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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 51870.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.k1 | 51870n4 | \([1, 1, 0, -60972, 5769384]\) | \(25068373942677879241/781754555400\) | \(781754555400\) | \([2]\) | \(196608\) | \(1.3785\) | |
| 51870.k2 | 51870n2 | \([1, 1, 0, -3972, 80784]\) | \(6933016959591241/1076198760000\) | \(1076198760000\) | \([2, 2]\) | \(98304\) | \(1.0320\) | |
| 51870.k3 | 51870n1 | \([1, 1, 0, -1092, -13104]\) | \(144215816802121/14341017600\) | \(14341017600\) | \([2]\) | \(49152\) | \(0.68539\) | \(\Gamma_0(N)\)-optimal |
| 51870.k4 | 51870n3 | \([1, 1, 0, 6948, 458616]\) | \(37085613814461239/111180103125000\) | \(-111180103125000\) | \([2]\) | \(196608\) | \(1.3785\) |
Rank
sage: E.rank()
The elliptic curves in class 51870.k have rank \(2\).
Complex multiplication
The elliptic curves in class 51870.k do not have complex multiplication.Modular form 51870.2.a.k
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.
