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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 3600bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3600.u7 | 3600bf1 | \([0, 0, 0, -75, 40250]\) | \(-1/15\) | \(-699840000000\) | \([2]\) | \(3072\) | \(0.95175\) | \(\Gamma_0(N)\)-optimal |
| 3600.u6 | 3600bf2 | \([0, 0, 0, -18075, 922250]\) | \(13997521/225\) | \(10497600000000\) | \([2, 2]\) | \(6144\) | \(1.2983\) | |
| 3600.u5 | 3600bf3 | \([0, 0, 0, -36075, -1219750]\) | \(111284641/50625\) | \(2361960000000000\) | \([2, 2]\) | \(12288\) | \(1.6449\) | |
| 3600.u4 | 3600bf4 | \([0, 0, 0, -288075, 59512250]\) | \(56667352321/15\) | \(699840000000\) | \([4]\) | \(12288\) | \(1.6449\) | |
| 3600.u2 | 3600bf5 | \([0, 0, 0, -486075, -130369750]\) | \(272223782641/164025\) | \(7652750400000000\) | \([2, 2]\) | \(24576\) | \(1.9915\) | |
| 3600.u8 | 3600bf6 | \([0, 0, 0, 125925, -9157750]\) | \(4733169839/3515625\) | \(-164025000000000000\) | \([2]\) | \(24576\) | \(1.9915\) | |
| 3600.u1 | 3600bf7 | \([0, 0, 0, -7776075, -8346199750]\) | \(1114544804970241/405\) | \(18895680000000\) | \([2]\) | \(49152\) | \(2.3380\) | |
| 3600.u3 | 3600bf8 | \([0, 0, 0, -396075, -180139750]\) | \(-147281603041/215233605\) | \(-10041939074880000000\) | \([2]\) | \(49152\) | \(2.3380\) |
Rank
sage: E.rank()
The elliptic curves in class 3600bf have rank \(1\).
Complex multiplication
The elliptic curves in class 3600bf do not have complex multiplication.Modular form 3600.2.a.bf
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
