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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 48510.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48510.e1 | 48510c4 | \([1, -1, 0, -2001120, 1019458880]\) | \(382704614800227/27778076480\) | \(64325282450335220160\) | \([2]\) | \(1990656\) | \(2.5473\) | |
| 48510.e2 | 48510c2 | \([1, -1, 0, -380445, -89935175]\) | \(1917114236485083/7117764500\) | \(22609742642833500\) | \([2]\) | \(663552\) | \(1.9980\) | |
| 48510.e3 | 48510c1 | \([1, -1, 0, -12945, -2690675]\) | \(-75526045083/943250000\) | \(-2996255319750000\) | \([2]\) | \(331776\) | \(1.6514\) | \(\Gamma_0(N)\)-optimal |
| 48510.e4 | 48510c3 | \([1, -1, 0, 115680, 69862400]\) | \(73929353373/954060800\) | \(-2209304538382233600\) | \([2]\) | \(995328\) | \(2.2007\) |
Rank
sage: E.rank()
The elliptic curves in class 48510.e have rank \(0\).
Complex multiplication
The elliptic curves in class 48510.e do not have complex multiplication.Modular form 48510.2.a.e
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.
