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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 5610.be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5610.be1 | 5610be4 | \([1, 0, 0, -178426, -25264420]\) | \(628200507126935410849/88124751829125000\) | \(88124751829125000\) | \([2]\) | \(82944\) | \(1.9777\) | |
| 5610.be2 | 5610be2 | \([1, 0, 0, -45586, 3738116]\) | \(10476561483361670689/13992628953600\) | \(13992628953600\) | \([6]\) | \(27648\) | \(1.4283\) | |
| 5610.be3 | 5610be1 | \([1, 0, 0, -2066, 91140]\) | \(-975276594443809/3037581803520\) | \(-3037581803520\) | \([6]\) | \(13824\) | \(1.0818\) | \(\Gamma_0(N)\)-optimal |
| 5610.be4 | 5610be3 | \([1, 0, 0, 18094, -2114364]\) | \(655127711084516831/2313151512408000\) | \(-2313151512408000\) | \([2]\) | \(41472\) | \(1.6311\) |
Rank
sage: E.rank()
The elliptic curves in class 5610.be have rank \(1\).
Complex multiplication
The elliptic curves in class 5610.be do not have complex multiplication.Modular form 5610.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
