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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 219450d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 219450.hl3 | 219450d1 | \([1, 0, 0, -14102053, -18455059903]\) | \(2481194036785531116697829/258949680664193563968\) | \(32368710083024195496000\) | \([2]\) | \(27264000\) | \(3.0543\) | \(\Gamma_0(N)\)-optimal |
| 219450.hl4 | 219450d2 | \([1, 0, 0, 18108147, -90509277303]\) | \(5253342688178294786187931/31352380501055976236952\) | \(-3919047562631997029619000\) | \([2]\) | \(54528000\) | \(3.4009\) | |
| 219450.hl1 | 219450d3 | \([1, 0, 0, -1963596578, 33490702261572]\) | \(6698391064416261144129516088949/2449861814311786119168\) | \(306232726788973264896000\) | \([10]\) | \(136320000\) | \(3.8590\) | |
| 219450.hl2 | 219450d4 | \([1, 0, 0, -1954585378, 33813312232772]\) | \(-6606594261153843534370179395189/128160831539202997006467072\) | \(-16020103942400374625808384000\) | \([10]\) | \(272640000\) | \(4.2056\) |
Rank
sage: E.rank()
The elliptic curves in class 219450d have rank \(0\).
Complex multiplication
The elliptic curves in class 219450d do not have complex multiplication.Modular form 219450.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
