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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 369600.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.cf1 | 369600cf3 | \([0, -1, 0, -493633, 133655137]\) | \(12990838708516/144375\) | \(147840000000000\) | \([2]\) | \(2359296\) | \(1.8724\) | |
| 369600.cf2 | 369600cf2 | \([0, -1, 0, -31633, 1985137]\) | \(13674725584/1334025\) | \(341510400000000\) | \([2, 2]\) | \(1179648\) | \(1.5258\) | |
| 369600.cf3 | 369600cf1 | \([0, -1, 0, -7133, -195363]\) | \(2508888064/396165\) | \(6338640000000\) | \([2]\) | \(589824\) | \(1.1792\) | \(\Gamma_0(N)\)-optimal |
| 369600.cf4 | 369600cf4 | \([0, -1, 0, 38367, 9475137]\) | \(6099383804/41507235\) | \(-42503408640000000\) | \([2]\) | \(2359296\) | \(1.8724\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.cf have rank \(2\).
Complex multiplication
The elliptic curves in class 369600.cf do not have complex multiplication.Modular form 369600.2.a.cf
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.
