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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 29040.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29040.b1 | 29040ch7 | \([0, -1, 0, -10325696, -12767623680]\) | \(16778985534208729/81000\) | \(587761422336000\) | \([2]\) | \(829440\) | \(2.4563\) | |
| 29040.b2 | 29040ch8 | \([0, -1, 0, -878016, -42806784]\) | \(10316097499609/5859375000\) | \(42517464000000000000\) | \([2]\) | \(829440\) | \(2.4563\) | |
| 29040.b3 | 29040ch6 | \([0, -1, 0, -645696, -199111680]\) | \(4102915888729/9000000\) | \(65306824704000000\) | \([2, 2]\) | \(414720\) | \(2.1097\) | |
| 29040.b4 | 29040ch5 | \([0, -1, 0, -558576, 160868160]\) | \(2656166199049/33750\) | \(244900592640000\) | \([2]\) | \(276480\) | \(1.9070\) | |
| 29040.b5 | 29040ch4 | \([0, -1, 0, -132656, -15973824]\) | \(35578826569/5314410\) | \(38563026919464960\) | \([2]\) | \(276480\) | \(1.9070\) | |
| 29040.b6 | 29040ch2 | \([0, -1, 0, -35856, 2379456]\) | \(702595369/72900\) | \(528985280102400\) | \([2, 2]\) | \(138240\) | \(1.5604\) | |
| 29040.b7 | 29040ch3 | \([0, -1, 0, -26176, -5325824]\) | \(-273359449/1536000\) | \(-11145698082816000\) | \([2]\) | \(207360\) | \(1.7631\) | |
| 29040.b8 | 29040ch1 | \([0, -1, 0, 2864, 180160]\) | \(357911/2160\) | \(-15673637928960\) | \([2]\) | \(69120\) | \(1.2138\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 29040.b have rank \(1\).
Complex multiplication
The elliptic curves in class 29040.b do not have complex multiplication.Modular form 29040.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
