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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 38025bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38025.cj7 | 38025bc1 | \([1, -1, 0, -792, -1381509]\) | \(-1/15\) | \(-824705568984375\) | \([2]\) | \(110592\) | \(1.5411\) | \(\Gamma_0(N)\)-optimal |
| 38025.cj6 | 38025bc2 | \([1, -1, 0, -190917, -31611384]\) | \(13997521/225\) | \(12370583534765625\) | \([2, 2]\) | \(221184\) | \(1.8876\) | |
| 38025.cj5 | 38025bc3 | \([1, -1, 0, -381042, 41966991]\) | \(111284641/50625\) | \(2783381295322265625\) | \([2, 2]\) | \(442368\) | \(2.2342\) | |
| 38025.cj4 | 38025bc4 | \([1, -1, 0, -3042792, -2042183259]\) | \(56667352321/15\) | \(824705568984375\) | \([2]\) | \(442368\) | \(2.2342\) | |
| 38025.cj8 | 38025bc5 | \([1, -1, 0, 1330083, 314035866]\) | \(4733169839/3515625\) | \(-193290367730712890625\) | \([2]\) | \(884736\) | \(2.5808\) | |
| 38025.cj2 | 38025bc6 | \([1, -1, 0, -5134167, 4476632616]\) | \(272223782641/164025\) | \(9018155396844140625\) | \([2, 2]\) | \(884736\) | \(2.5808\) | |
| 38025.cj3 | 38025bc7 | \([1, -1, 0, -4183542, 6184905741]\) | \(-147281603041/215233605\) | \(-11833623511738881328125\) | \([2]\) | \(1769472\) | \(2.9274\) | |
| 38025.cj1 | 38025bc8 | \([1, -1, 0, -82134792, 286529921991]\) | \(1114544804970241/405\) | \(22267050362578125\) | \([2]\) | \(1769472\) | \(2.9274\) |
Rank
sage: E.rank()
The elliptic curves in class 38025bc have rank \(0\).
Complex multiplication
The elliptic curves in class 38025bc do not have complex multiplication.Modular form 38025.2.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.
