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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 51870.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.j1 | 51870j4 | \([1, 1, 0, -13903218, -19959387228]\) | \(297214339265273649756432169/488484917902800\) | \(488484917902800\) | \([2]\) | \(2359296\) | \(2.5104\) | |
| 51870.j2 | 51870j2 | \([1, 1, 0, -869218, -311935628]\) | \(72629093972969564016169/93022316019360000\) | \(93022316019360000\) | \([2, 2]\) | \(1179648\) | \(2.1638\) | |
| 51870.j3 | 51870j3 | \([1, 1, 0, -635218, -483364028]\) | \(-28346090452899214800169/84418326220247182800\) | \(-84418326220247182800\) | \([4]\) | \(2359296\) | \(2.5104\) | |
| 51870.j4 | 51870j1 | \([1, 1, 0, -69218, -2015628]\) | \(36676733979624816169/19519718400000000\) | \(19519718400000000\) | \([2]\) | \(589824\) | \(1.8172\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51870.j have rank \(0\).
Complex multiplication
The elliptic curves in class 51870.j do not have complex multiplication.Modular form 51870.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
