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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 39675bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39675.bk7 | 39675bc1 | \([1, 0, 1, -276, -283427]\) | \(-1/15\) | \(-34695911484375\) | \([2]\) | \(76032\) | \(1.2770\) | \(\Gamma_0(N)\)-optimal |
| 39675.bk6 | 39675bc2 | \([1, 0, 1, -66401, -6499177]\) | \(13997521/225\) | \(520438672265625\) | \([2, 2]\) | \(152064\) | \(1.6236\) | |
| 39675.bk5 | 39675bc3 | \([1, 0, 1, -132526, 8577323]\) | \(111284641/50625\) | \(117098701259765625\) | \([2, 2]\) | \(304128\) | \(1.9702\) | |
| 39675.bk4 | 39675bc4 | \([1, 0, 1, -1058276, -419119177]\) | \(56667352321/15\) | \(34695911484375\) | \([2]\) | \(304128\) | \(1.9702\) | |
| 39675.bk8 | 39675bc5 | \([1, 0, 1, 462599, 64519073]\) | \(4733169839/3515625\) | \(-8131854254150390625\) | \([2]\) | \(608256\) | \(2.3168\) | |
| 39675.bk2 | 39675bc6 | \([1, 0, 1, -1785651, 917796073]\) | \(272223782641/164025\) | \(379399792081640625\) | \([2, 2]\) | \(608256\) | \(2.3168\) | |
| 39675.bk3 | 39675bc7 | \([1, 0, 1, -1455026, 1268258573]\) | \(-147281603041/215233605\) | \(-497848407169528828125\) | \([2]\) | \(1216512\) | \(2.6633\) | |
| 39675.bk1 | 39675bc8 | \([1, 0, 1, -28566276, 58763946073]\) | \(1114544804970241/405\) | \(936789610078125\) | \([2]\) | \(1216512\) | \(2.6633\) |
Rank
sage: E.rank()
The elliptic curves in class 39675bc have rank \(1\).
Complex multiplication
The elliptic curves in class 39675bc do not have complex multiplication.Modular form 39675.2.a.bc
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
