\) | \(1922320977757680825\) | \([2]\) | \(2359296\) | \(2.2069\) | \(\Gamma_0(N)\)-optimal |
| 169065.u4 | 169065bb4 | \([1, -1, 0, 322470, 4802592901]\) | \(210751100351/566398828125\) | \(-9966516789836633203125\) | \([2]\) | \(9437184\) | \(2.9001\) |
Rank
sage: E.rank()
The elliptic curves in class 169065.u have rank \(1\).
Complex multiplication
The elliptic curves in class 169065.u do not have complex multiplication.Modular form 169065.2.a.u
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 LMFDB 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 LMFDB labels.
